- book metadata
European Mathematical Society Publishing House
2017-04-23 00:06:00
171
http://www.ems-ph.org/meta/bmeta-stream.php?update_since=2017-04-23
Functional Analysis and Operator Theory for Quantum Physics
Pavel Exner Anniversary Volume
Jaroslav
Dittrich
Czech Academy of Sciences, Rez-Prague, Czech Republic
Hynek
Kovařík
Università degli Studi di Brescia, Italy
Ari
Laptev
Imperial College London, UK
Quantum theory
Partial differential equations
81Q37, 81Q35, 35P15, 35P25
Schrödinger operators, point interactions, metric graphs, quantum waveguides, eigenvalue estimates, operator-valued functions, Cayley–Hamilton theorem, adiabatic theorem
This volume is dedicated to Pavel Exner on the occasion of his 70th anniversary. It collects contributions by numerous scientists with expertise in mathematical physics and in particular in problems arising from quantum mechanics. The questions addressed in the contributions cover a large range of topics. A lot of attention was paid to differential operators with zero range interactions, which are often used as models in quantum mechanics. Several authors considered problems related to systems with mixed-dimensions such as quantum waveguides, quantum layers and quantum graphs. Eigenvalues and eigenfunctions of Laplace and Schrödinger operators are discussed too, as well as systems with adiabatic time evolution. Although most of the problems treated in the book have a quantum mechanical background, some contributions deal with issues which go well beyond this framework; for example the Cayley–Hamilton theorem, approximation formulae for contraction semigroups or factorization of analytic operator-valued Fredholm functions. As for the mathematical tools involved, the book provides a wide variety of techniques from functional analysis and operator theory. Altogether the volume presents a collection of research papers which will be of interest to any active scientist working in one of the above mentioned fields.
5
10
2017
978-3-03719-175-0
978-3-03719-675-5
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/175
http://www.ems-ph.org/doi/10.4171/175
EMS Series of Congress Reports
Relative partition function of Coulomb plus delta interaction
Sergio
Albeverio
Universität Bonn, BONN, GERMANY
Claudio
Cacciapuoti
Università dell'Insubria, COMO, ITALY
Mauro
Spreafico
Università del Salento & INFN, LECCE, ITALY
Partial differential equations
General
1
29
1
10.4171/175-1/1
http://www.ems-ph.org/doi/10.4171/175-1/1
Inequivalence of quantum Dirac fields of different masses and the underlying general structures involved
Asao
Arai
Hokkaido University, SAPPORO, JAPAN
Partial differential equations
General
31
53
1
10.4171/175-1/2
http://www.ems-ph.org/doi/10.4171/175-1/2
On a class of Schrödinger operators exhibiting spectral transition
Diana
Barseghyan
University of Ostrava, OSTRAVA, CZECH REPUBLIC
Olga
Rossi
University of Ostrava, OSTRAVA, CZECH REPUBLIC
Partial differential equations
General
55
70
1
10.4171/175-1/3
http://www.ems-ph.org/doi/10.4171/175-1/3
On the quantum mechanical three-body problem with zero-range interactions
Giulia
Basti
Università di Roma La Sapienza, ROMA, ITALY
Alessandro
Teta
Università di Roma La Sapienza, ROMA, ITALY
Quantum theory
Partial differential equations
General
71
93
1
10.4171/175-1/4
http://www.ems-ph.org/doi/10.4171/175-1/4
On the index of meromorphic operator-valued functions and some applications
Jussi
Behrndt
TU Graz, GRAZ, AUSTRIA
Fritz
Gesztesy
Baylor University, WACO, UNITED STATES
Helge
Holden
University of Trondheim, TRONDHEIM, NORWAY
Roger
Nichols
The University of Tennessee at Chattanooga, CHATTANOOGA, UNITED STATES
Partial differential equations
Quantum theory
General
95
127
1
10.4171/175-1/5
http://www.ems-ph.org/doi/10.4171/175-1/5
Trace formulae for Schrödinger operators with singular interactions
Jussi
Behrndt
TU Graz, GRAZ, AUSTRIA
Matthias
Langer
University of Strathclyde, GLASGOW, UNITED KINGDOM
Vladimir
Lotoreichik
Nuclear Physics Institute, ŘEŽ - PRAGUE, CZECH REPUBLIC
Partial differential equations
Quantum theory
General
129
152
1
10.4171/175-1/6
http://www.ems-ph.org/doi/10.4171/175-1/6
An improved bound for the non-existence of radial solutions of the Brezis–Nirenberg problem in $\mathbb H^n$
Rafael
Benguria
Pontificia Universidad Católica de Chile, SANTIAGO DE CHILE, CHILE
Soledad
Benguria
University of Wisconsin, MADISON, UNITED STATES
Partial differential equations
Quantum theory
General
153
160
1
10.4171/175-1/7
http://www.ems-ph.org/doi/10.4171/175-1/7
Twisted waveguide with a Neumann window
Philippe
Briet
Université de Toulon et du Var, LA GARDE CEDEX, FRANCE
Hiba
Hammedi
Université de Toulon et du Var, LA GARDE CEDEX, FRANCE
Partial differential equations
Quantum theory
General
161
175
1
10.4171/175-1/8
http://www.ems-ph.org/doi/10.4171/175-1/8
Example of a periodic Neumann waveguide with a gap in its spectrum
Giuseppe
Cardone
Università del Sannio, BENEVENTO, ITALY
Andrii
Khrabustovskyi
Karlsruher Institut für Technologie, KARLSRUHE, GERMANY
Partial differential equations
Quantum theory
General
177
187
1
10.4171/175-1/9
http://www.ems-ph.org/doi/10.4171/175-1/9
Two-dimensional time-dependent point interactions
Raffaele
Carlone
Università degli Studi di Napoli Federico II, NAPOLI, ITALY
Michele
Correggi
Università degli studi Roma Tre, ROMA, ITALY
Rodolfo
Figari
Università degli Studi di Napoli Federico II, NAPOLI, ITALY
Partial differential equations
Quantum theory
General
189
211
1
10.4171/175-1/10
http://www.ems-ph.org/doi/10.4171/175-1/10
On resonant spectral gaps in quantum graphs
Ngoc
Do
Texas A&M University, COLLEGE STATION, UNITED STATES
Peter
Kuchment
Texas A&M University, COLLEGE STATION, UNITED STATES
Beng
Ong
CGG, HOUSTON, UNITED STATES
Partial differential equations
General
213
222
1
10.4171/175-1/11
http://www.ems-ph.org/doi/10.4171/175-1/11
Adiabatic theorem for a class of stochastic differential equations on a Hilbert space
Martin
Fraas
, ALBANY, UNITED STATES
Partial differential equations
Quantum theory
General
223
243
1
10.4171/175-1/12
http://www.ems-ph.org/doi/10.4171/175-1/12
Eigenvalues of Schrödinger operators with complex surface potentials
Rupert
Frank
Caltech, PASADENA, UNITED STATES
Partial differential equations
Quantum theory
General
245
259
1
10.4171/175-1/13
http://www.ems-ph.org/doi/10.4171/175-1/13
A lower bound to the spectral threshold in curved quantum layers
Pedro
Freitas
Universidade de Lisboa - GFM, LISBOA, PORTUGAL
David
Krejčiřík
Czech Technical University in Prague, PRAGUE 2, CZECH REPUBLIC
Partial differential equations
Quantum theory
General
261
269
1
10.4171/175-1/14
http://www.ems-ph.org/doi/10.4171/175-1/14
To the spectral theory of vector-valued Sturm–Liouville operators with summable potentials and point interactions
Yaroslav
Granovskyi
National Academy of Science of Ukraine, SLAVYANSK, UKRAINE
Mark
Malamud
National Academy of Science of Ukraine, SLAVYANSK, UKRAINE
Hagen
Neidhardt
Karl-Weierstraß-Institut für Mathematik, BERLIN, GERMANY
Andrea
Posilicano
Università dell'Insubria, COMO, ITALY
Partial differential equations
Quantum theory
General
271
313
1
10.4171/175-1/15
http://www.ems-ph.org/doi/10.4171/175-1/15
Spectral asymptotics for the Dirichlet Laplacian with a Neumann window via a Birman–Schwinger analysis of the Dirichlet-to-Neumann operator
André
Hänel
Leibniz Universität Hannover, HANNOVER, GERMANY
Timo
Weidl
Universität Stuttgart, STUTTGART, GERMANY
Partial differential equations
Quantum theory
General
315
352
1
10.4171/175-1/16
http://www.ems-ph.org/doi/10.4171/175-1/16
Dirichlet eigenfunctions in the cube, sharpening the Courant nodal inequality
Bernard
Helffer
Université de Nantes, NANTES, FRANCE
Rola
Kiwan
American University in Dubai, DUBAI, UNITED ARAB EMIRATES
Partial differential equations
Quantum theory
General
353
371
1
10.4171/175-1/17
http://www.ems-ph.org/doi/10.4171/175-1/17
A mathematical modeling of electron–phonon interaction for small wave numbers close to zero
Masao
Hirokawa
Hiroshima University, HIROSHIMA, JAPAN
Quantum theory
General
373
399
1
10.4171/175-1/18
http://www.ems-ph.org/doi/10.4171/175-1/18
The modified unitary Trotter–Kato and Zeno product formulas revisited
Takashi
Ichinose
Kanazawa University, KANAZAWA, JAPAN
Partial differential equations
Quantum theory
General
401
417
1
10.4171/175-1/19
http://www.ems-ph.org/doi/10.4171/175-1/19
Spectral asymptotics induced by approaching and diverging planar circles
Sylwia
Kondej
University of Zielona Góra, ZIELONA GORA, POLAND
Partial differential equations
Quantum theory
General
419
432
1
10.4171/175-1/20
http://www.ems-ph.org/doi/10.4171/175-1/20
Spectral estimates for the Heisenberg Laplacian on cylinders
Hynek
Kovařík
Università degli Studi di Brescia, BRESCIA, ITALY
Bartosch
Ruszkowski
Universität Stuttgart, STUTTGART, GERMANY
Timo
Weidl
Universität Stuttgart, STUTTGART, GERMANY
Partial differential equations
Quantum theory
General
433
446
1
10.4171/175-1/21
http://www.ems-ph.org/doi/10.4171/175-1/21
Variational proof of the existence of eigenvalues for star graphs
Konstantin
Pankrashkin
Université Paris-Sud, ORSAY CEDEX, FRANCE
Partial differential equations
Quantum theory
General
447
458
1
10.4171/175-1/22
http://www.ems-ph.org/doi/10.4171/175-1/22
On the boundedness and compactness of weighted Green operators of second-order elliptic operators
Yehuda
Pinchover
Technion - Israel Institute of Technology, HAIFA, ISRAEL
Partial differential equations
Quantum theory
General
459
489
1
10.4171/175-1/23
http://www.ems-ph.org/doi/10.4171/175-1/23
Abstract graph-like spaces and vector-valued metric graphs
Olaf
Post
Universität Trier, TRIER, GERMANY
Partial differential equations
Quantum theory
General
491
524
1
10.4171/175-1/24
http://www.ems-ph.org/doi/10.4171/175-1/24
A Cayley–Hamiltonian theorem for periodic finite band matrices
Barry
Simon
California Institute of Technology, PASADENA, UNITED STATES
Partial differential equations
Quantum theory
General
525
529
1
10.4171/175-1/25
http://www.ems-ph.org/doi/10.4171/175-1/25
Path topology dependence of adiabatic time evolution
Atushi
Tanaka
Tokyo Metropolitan University, TOKYO, JAPAN
Taksu
Cheon
Kochi University of Technology, KOCHI, JAPAN
Quantum theory
Partial differential equations
General
531
542
1
10.4171/175-1/26
http://www.ems-ph.org/doi/10.4171/175-1/26
On quantum graph filters with flat passbands
Ondřej
Turek
Czech Academy of Sciences, ŘEŽ - PRAGUE, CZECH REPUBLIC
Quantum theory
Partial differential equations
General
543
563
1
10.4171/175-1/27
http://www.ems-ph.org/doi/10.4171/175-1/27
Comments on the Chernoff $\sqrt n$-lemma
Valentin
Zagrebnov
Centre de Mathématiques et Informatique, MARSEILLE CEDEX 13, FRANCE
Partial differential equations
General
565
573
1
10.4171/175-1/28
http://www.ems-ph.org/doi/10.4171/175-1/28
Asymptotic Theory of Transaction Costs
Walter
Schachermayer
Universität Wien, Austria
Statistics
Probability theory and stochastic processes
Game theory, economics, social and behavioral sciences
Primary: 62P05, 91G10; Secondary: 60G44
Portfolio optimization, transaction costs, shadow price, semimartingale, fractional Brownian motion
A classical topic in Mathematical Finance is the theory of portfolio optimization. Robert Merton's work from the early seventies had enormous impact on academic research as well as on the paradigms guiding practitioners. One of the ramifications of this topic is the analysis of (small) proportional transaction costs, such as a Tobin tax. The lecture notes present some striking recent results of the asymptotic dependence of the relevant quantities when transaction costs tend to zero. An appealing feature of the consideration of transaction costs is that it allows for the first time to reconcile the no arbitrage paradigm with the use of non-semimartingale models, such as fractional Brownian motion. This leads to the culminating theorem of the present lectures which roughly reads as follows: for a fractional Brownian motion stock price model we always find a shadow price process for given transaction costs. This process is a semimartingale and can therefore be dealt with using the usual machinery of mathematical finance.
3
31
2017
978-3-03719-173-6
978-3-03719-673-1
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/173
http://www.ems-ph.org/doi/10.4171/173
Zurich Lectures in Advanced Mathematics
PDE Models for Chemotaxis and Hydrodynamics in Supercritical Function Spaces
Hans
Triebel
University of Jena, Germany
Partial differential equations
35–02, 46–02, 76–02, 92–02; 35K05, 35Q30, 35Q92, 42B35, 46E35, 76D05, 92C15, 92C17
Function spaces of Besov–Sobolev type, chemotaxis, hydrodynamics, heat equations, Keller–Segel equations, Navier–Stokes equations
This book deals with PDE models for chemotaxis (the movement of biological cells or organisms in response of chemical gradients) and hydrodynamics (viscous, homogeneous, and incompressible fluid filling the entire space). The underlying Keller–Segel equations (chemotaxis), Navier–Stokes equations (hydrodynamics), and their numerous modifications and combinations are treated in the context of inhomogeneous spaces of Besov–Sobolev type paying special attention to mapping properties of related nonlinearities. Further models are considered, including (deterministic) Fokker–Planck equations and chemotaxis Navier–Stokes equations. These notes are addressed to graduate students and mathematicians having a working knowledge of basic elements of the theory of function spaces, especially of Besov-Sobolev type and interested in mathematical biology and physics.
3
31
2017
978-3-03719-172-9
978-3-03719-672-4
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/172
http://www.ems-ph.org/doi/10.4171/172
EMS Series of Lectures in Mathematics
Representation Theory – Current Trends and Perspectives
Henning
Krause
Universität Bielefeld, Germany
Peter
Littelmann
Universität Köln, Germany
Gunter
Malle
Universität Kaiserslautern, Germany
Karl-Hermann
Neeb
FAU Erlangen-Nürnberg, Germany
Christoph
Schweigert
Universität Hamburg, Germany
Algebraic geometry
Associative rings and algebras
Nonassociative rings and algebras
Category theory; homological algebra
Primary: 14Mxx,16Gxx, 17Bxx, 18Exx, 20Gxx, 22Exx; secondary: 58Cxx, 81Txx.
Algebraic groups, bounded and semibounded representations, categorification, character formulae, cluster algebras, Deligne-Lusztig theory, flat degenerations, geometrization, higher representation theory, highest weight categories, infinite dimensional Lie groups, local-global conjectures, special varieties, topological field theory
From April 2009 until March 2016, the German Science Foundation supported generously the Priority Program SPP 1388 in Representation Theory. The core principles of the projects realized in the framework of the priority program have been categorification and geometrization, this is also reflected by the contributions to this volume. Apart from the articles by former postdocs supported by the priority program, the volume contains a number of invited research and survey articles, many of them are extended versions of talks given at the last joint meeting of the priority program in Bad Honnef in March 2015. This volume is covering current research topics from the representation theory of finite groups, of algebraic groups, of Lie superalgebras, of finite dimensional algebras and of infinite dimensional Lie groups. Graduate students and researchers in mathematics interested in representation theory will find this volume inspiring. It contains many stimulating contributions to the development of this broad and extremely diverse subject.
1
31
2017
978-3-03719-171-2
978-3-03719-671-7
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/171
http://www.ems-ph.org/doi/10.4171/171
EMS Series of Congress Reports
Symmetric superspaces: slices, radial parts, and invariant functions
Alexander
Alldridge
Universität Köln, KÖLN, GERMANY
Global analysis, analysis on manifolds
Nonassociative rings and algebras
Differential geometry
General
We study the restriction of invariant polynomials on the tangent space of a Riemannian symmetric supermanifold to a Cartan subspace. We survey known results in the case the symmetric space is a Lie supergroup, and more generally, where the Cartan subspace is even. We then describe an approach to this problem, developed in joint work in progress with K. Coulembier, based on the study of radial parts of di fferential operators. This leads to a characterisation of the invariant functions for an arbitrary linear isometric action, and as a special case, to a Chevalley restriction theorem valid for the isotropy representation of any contragredient Riemannian symmetric superspace.
1
11
1
10.4171/171-1/1
http://www.ems-ph.org/doi/10.4171/171-1/1
Geometry of quiver Grassmannians of Dynkin type with applications to cluster algebras
Giovanni
Cerulli Irelli
Università di Roma La Sapienza, ROMA, ITALY
Associative rings and algebras
Algebraic geometry
General
The paper includes a new proof of the fact that quiver Grassmannians associated with rigid representations of Dynkin quivers do not have cohomology in odd degrees. Moreover, it is shown that they do not have torsion in homology. A new proof of the Caldero–Chapoton formula is provided. As a consequence a new proof of the positivity of cluster monomials in the acyclic clusters associated with Dynkin quivers is obtained. The methods used here are based on joint works with Markus Reineke and Evgeny Feigin.
13
45
1
10.4171/171-1/2
http://www.ems-ph.org/doi/10.4171/171-1/2
Spherical varieties and perspectives in representation theory
Stéphanie
Cupit-Foutou
Ruhr-Universität Bochum, BOCHUM, GERMANY
Algebraic geometry
Group theory and generalizations
General
This paper is a brief overview on recent classi cation results and related problems concerning spherical varieties. The emphasis is made on the representation theoretical aspects of these objects.
47
57
1
10.4171/171-1/3
http://www.ems-ph.org/doi/10.4171/171-1/3
Categorical actions from Lusztig induction and restriction on finite general linear groups
Olivier
Dudas
Université Paris Diderot Paris 7, PARIS CEDEX 13, FRANCE
Michela
Varagnolo
Université de Cergy-Pontoise, CERGY-PONTOISE, FRANCE
Éric
Vasserot
Université Paris Diderot Paris 7, PARIS CEDEX 13, FRANCE
Group theory and generalizations
General
In this note we explain how Lusztig's induction and restriction functors yield categorical actions of Kac–Moody algebras on the derived category of unipotent representations. We focus on the example of finite general linear groups and induction/restriction associated with split Levi subgroups, providing a derived analogue of Harish–Chandra induction/restriction as studied by Chuang–Rouquier in [5].
59
74
1
10.4171/171-1/4
http://www.ems-ph.org/doi/10.4171/171-1/4
Homological mirror symmetry for singularities
Wolfgang
Ebeling
Universität Hannover, HANNOVER, GERMANY
Algebraic geometry
Associative rings and algebras
Several complex variables and analytic spaces
Differential geometry
We give a survey on results related to the Berglund–Hubsch duality of invertible polynomials and the homological mirror symmetry conjecture for singularities.
75
107
1
10.4171/171-1/5
http://www.ems-ph.org/doi/10.4171/171-1/5
On the category of finite-dimensional representations of OSp$(r|2n)$: Part I
Michael
Ehrig
Universität Bonn, BONN, GERMANY
Catharina
Stroppel
Universität Bonn, BONN, GERMANY
Nonassociative rings and algebras
General
We study the combinatorics of the category $\mathcal F$ of fi nite-dimensional modules for the orthosymplectic Lie supergroup OSp$(r|2n)$. In particular we present a positive counting formula for the dimension of the space of homomorphisms between two projective modules. This refi nes earlier results of Gruson and Serganova. For each block $\mathcal B$ we construct an algebra $A_\mathcal B$ whose module category shares the combinatorics with $\mathcal B$. It arises as a subquotient of a suitable limit of type D Khovanov algebras. It turns out that $A_\mathcal B$ is isomorphic to the endomorphism algebra of a minimal projective generator of $\mathcal B$. This provides a direct link from $\mathcal F$ to parabolic categories $\mathcal O$ of type B/D, with maximal parabolic of type A, to the geometry of isotropic Grassmannians of types B/D and to Springer fi bres of type C/D. We also indicate why $\mathcal F$ is not highest weight in general.
109
170
1
10.4171/171-1/6
http://www.ems-ph.org/doi/10.4171/171-1/6
On cubes of Frobenius extensions
Ben
Elias
University of Oregon, EUGENE, UNITED STATES
Noah
Snyder
Indiana University, BLOOMINGTON, UNITED STATES
Geordie
Williamson
Max-Planck-Institut für Mathematik, BONN, GERMANY
$K$-theory
Nonassociative rings and algebras
Category theory; homological algebra
General
Given a hypercube of Frobenius extensions between commutative algebras, we provide a diagrammatic description of some natural transformations between compositions of induction and restriction functors, in terms of colored transversely-intersecting planar 1-manifolds. The relations arise in the first and third author’s work on (singular) Soergel bimodules.
171
186
1
10.4171/171-1/7
http://www.ems-ph.org/doi/10.4171/171-1/7
On toric degenerations of flag varieties
Xin
Fang
Universität Köln, KÖLN, GERMANY
Ghislain
Fourier
University of Glasgow, GLASGOW, UNITED KINGDOM
Peter
Littelmann
Universität Köln, KÖLN, GERMANY
Algebraic geometry
Convex and discrete geometry
General
Following the historical track in pursuing $T$-equivariant flat toric degenerations of flag varieties and spherical varieties, we explain how powerful tools in algebraic geometry and representation theory, such as canonical bases, Newton–Okounkov bodies, PBW- filtrations and cluster algebras come to push the subject forward.
187
232
1
10.4171/171-1/8
http://www.ems-ph.org/doi/10.4171/171-1/8
Subquotient categories of the affine category $\mathcal O$ at the critical level
Peter
Fiebig
Universität Erlangen-Nürnberg, ERLANGEN, GERMANY
Nonassociative rings and algebras
Quantum theory
General
We introduce subquotient categories of the restricted category $\mathcal O$ over an affi ne Kac–Moody algebra at the critical level and show that some of them have a realization in terms of moment graph sheaves.
233
253
1
10.4171/171-1/9
http://www.ems-ph.org/doi/10.4171/171-1/9
Low-dimensional topology, low-dimensional field theory and representation theory
Jürgen
Fuchs
Karlstads Universitet, KARLSTAD, SWEDEN
Christoph
Schweigert
Universität Hamburg, HAMBURG, GERMANY
Quantum theory
Manifolds and cell complexes
General
Structures in low-dimensional topology and low-dimensional geometry – often combined with ideas from (quantum) fi eld theory – can explain and inspire concepts in algebra and in representation theory and their categori ed versions. We present a personal view on some of these instances which have appeared within the Research Priority Programme SPP 1388 "Representation theory".
255
267
1
10.4171/171-1/10
http://www.ems-ph.org/doi/10.4171/171-1/10
Derived categories of quasi-hereditary algebras and their derived composition series
Martin
Kalck
University of Edinburgh, EDINBURGH, UNITED KINGDOM
Category theory; homological algebra
Associative rings and algebras
General
We study composition series of derived module categories in the sense of Angeleri Hügel, König & Liu for quasi-hereditary algebras. More precisely, we show that having a composition series with all factors being derived categories of vector spaces does not characterise derived categories of quasi-hereditay algebras. This gives a negative answer to a question of Liu & Yang and the proof also confi rms part of a conjecture of Bobi nski & Malicki. In another direction, we show that derived categories of quasi-hereditary algebras can have composition series with lots of diff erent lengths and composition factors. In other words, there is no Jordan–Hölder property for composition series of derived categories of quasi-hereditary algebras.
269
308
1
10.4171/171-1/11
http://www.ems-ph.org/doi/10.4171/171-1/11
Dominant dimension and applications
Steffen
Koenig
Universität Stuttgart, STUTTGART, GERMANY
Associative rings and algebras
General
Dominant dimension is a little known homological dimension, which is, however, crucial in many respects, both for abstractly studying fi nite dimensional algebras and their representation theory, and for applications to group algebras or in algebraic Lie theory. Various aspects and recent applications of dominant dimension will be outlined and illustrated.
309
330
1
10.4171/171-1/12
http://www.ems-ph.org/doi/10.4171/171-1/12
Highest weight categories and strict polynomial functors. With an appendix by Cosima Aquilino
Henning
Krause
Universität Bielefeld, BIELEFELD, GERMANY
Group theory and generalizations
Category theory; homological algebra
General
Highest weight categories are described in terms of standard objects and recollements of abelian categories, working over an arbitrary commutative base ring. Then the highest weight structure for categories of strict polynomial functors is explained, using the theory of Schur and Weyl functors. A consequence is the well-known fact that Schur algebras are quasi-hereditary.
331
373
1
10.4171/171-1/13
http://www.ems-ph.org/doi/10.4171/171-1/13
In the bocs seat: Quasi-hereditary algebras and representation type
Julian
Külshammer
Universität Stuttgart, STUTTGART, GERMANY
Associative rings and algebras
Nonassociative rings and algebras
Category theory; homological algebra
General
This paper surveys bocses, quasi-hereditary algebras and their relationship which was established in a recent result by Koenig, Ovsienko, and the author. Particular emphasis is placed on applications of this result to the representation type of the category filtered by standard modules for a quasi-hereditary algebra. In this direction, joint work with Thiel is presented showing that the subcategory of modules fi ltered by Weyl modules for tame Schur algebras is of fi nite representation type. The paper also includes a new proof for the classi cation of quasi-hereditary algebras with two simple modules, a result originally obtained by Membrillo–Hernández in [70].
375
426
1
10.4171/171-1/14
http://www.ems-ph.org/doi/10.4171/171-1/14
From groups to clusters
Sefi
Ladkani
University of Haifa, HAIFA, ISRAEL
Associative rings and algebras
Commutative rings and algebras
Category theory; homological algebra
Group theory and generalizations
We construct a new class of symmetric algebras of tame representation type that are also the endomorphism algebras of cluster-tilting objects in 2-Calabi–Yau triangulated categories, hence all their non-projective indecomposable modules are $\Omega$-periodic of period dividing 4. Our construction is based on the combinatorial notion of triangulation quivers, which arise naturally from triangulations of oriented surfaces with marked points. This class of algebras contains the algebras of quaternion type introduced and studied by Erdmann with relation to certain blocks of group algebras. On the other hand, it contains also the Jacobian algebras of the quivers with potentials associated by Fomin–Shapiro–Thurston and Labardini–Fragoso to triangulations of closed surfaces with punctures, hence our construction may serve as a bridge between the modular representation theory of nite groups and the theory of cluster algebras.
427
500
1
10.4171/171-1/15
http://www.ems-ph.org/doi/10.4171/171-1/15
Semi-infinite combinatorics in representation theory
Martina
Lanini
Edinburgh University, EDINBURGH, UNITED KINGDOM
Nonassociative rings and algebras
Group theory and generalizations
General
In this work we discuss some appearances of semi-infi nite combinatorics in representation theory. We propose a semi-in finite moment graph theory and we motivate it by considering the (not yet rigorously de fined) geometric side of the story. We show that it is possible to compute stalks of the local intersection cohomology of the semi-infi nite flag variety, and hence of spaces of quasi maps, by performing an algorithm due to Braden and MacPherson.
501
518
1
10.4171/171-1/16
http://www.ems-ph.org/doi/10.4171/171-1/16
Local-global conjectures in the representation theory of finite groups
Gunter
Malle
Universität Kaiserslautern, KAISERSLAUTERN, GERMANY
Group theory and generalizations
General
We give a survey of recent developments in the investigation of the various local-global conjectures for representations of nite groups.
519
539
1
10.4171/171-1/17
http://www.ems-ph.org/doi/10.4171/171-1/17
Bounded and semibounded representations of infinite dimensional Lie groups
Karl-Hermann
Neeb
Friedrich-Alexander-Universität Erlangen-Nürnberg, ERLANGEN, GERMANY
Topological groups, Lie groups
General
In this note we describe the recent progress in the classi fication of bounded and semibounded representations of in finite dimensional Lie groups. We start with a discussion of the semiboundedness condition and how the new concept of a smoothing operator can be used to construct $C*$ -algebras (so called host algebras) whose representations are in one-to-one correspondence with certain semibounded representations of an infi nite dimensional Lie group $G$. This makes the full power of $C*$ -theory available in this context. Then we discuss the classi cation of bounded representations of several types of unitary groups on Hilbert spaces and of gauge groups. After explaining the method of holomorphic induction as a means to pass from bounded representations to semibounded ones, we describe the classifi cation of semibounded representations for hermitian Lie groups of operators, loop groups (with infi nite dimensional targets), the Virasoro group and certain in finite dimensional oscillator groups.
541
563
1
10.4171/171-1/18
http://www.ems-ph.org/doi/10.4171/171-1/18
On ideals in $\operatorname{U}(\mathfrak {sl} (\infty)), \operatorname{U}(\mathfrak {o} (\infty)), \operatorname{U}(\mathfrak {sp} (\infty))$
Ivan
Penkov
Jacobs-Universität Bremen, BREMEN, GERMANY
Alexey
Petukhov
The University of Manchester, MANCHESTER, UNITED KINGDOM
Nonassociative rings and algebras
General
We provide a review of results on two-sided ideals in the enveloping algebra $\operatorname{U}(\mathfrak g(\infty))$ of a locally simple Lie algebra $\mathfrak g(\infty)$. We pay special attention to the case when $\mathfrak g(\infty)$ is one of the finitary Lie algebras $\mathfrak{sl}(\infty), \mathfrak o(\infty), \mathfrak{sp}(\infty)$. The main results include a description of all integrable ideals in $\operatorname{U}(\mathfrak g(\infty))$, as well as a criterion for the annihilator of an arbitrary (not necessarily integrable) simple highest weight module to be nonzero. This criterion is new for $\mathfrak g(\infty)=\mathfrak o(\infty), \mathfrak{sp}(\infty)$. All annihilators of simple highest weight modules are integrable ideals for $\mathfrak g(\infty)=\mathfrak{sl}(\infty),$ $\mathfrak o(\infty)$. Finally, we prove that the lattices of ideals in $\operatorname{U}(\mathfrak o(\infty))$ and $\operatorname{U}(\mathfrak{sp}(\infty))$ are isomorphic.
565
602
1
10.4171/171-1/19
http://www.ems-ph.org/doi/10.4171/171-1/19
Spherical varieties: applications and generalizations
Guido
Pezzini
Friedrich-Alexander-Universität Erlangen-Nürnberg, ERLANGEN, GERMANY
Algebraic geometry
Group theory and generalizations
General
In this short note we review some applications of the theory of spherical varieties in related fields, some generalizations of this theory, and present some open problems.
603
612
1
10.4171/171-1/20
http://www.ems-ph.org/doi/10.4171/171-1/20
Quiver moduli and small desingularizations of some GIT quotients
Markus
Reineke
Ruhr-Universität Bochum, BOCHUM, GERMANY
Associative rings and algebras
Algebraic geometry
Algebraic topology
General
We construct small desingularizations of moduli spaces of semistable quiver representations for indivisible dimension vectors using deformations of stabilities and a dimension estimate for nullcones. We apply this construction to several classes of GIT quotients.
613
635
1
10.4171/171-1/21
http://www.ems-ph.org/doi/10.4171/171-1/21
Geometric invariant theory for principal three-dimensional subgroups acting on flag varieties
Henrik
Seppänen
Georg-August Universität Göttingen, GÖTTINGEN, GERMANY
Valdemar
Tsanov
Georg-August Universität Göttingen, GÖTTINGEN, GERMANY
Algebraic geometry
Nonassociative rings and algebras
General
Let $G$ be a semisimple complex Lie group. In this article, we study Geometric Invariant Theory on a flag variety $G/B$ with respect to the action of a principal 3-dimensional simple subgroup $S\subset G$. We determine explicitly the GIT-equivalence classes of $S$-ample line bundles on $G/B$. We show that, under mild assumptions, among the GIT-classes there are chambers, in the sense of Dolgachev-Hu. The GIT-quotients with respect to various chambers form a family of Mori dream spaces, canonically associated with $G$. We are able to determine the three important cones in the Picard group of any of these quotients: the pseudoeffective-, the movable-, and the nef cones.
637
663
1
10.4171/171-1/22
http://www.ems-ph.org/doi/10.4171/171-1/22
Inductive conditions for counting conjectures via character triples
Britta
Späth
Bergische Universität Wuppertal, WUPPERTAL, GERMANY
Group theory and generalizations
General
In recent years several global/local conjectures in the representation theory of fi nite groups have been reduced to conditions on quasi-simple groups. We reformulate the inductive conditions for the conjectures by Alperin and McKay using (new) order relations between ordinary, respectively modular character triples. This allows to clarify the similarities and di fferences between those conditions.
665
680
1
10.4171/171-1/23
http://www.ems-ph.org/doi/10.4171/171-1/23
Restricted rational Cherednik algebras
Ulrich
Thiel
Universität Stuttgart, STUTTGART, GERMANY
Associative rings and algebras
Group theory and generalizations
General
We give an overview of the representation theory of restricted rational Cherednik algebras. These are certain finite-dimensional quotients of rational Cherednik algebras at $t = 0$. Their representation theory is connected to the geometry of the Calogero–Moser space, and there is a lot of evidence that they contain certain information about Hecke algebras even though the precise connection is so far unclear. We outline the basic theory along with some open problems and conjectures, and give explicit results in the cyclic and dihedral cases.
681
745
1
10.4171/171-1/24
http://www.ems-ph.org/doi/10.4171/171-1/24
On the existence of regular vectors
Christoph
Zellner
Friedrich-Alexander-Universität Erlangen-Nürnberg, ERLANGEN, GERMANY
Topological groups, Lie groups
General
Let $G$ be a locally convex Lie group and $\pi:G \to \mathrm{U}(\mathcal H)$ be a continuous unitary representation. $\pi$ is called smooth if the space of $\pi$-smooth vectors $\mathcal H^\infty\subset \mathcal H$ is dense. In this article we show that under certain conditions, concerning in particular the structure of the Lie algebra $\mathfrak{g}$ of $G$, a continuous unitary representation of $G$ is automatically smooth. As an application, this yields a dense space of smooth vectors for continuous positive energy representations of oscillator groups, double extensions of loop groups and the Virasoro group. Moreover we show the existence of a dense space of analytic vectors for the class of semibounded representations of Banach–Lie groups. Here $\pi$ is called semibounded, if $\pi$ is smooth and there exists a non-empty open subset $U\subset\mathfrak{g}$ such that the operators $i \mathrm d\ pi(x)$ from the derived representation are uniformly bounded from above for $x \in U$.
747
763
1
10.4171/171-1/25
http://www.ems-ph.org/doi/10.4171/171-1/25
The Monge-Ampère Equation and Its Applications
Alessio
Figalli
ETH Zürich, Switzerland
Partial differential equations
Differential geometry
Primary: 35J96; Secondary: 35B65, 35J60, 35J66, 35B45, 35B50, 35D05, 35D10, 35J65, 53A15, 53C45
Monge–Ampère equation, weak and strong solutions, existence, uniqueness, regularity
The Monge-Ampère equation is one of the most important partial differential equations, appearing in many problems in analysis and geometry. This monograph is a comprehensive introduction to the existence and regularity theory of the Monge-Ampère equation and some selected applications; the main goal is to provide the reader with a wealth of results and techniques he or she can draw from to understand current research related to this beautiful equation. The presentation is essentially self-contained, with an appendix wherein one can find precise statements of all the results used from different areas (linear algebra, convex geometry, measure theory, nonlinear analysis, and PDEs). This book is intended for graduate students and researchers interested in nonlinear PDEs: explanatory figures, detailed proofs, and heuristic arguments make this book suitable for self-study and also as a reference.
1
31
2017
978-3-03719-170-5
978-3-03719-670-0
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/170
http://www.ems-ph.org/doi/10.4171/170
Zurich Lectures in Advanced Mathematics
Bound States of the Magnetic Schrödinger Operator
Nicolas
Raymond
Université de Rennes, France
Partial differential equations
Calculus of variations and optimal control; optimization
Quantum theory
35P15, 35P20, 49R05, 81Q10, 81Q20
Magnetic Schrödinger equation, discrete spectrum, semiclassical analysis, magnetic harmonic approximation
This book is a synthesis of recent advances in the spectral theory of the magnetic Schrödinger operator. It can be considered a catalog of concrete examples of magnetic spectral asymptotics. Since the presentation involves many notions of spectral theory and semiclassical analysis, it begins with a concise account of concepts and methods used in the book and is illustrated by many elementary examples. Assuming various points of view (power series expansions, Feshbach–Grushin reductions, WKB constructions, coherent states decompositions, normal forms) a theory of Magnetic Harmonic Approximation is then established which allows, in particular, accurate descriptions of the magnetic eigenvalues and eigenfunctions. Some parts of this theory, such as those related to spectral reductions or waveguides, are still accessible to advanced students while others (e.g., the discussion of the Birkhoff normal form and its spectral consequences, or the results related to boundary magnetic wells in dimension three) are intended for seasoned researchers.
1
31
2017
978-3-03719-169-9
978-3-03719-669-4
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/169
http://www.ems-ph.org/doi/10.4171/169
EMS Tracts in Mathematics
27
Dynamics Done with Your Bare Hands
Lecture notes by Diana Davis, Bryce Weaver, Roland K. W. Roeder, Pablo Lessa
Françoise
Dal’Bo
Université de Rennes I, France
François
Ledrappier
University of Notre Dame, USA
Amie
Wilkinson
University of Chicago, USA
Dynamical systems and ergodic theory
Differential geometry
37A, 37B, 37D,37F, 37H, 53A
Dynamical systems, geometry, ergodic theory, billards, complex dynamics, random walk, group theory
This book arose from 4 lectures given at the Undergraduate Summer School of the Thematic Program Dynamics and Boundaries held at the University of Notre Dame. It is intended to introduce (under)graduate students to the field of dynamical systems by emphasizing elementary examples, exercises and bare hands constructions. The lecture of Diana Davis is devoted to billiard flows on polygons, a simple-sounding class of continuous time dynamical system for which many problems remain open. Bryce Weaver focuses on the dynamics of a 2x2 matrix acting on the flat torus. This example introduced by Vladimir Arnold illustrates the wide class of uniformly hyperbolic dynamical systems, including the geodesic flow for negatively curved, compact manifolds. Roland Roeder considers a dynamical system on the complex plane governed by a quadratic map with a complex parameter. These maps exhibit complicated dynamics related to the Mandelbrot set defined as the set of parameters for which the orbit remains bounded. Pablo Lessa deals with a type of non-deterministic dynamical system: a simple walk on an infinite graph, obtained by starting at a vertex and choosing a random neighbor at each step. The central question concerns the recurrence property. When the graph is a Cayley graph of a group, the behavior of the walk is deeply related to algebraic properties of the group.
11
30
2016
978-3-03719-168-2
978-3-03719-668-7
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/168
http://www.ems-ph.org/doi/10.4171/168
EMS Series of Lectures in Mathematics
Lines in positive genus: An introduction to flat surfaces
Diana
Davis
Williams College, WILLIAMSTOWN, UNITED STATES
Dynamical systems and ergodic theory
General
1
55
1
10.4171/168-1/1
http://www.ems-ph.org/doi/10.4171/168-1/1
Introduction to complicated behavior and periodic orbits
Bryce
Weaver
James Madison University, HARRISONBURG, UNITED STATES
Dynamical systems and ergodic theory
General
57
100
1
10.4171/168-1/2
http://www.ems-ph.org/doi/10.4171/168-1/2
Around the boundary of complex dynamics
Roland
Roeder
Indiana University Purdue University Indianapolis, INDIANAPOLIS, UNITED STATES
Dynamical systems and ergodic theory
General
101
155
1
10.4171/168-1/3
http://www.ems-ph.org/doi/10.4171/168-1/3
Recurrence vs transience: An introduction to random walks
Pablo
Lessa
Universidad de la República, MONTEVIDEO, URUGUAY
Dynamical systems and ergodic theory
General
157
201
1
10.4171/168-1/4
http://www.ems-ph.org/doi/10.4171/168-1/4
Degenerate Complex Monge–Ampère Equations
Vincent
Guedj
Université Paul Sabatier, Toulouse, France
Ahmed
Zeriahi
Université Paul Sabatier, Toulouse, France
Several complex variables and analytic spaces
32W20, 32U20, 32Q20, 35D30, 53C55, 32U15, 32U40, 35B65
Pluripotential theory, plurisubharmonic functions, complex Monge–Ampère operators, generalized capacities, weak solutions, a priori estimates, canonical Kähler metrics, singular varieties
Winner of the 2016 EMS Monograph Award! Complex Monge–Ampère equations have been one of the most powerful tools in Kähler geometry since Aubin and Yau’s classical works, culminating in Yau’s solution to the Calabi conjecture. A notable application is the construction of Kähler-Einstein metrics on some compact Kähler manifolds. In recent years degenerate complex Monge–Ampère equations have been intensively studied, requiring more advanced tools. The main goal of this book is to give a self-contained presentation of the recent developments of pluripotential theory on compact Kähler manifolds and its application to Kähler–Einstein metrics on mildly singular varieties. After reviewing basic properties of plurisubharmonic functions, Bedford–Taylor’s local theory of complex Monge–Ampère measures is developed. In order to solve degenerate complex Monge–Ampère equations on compact Kähler manifolds, fine properties of quasi-plurisubharmonic functions are explored, classes of finite energies defined and various maximum principles established. After proving Yau’s celebrated theorem as well as its recent generalizations, the results are then used to solve the (singular) Calabi conjecture and to construct (singular) Kähler–Einstein metrics on some varieties with mild singularities. The book is accessible to advanced students and researchers of complex analysis and differential geometry.
1
31
2017
978-3-03719-167-5
978-3-03719-667-0
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/167
http://www.ems-ph.org/doi/10.4171/167
EMS Tracts in Mathematics
26
Metric Geometry of Locally Compact Groups
Yves
Cornulier
Université Paris-Sud, Orsay, France
Pierre
de la Harpe
Université de Genève, Switzerland
Group theory and generalizations
Topological groups, Lie groups
Geometry
Manifolds and cell complexes
Primary: 20F65; Secondary: 20F05, 22D05, 51F99, 54E35, 57M07, 57T20
Locally compact groups, left-invariant metrics, $\sigma$-compactness, second countability, compact generation, compact presentation, metric coarse equivalence, quasi-isometry, coarse connectedness, coarse simple connectedness, growth, amenability
The main aim of this book is the study of locally compact groups from a geometric perspective, with an emphasis on appropriate metrics that can be defined on them. The approach has been successful for finitely generated groups, and can favourably be extended to locally compact groups. Parts of the book address the coarse geometry of metric spaces, where ‘coarse’ refers to that part of geometry concerning properties that can be formulated in terms of large distances only. This point of view is instrumental in studying locally compact groups. Basic results in the subject are exposed with complete proofs, others are stated with appropriate references. Most importantly, the development of the theory is illustrated by numerous examples, including matrix groups with entries in the the field of real or complex numbers, or other locally compact fields such as p-adic fields, isometry groups of various metric spaces, and, last but not least, discrete group themselves. The book is aimed at graduate students and advanced undergraduate students, as well as mathematicians who wish some introduction to coarse geometry and locally compact groups.
9
30
2016
978-3-03719-166-8
978-3-03719-666-3
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/166
http://www.ems-ph.org/doi/10.4171/166
EMS Tracts in Mathematics
25
Free Probability and Operator Algebras
Dan-Virgil
Voiculescu
University of California, Berkeley, USA
Nicolai
Stammeier
University of Oslo, Norway
Moritz
Weber
Universität Saarbrücken, Germany
Functional analysis
Group theory and generalizations
Operator theory
Probability theory and stochastic processes
Primary 46L54; secondary 60B20, 47C15, 20G42
Free probability, operator algebras, random matrices, free monotone transport, free group factors, free convolution, compact quantum groups, easy quantum groups, noncrossing partitions, free independence, entropy, max-stable laws, exchangeability
Free probability is a probability theory dealing with variables having the highest degree of noncommutativity, an aspect found in many areas (quantum mechanics, free group algebras, random matrices etc). Thirty years after its foundation, it is a well-established and very active field of mathematics. Originating from Voiculescu’s attempt to solve the free group factor problem in operator algebras, free probability has important connections with random matrix theory, combinatorics, harmonic analysis, representation theory of large groups, and wireless communication. These lecture notes arose from a masterclass in Münster, Germany and present the state of free probability from an operator algebraic perspective. This volume includes introductory lectures on random matrices and combinatorics of free probability (Speicher), free monotone transport (Shlyakhtenko), free group factors (Dykema), free convolution (Bercovici), easy quantum groups (Weber), and a historical review with an outlook (Voiculescu). In order to make it more accessible, the exposition features a chapter on basics in free probability, and exercises for each part. This book is aimed at master students to early career researchers familiar with basic notions and concepts from operator algebras.
7
31
2016
978-3-03719-165-1
978-3-03719-665-6
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/165
http://www.ems-ph.org/doi/10.4171/165
Münster Lectures in Mathematics
Background and outlook
Dan-Virgil
Voiculescu
University of California, BERKELEY, UNITED STATES
Functional analysis
General
1
6
1
10.4171/165-1/1
http://www.ems-ph.org/doi/10.4171/165-1/1
Basics in free probability
Moritz
Weber
Universität des Saarlandes, SAARBRÜCKEN, GERMANY
Probability theory and stochastic processes
General
7
16
1
10.4171/165-1/2
http://www.ems-ph.org/doi/10.4171/165-1/2
Random matrices and combinatorics
Roland
Speicher
Universität des Saarlandes, SAARBRÜCKEN, GERMANY
Probability theory and stochastic processes
General
17
37
1
10.4171/165-1/3
http://www.ems-ph.org/doi/10.4171/165-1/3
Free monotone transport
Dimitri
Shlyakhtenko
University of California Los Angeles, LOS ANGELES, UNITED STATES
Probability theory and stochastic processes
General
39
56
1
10.4171/165-1/4
http://www.ems-ph.org/doi/10.4171/165-1/4
Free group factors
Ken
Dykema
Texas A&M University, COLLEGE STATION, UNITED STATES
Functional analysis
General
57
72
1
10.4171/165-1/5
http://www.ems-ph.org/doi/10.4171/165-1/5
Free convolution
Hari
Bercovici
Indiana University, BLOOMINGTON, UNITED STATES
Operator theory
General
73
93
1
10.4171/165-1/6
http://www.ems-ph.org/doi/10.4171/165-1/6
Easy quantum groups
Moritz
Weber
Universität des Saarlandes, SAARBRÜCKEN, GERMANY
Operator theory
General
95
121
1
10.4171/165-1/7
http://www.ems-ph.org/doi/10.4171/165-1/7
Mathematics and Society
Wolfgang
König
WIAS Berlin and Technical University Berlin, Germany
General
00-XX
Mathematics in the public; mathematics in architecture, biology, climate, encryption, engineering, finance, industry, nature shapes, physics, telecommunication, and voting systems; experimental mathematics, mathematics museums, mathematics for complex data
The ubiquity and importance of mathematics in our complex society is generally not in doubt. However, even a scientifically interested layman would be hard pressed to point out aspects of our society where contemporary mathematical research is essential. Most popular examples are finance, engineering, wheather and industry, but the way mathematics comes into play is widely unknown in the public. And who thinks of application fields like biology, encryption, architecture, or voting systems? This volume comprises a number of success stories of mathematics in our society – important areas being shaped by cutting edge mathematical research. The authors are eminent mathematicians with a high sense for public presentation, addressing scientifically interested laymen as well as professionals in mathematics and its application disciplines.
7
4
2016
978-3-03719-164-4
978-3-03719-664-9
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/164
http://www.ems-ph.org/doi/10.4171/164
The truth, the whole truth and nothing but the truth: The challenges of reporting on mathematics
George
Szpiro
, NEW YORK, UNITED STATES
General
1
5
1
10.4171/164-1/1
http://www.ems-ph.org/doi/10.4171/164-1/1
Experimental mathematics in the society of the future
David
Bailey
University of California at Davis, DAVIS, UNITED STATES
Jonathan
Borwein
The University of Newcastle, CALLAGHAN, AUSTRALIA
General
Computer-based tools for mathematics are changing how mathematics is researched, taught and communicated to society. Future technology trends point to ever-more powerful tools in the future. Computation in mathematics is thus giving rise to a new mode of mathematical research, where algorithms, datasets and public databases are as significant as the resulting theorems, and even the definition of what constitutes secure mathematical knowledge is seen in a new light.
7
25
1
10.4171/164-1/2
http://www.ems-ph.org/doi/10.4171/164-1/2
What is the impact of interactive mathematical experiments?
Albrecht
Beutelspacher
Mathematikum Giessen e.V., GIESSEN, GERMANY
General
In this article we look at mathematical experiments, in particular those shown in mathematical science centers. Although some mathematicians have the sneaking suspicion that such experiments are far too superficial and do not correspond to proper mathematics, we try to show that in fact mathematical experiments provide an ideal first step for the general public into mathematics.
27
35
1
10.4171/164-1/3
http://www.ems-ph.org/doi/10.4171/164-1/3
Mathematics and finance
Walter
Schachermayer
Universität Wien, WIEN, AUSTRIA
General
This article consists of two parts. The first briefly discusses the history and the basic ideas of option pricing. Based on this background, in the second part we critically analyze the role of academic research in Mathematical Finance relating to the emergence of the 2007–2008 financial crisis.
37
50
1
10.4171/164-1/4
http://www.ems-ph.org/doi/10.4171/164-1/4
Statistics in high dimensions
Aad
van der Vaart
Universiteit Leiden, LEIDEN, NETHERLANDS
Wessel
van Wieringen
VU University Medical Center, AMSTERDAM, NETHERLANDS
Statistics
General
High-dimensional data and models are central to modern statistics. We review some key concepts, some intriguing connections to ideas of the past, and some methods and theoretical results developed in the past decade. We illustrate these results in the context of genomic data from cancer research.
51
70
1
10.4171/164-1/5
http://www.ems-ph.org/doi/10.4171/164-1/5
Filtering theory: Mathematics in engineering, from Gauss to particle filters
Ofer
Zeitouni
Weizmann Institute of Science, REHOVOT, ISRAEL
General
The evolution of engineering needs, especially in the areas of estimation and system theory, has triggered the development of mathematical tools which, in turns, have had a profound influence on engineering practice. We describe this interaction through one example, the evolution of filtering theory.
71
80
1
10.4171/164-1/6
http://www.ems-ph.org/doi/10.4171/164-1/6
Mathematical models for population dynamics: Randomness versus determinism
Jean
Bertoin
Universität Zürich, ZÜRICH, SWITZERLAND
General
Mathematical models are used more and more frequently in Life Sciences. These may be deterministic, or stochastic. We present some classical models for population dynamics and discuss in particular the averaging effect in the setting of large populations, to point at circumstances where randomness prevails nonetheless.
81
99
1
10.4171/164-1/7
http://www.ems-ph.org/doi/10.4171/164-1/7
The quest for laws and structure
Jürg
Fröhlich
ETH Zürich, ZÜRICH, SWITZERLAND
General
The purpose of this paper is to illustrate, on some concrete examples, the quest of theoretical physicists for new laws of Nature and for appropriate mathematical structures that enables them to formulate and analyze new laws in as simple terms as possible and to derive consequences therefrom. The examples are taken from thermodynamics, atomism and quantum theory.
101
129
1
10.4171/164-1/8
http://www.ems-ph.org/doi/10.4171/164-1/8
Geometry and freeform architecture
Helmut
Pottmann
Technische Universität Wien, WIEN, AUSTRIA
Johannes
Wallner
Technische Universität Graz, GRAZ, AUSTRIA
Geometry
General
During the last decade, the geometric aspects of freeform architecture have defined a field of applications which is systematically explored and which conversely serves as inspiration for new mathematical research. This paper discusses topics relevant to the realization of freeform skins by various means (flat and curved panels, straight and curved members, masonry, etc.) and illuminates the interrelations of those questions with theory, in particular discrete differential geometry and discrete conformal geometry.
131
151
1
10.4171/164-1/9
http://www.ems-ph.org/doi/10.4171/164-1/9
Some geometries to describe nature
Christiane
Rousseau
University of Montreal, MONTRÉAL, CANADA
Geometry
General
Since ancient times, the development of mathematics has been inspired, at least in part, by the need to provide models in other sciences, and that of describing and understanding the world around us. In this note, we concentrate on the shapes of nature and introduce two related geometries that play an important role in contemporary science. Fractal geometry allows describing a wide range of shapes in nature. In 1973, Harry Blum introduced a new geometry well suited to describe animal morphology.
153
165
1
10.4171/164-1/10
http://www.ems-ph.org/doi/10.4171/164-1/10
Mathematics in industry
Helmut
Neunzert
ITWM, KAISERSLAUTERN, GERMANY
General
Industrial mathematics has become a fashionable subject in the last three decades. Today, there are many university groups and even research institutes dedicated to industrial and applied mathematics worldwide, proving that mathematics has indeed become a key technology. In this paper we first give a short account of the history of industrial mathematics. Using experiences from the Fraunhofer Institute for Industrial Mathematics (ITWM), we try to characterize the specific problem driven work of industrial mathematicians and take a look at future challenges.
167
183
1
10.4171/164-1/11
http://www.ems-ph.org/doi/10.4171/164-1/11
Mathematics of signal design for communication systems
Holger
Boche
Technische Universität München, MÜNCHEN, GERMANY
Ezra
Tampubolon
Technische Universität München, MÜNCHEN, GERMANY
General
Orthogonal transmission schemes constitute the foundations of both our present and future communication standards. One of the major drawback of orthogonal transmission schemes is their high dynamical behaviour, which can be measured by the so called Peakto–Average power value – the ratio between the peak value (i.e. $L^\infty$-norm) and the average power (i.e., $L^2$-norm) of a signal. This undesired behaviour of orthogonal schemes has remarkable negative impacts on the performance, the energy-efficiency, and the maintain cost of the transmission systems. In this work, we give some discussions concerning to the problem of reduction of the high dynamics of an orthogonal transmission scheme. We show that this problem is connected with some mathematical fields, such as functional analysis (Hahn-Banach Theorem and Baire Category), additive combinatorics (Szeméredi Theorem, Green-Tao Theorem on arithmetic progressions in the primes, sparse Szeméredi type Theorems by Conlon and Gowers, and the famous Erd˝os problem on arithmetic progressions), and both trigonometric and non-trigonometric harmonic analysis.
185
220
1
10.4171/164-1/12
http://www.ems-ph.org/doi/10.4171/164-1/12
Cryptology: Methods, applications and challenges
Claus
Diem
Universität Leipzig, LEIPZIG, GERMANY
General
Information processing by electronic devices leads to a multitude of security-relevant challenges. With the help of cryptography, many of these challenges can be solved and new applications can be made possible. What methods are hereby used? On which mathematical foundations do they rest? How did the prevailing ideas and methods come about? What are the current developments, what challenges exist and which future challenges can be predicted?
221
250
1
10.4171/164-1/13
http://www.ems-ph.org/doi/10.4171/164-1/13
A mathematical view on voting and power
Werner
Kirsch
FernUniversität Hagen, HAGEN, GERMANY
General
In this article we describe some concepts, ideas and results from the mathematical theory of voting. We give a mathematical description of voting systems and introduce concepts to measure the power of a voter. We also describe and investigate two-tier voting systems, for example the Council of the European Union. In particular, we prove criteria which give the optimal voting weights in such systems.
251
279
1
10.4171/164-1/14
http://www.ems-ph.org/doi/10.4171/164-1/14
Numerical methods and scientific computing for climate and geosciences
Jörn
Behrens
Universität Hamburg, HAMBURG, GERMANY
General
Studying the climate, weather or other geoscientific phenomena is strongly related to simulation based knowledge gain, since the climate system, for example, is not assessable by laboratory experiments. In these simulations, mathematical models as well as numerical methods play a crucial role in many aspects of the knowledge work-flow. We will describe the general set-up of geoscientific models, and explore some of the applied mathematical methods involved in solving such models. One of the paramount problems of geoscientific simulation applications is the large span of scales that interact.
281
293
1
10.4171/164-1/15
http://www.ems-ph.org/doi/10.4171/164-1/15
Geometry, Analysis and Dynamics on sub-Riemannian Manifolds
Volume II
Davide
Barilari
Université Paris 7 Denis Diderot, Paris, France
Ugo
Boscain
École Polytechnique, Palaiseau, France
Mario
Sigalotti
École Polytechnique, Palaiseau, France
Differential geometry
Partial differential equations
Calculus of variations and optimal control; optimization
Probability theory and stochastic processes
53C17, 35H10, 60H30, 49J15
Sub-Riemannian geometry, hypoelliptic operators, non-holonomic constraints, optimal control, rough paths
Sub-Riemannian manifolds model media with constrained dynamics: motion at any point is only allowed along a limited set of directions, which are prescribed by the physical problem. From the theoretical point of view, sub-Riemannian geometry is the geometry underlying the theory of hypoelliptic operators and degenerate diffusions on manifolds. In the last twenty years, sub-Riemannian geometry has emerged as an independent research domain, with extremely rich motivations and ramifications in several parts of pure and applied mathematics, such as geometric analysis, geometric measure theory, stochastic calculus and evolution equations together with applications in mechanics, optimal control and biology. The aim of the lectures collected here is to present sub-Riemannian structures for the use of both researchers and graduate students.
10
25
2016
978-3-03719-163-7
978-3-03719-663-2
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/163
http://www.ems-ph.org/doi/10.4171/163
EMS Series of Lectures in Mathematics
Introduction to geodesics in sub-Riemannian geometry
Andrei
Agrachev
SISSA, TRIESTE, ITALY
Davide
Barilari
Université Paris 7, Denis Diderot, PARIS 13 CEDEX, FRANCE
Ugo
Boscain
Ecole Polytechnique, PALAISEAU CEDEX, FRANCE
General
1
83
1
10.4171/163-1/1
http://www.ems-ph.org/doi/10.4171/163-1/1
Geometry of subelliptic diffusions
Anton
Thalmaier
Université du Luxembourg, LUXEMBOURG, LUXEMBOURG
General
85
169
1
10.4171/163-1/2
http://www.ems-ph.org/doi/10.4171/163-1/2
Geometric foundations of rough paths
Peter
Friz
Technische Universität Berlin, BERLIN, GERMANY
Paul
Gassiat
Université Paris-Dauphine, PARIS CEDEX 16, FRANCE
General
171
210
1
10.4171/163-1/3
http://www.ems-ph.org/doi/10.4171/163-1/3
Sobolev and bounded variation functions on metric measure spaces
Luigi
Ambrosio
Scuola Normale Superiore, PISA, ITALY
Roberta
Ghezzi
Université de Bourgogne, DIJON CEDEX, FRANCE
General
211
273
1
10.4171/163-1/4
http://www.ems-ph.org/doi/10.4171/163-1/4
Singularities of vector distributions
Michail
Zhitomirskii
Technion - Israel Institute of Technology, HAIFA, ISRAEL
General
275
295
1
10.4171/163-1/5
http://www.ems-ph.org/doi/10.4171/163-1/5
Geometry, Analysis and Dynamics on sub-Riemannian Manifolds
Volume I
Davide
Barilari
Université Paris 7 Denis Diderot, Paris, France
Ugo
Boscain
École Polytechnique, Palaiseau, France
Mario
Sigalotti
École Polytechnique, Palaiseau, France
Differential geometry
Partial differential equations
Calculus of variations and optimal control; optimization
Probability theory and stochastic processes
53C17, 35H10, 60H30, 49J15
Sub-Riemannian geometry, hypoelliptic operators, non-holonomic constraints, optimal control, rough paths
Sub-Riemannian manifolds model media with constrained dynamics: motion at any point is only allowed along a limited set of directions, which are prescribed by the physical problem. From the theoretical point of view, sub-Riemannian geometry is the geometry underlying the theory of hypoelliptic operators and degenerate diffusions on manifolds. In the last twenty years, sub-Riemannian geometry has emerged as an independent research domain, with extremely rich motivations and ramifications in several parts of pure and applied mathematics, such as geometric analysis, geometric measure theory, stochastic calculus and evolution equations together with applications in mechanics, optimal control and biology. The aim of the lectures collected here is to present sub-Riemannian structures for the use of both researchers and graduate students.
6
30
2016
978-3-03719-162-0
978-3-03719-662-5
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/162
http://www.ems-ph.org/doi/10.4171/162
EMS Series of Lectures in Mathematics
Some topics of geometric measure theory in Carnot groups
Francesco
Serra Cassano
Università di Trento, TRENTO (TN), ITALY
Differential geometry
General
1
121
1
10.4171/162-1/1
http://www.ems-ph.org/doi/10.4171/162-1/1
Hypoelliptic operators and some aspects of analysis and geometry of sub-Riemannian spaces
Nicola
Garofalo
Università di Padova, PADOVA, ITALY
Differential geometry
General
123
257
1
10.4171/162-1/2
http://www.ems-ph.org/doi/10.4171/162-1/2
Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations
Fabrice
Baudoin
Purdue University, WEST LAFAYETTE, UNITED STATES
Differential geometry
General
259
321
1
10.4171/162-1/3
http://www.ems-ph.org/doi/10.4171/162-1/3
Handbook of Teichmüller Theory, Volume VI
Athanase
Papadopoulos
Functions of a complex variable
Primary 30-00, 32-00, 57-00, 32G13, 32G15, 30F60. Secondary 11F06, 11F75, 14D20, 11G32, 14C05, 14H15, 14H30, 14H15, 14H60, 14H55, 14J60, 18A22, 20F14, 20F28, 20F38, 20F65, 20F67, 20H10, 22E46, 30-03, 30C62, 30F20, 30F25, 30F10, 30F15, 30F30, 30F35, 30F40, 30F45, 32-03, 32S30, 32G13, 32G15, 37-99, 53A35, 53B35, 53C35, 53C50, 53C80, 53D55, 53Z05, 57M07, 57M20, 57M27, 57M50, 57M60, 57N16
This volume is the sixth in a series dedicated to Teichmüller theory in a broad sense, including various moduli and deformation spaces, and the study of mapping class groups. It is divided into five parts: Part A: The metric and the analytic theory. Part B: The group theory. Part C: Representation theory and generalized structures. Part D: The Grothendieck–Teichmüller theory. Part D: Sources. The topics surveyed include Grothendieck’s construction of the analytic structure of Teichmüller space, identities on the geodesic length spectrum of hyperbolic surfaces (including Mirzakhani’s application to the computation of Weil–Petersson volumes), moduli spaces of configurations spaces, the Teichmüller tower with the action of the Galois group on dessins d’enfants, and several others actions related to surfaces. The last part contains three papers by Teichmüller, translated into English with mathematical commentaries, and a document that contains H. Grötzsch’s comments on Teichmüller’s famous paper Extremale quasikonforme Abbildungen und quadratische Differentiale. The handbook is addressed to researchers and to graduate students.
5
31
2016
978-3-03719-161-3
978-3-03719-661-8
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/161
http://www.ems-ph.org/doi/10.4171/161
IRMA Lectures in Mathematics and Theoretical Physics
27
Introduction to Teichmüller theory, old and new, VI
Athanase
Papadopoulos
Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE
General
1
29
1
10.4171/161-1/1
http://www.ems-ph.org/doi/10.4171/161-1/1
Alexander Grothendieck
Valentin
Poenaru
Le Bois-Marie, BURES-SUR-YVETTE, FRANCE
History and biography
General
31
32
1
10.4171/161-1/2
http://www.ems-ph.org/doi/10.4171/161-1/2
On Grothendieck’s construction of Teichmüller space
Norbert
A’Campo
Universität Basel, BASEL, SWITZERLAND
Lizhen
Ji
University of Michigan, ANN ARBOR, UNITED STATES
Athanase
Papadopoulos
Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE
General
35
69
1
10.4171/161-1/3
http://www.ems-ph.org/doi/10.4171/161-1/3
Null-set compactifications of Teichmüller spaces
Vincent
Alberge
Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE
Hideki
Miyachi
Osaka University, OSAKA, JAPAN
Ken’ichi
Ohshika
Osaka University Graduate School of Science, OSAKA, JAPAN
General
71
94
1
10.4171/161-1/4
http://www.ems-ph.org/doi/10.4171/161-1/4
Mirzakhani’s recursion formula on Weil–Petersson volume and applications
Yi
Huang
University of Melbourne, MELBOURNE, Victoria, AUSTRALIA
General
95
127
1
10.4171/161-1/5
http://www.ems-ph.org/doi/10.4171/161-1/5
Rigidity phenomena in the mapping class group
Javier
Aramayona
Universidad Autónoma de Madrid. , MADRID, SPAIN
Juan
Souto
Université de Rennes 1, RENNES CEDEX, FRANCE
General
131
165
1
10.4171/161-1/6
http://www.ems-ph.org/doi/10.4171/161-1/6
Harmonic volume and its applications
Yuuki
Tadokoro
Kisarazu National College of Technology, CHIBA, JAPAN
General
167
193
1
10.4171/161-1/7
http://www.ems-ph.org/doi/10.4171/161-1/7
Torus bundles and 2-forms on the universal family of Riemann surfaces
Robin
de Jong
Universiteit Leiden, LEIDEN, NETHERLANDS
General
195
227
1
10.4171/161-1/8
http://www.ems-ph.org/doi/10.4171/161-1/8
Cubic Differentials in the Differential Geometry of Surfaces
John
Loftin
Rutgers University, NEWARK, UNITED STATES
Ian
McIntosh
University of York, YORK, UNITED KINGDOM
General
231
274
1
10.4171/161-1/9
http://www.ems-ph.org/doi/10.4171/161-1/9
Two-generator groups acting on the complex hyperbolic plane
Pierre
Will
Université de Grenoble I, SAINT-MARTIN D'HÈRES, FRANCE
Functions of a complex variable
General
275
334
1
10.4171/161-1/10
http://www.ems-ph.org/doi/10.4171/161-1/10
Configuration spaces of planar linkages
Alexey
Sossinsky
Independent University of Moscow, MOSCOW, RUSSIAN FEDERATION
Functions of a complex variable
General
335
373
1
10.4171/161-1/11
http://www.ems-ph.org/doi/10.4171/161-1/11
Quasiconformal mappings on the Heisenberg group: An overview
Ioannis
Platis
University of Crete, HERAKLION, GREECE
Functions of a complex variable
General
375
393
1
10.4171/161-1/12
http://www.ems-ph.org/doi/10.4171/161-1/12
Actions of the absolute Galois group
Norbert
A’Campo
Universität Basel, BASEL, SWITZERLAND
Lizhen
Ji
University of Michigan, ANN ARBOR, UNITED STATES
Athanase
Papadopoulos
Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE
Functions of a complex variable
General
397
435
1
10.4171/161-1/13
http://www.ems-ph.org/doi/10.4171/161-1/13
A primer on dessins
Pierre
Guillot
Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE
Functions of a complex variable
General
437
466
1
10.4171/161-1/14
http://www.ems-ph.org/doi/10.4171/161-1/14
Hypergeometric Galois Actions
A. Muhammed
Uludağ
Galatasaray University, ISTANBUL, TURKEY
İsmail
Sağlam
Galatasaray University, ISTANBUL, TURKEY
Functions of a complex variable
General
467
500
1
10.4171/161-1/15
http://www.ems-ph.org/doi/10.4171/161-1/15
A panaroma of the fundamental group of the modular orbifold
A. Muhammed
Uludağ
Galatasaray University, ISTANBUL, TURKEY
Ayberk
Zeytin
Galatasaray University, ISTANBUL, TURKEY
Functions of a complex variable
General
501
519
1
10.4171/161-1/16
http://www.ems-ph.org/doi/10.4171/161-1/16
On Grothendieck’s tame topology
Norbert
A’Campo
Universität Basel, BASEL, SWITZERLAND
Lizhen
Ji
University of Michigan, ANN ARBOR, UNITED STATES
Athanase
Papadopoulos
Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE
Functions of a complex variable
General
521
533
1
10.4171/161-1/17
http://www.ems-ph.org/doi/10.4171/161-1/17
Some historical commentaries on Teichmüller’s paper Extremale quasikonforme Abbildungen und quadratische Differentiale
Reiner
Kühnau
Martin-Luther-Universität Halle-Wittenberg, HALLE, GERMANY
Functions of a complex variable
General
537
546
1
10.4171/161-1/18
http://www.ems-ph.org/doi/10.4171/161-1/18
Complete solution of an extremal problem of the quasiconformal mapping
Oswald
Teichmüller
, Berlin, GERMANY
Functions of a complex variable
General
547
560
1
10.4171/161-1/19
http://www.ems-ph.org/doi/10.4171/161-1/19
A Commentary on Teichmüller’s paper Vollständige Lösung einer Extremalaufgabe der quasikonformen Abbildung
Vincent
Alberge
Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE
Athanase
Papadopoulos
Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE
Functions of a complex variable
General
561
567
1
10.4171/161-1/20
http://www.ems-ph.org/doi/10.4171/161-1/20
On extremal problems of conformal geometry
Oswald
Teichmüller
, Berlin, GERMANY
Functions of a complex variable
General
569
596
1
10.4171/161-1/21
http://www.ems-ph.org/doi/10.4171/161-1/21
A Commentary on Teichmüller’s paper Über Extremalprobleme der konformen Geometrie
Norbert
A’Campo
Universität Basel, BASEL, SWITZERLAND
Athanase
Papadopoulos
Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE
Functions of a complex variable
General
597
603
1
10.4171/161-1/22
http://www.ems-ph.org/doi/10.4171/161-1/22
A displacement theorem of quasiconformal mapping
Oswald
Teichmüller
, Berlin, GERMANY
Functions of a complex variable
General
605
612
1
10.4171/161-1/23
http://www.ems-ph.org/doi/10.4171/161-1/23
A Commentary on Teichmüller’s paper Ein Verschiebungssatz der quasikonformen Abbildung
Vincent
Alberge
Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE
Functions of a complex variable
General
613
629
1
10.4171/161-1/24
http://www.ems-ph.org/doi/10.4171/161-1/24
Handbook of Teichmüller Theory, Volume V
Athanase
Papadopoulos
Université de Strasbourg, France
Functions of a complex variable
Primary 30-00, 32-00, 57-00, 32G13, 32G15, 30F60; Secondary 11F06, 11F75, 14D20, 14H15, 14H60, 14H55, 14J60, 20F14, 20F28, 20F38, 20F65, 20F67, 20H10, 22E46, 30-03, 30C62, 30F20, 30F25, 30F10, 30F15, 30F30, 30F35, 30F40, 30F45, 32-03, 32S30, 37-99, 53A35, 53B35, 53C35, 53C50, 53C80, 53D55, 53Z05, 57M07, 57M20, 57M27, 57M50, 57M60, 57N16
This volume is the fifth in a series dedicated to Teichmüller theory in a broad sense, including the study of various deformation spaces and of mapping class group actions. It is divided into four parts: Part A: The metric and the analytic theory Part B: The group theory Part C: Representation theory and generalized structures Part D: Sources The topics that are covered include identities for the hyperbolic geodesic length spectrum, Thurston's metric, the cohomology of moduli space and mapping class groups, the Johnson homomorphisms, Higgs bundles, dynamics on character varieties, and there are many others. Besides surveying important parts of the theory, several chapters contain conjectures and open problems. The last part contains two fundamental papers by Teichmüller, translated into English and accompanied by mathematical commentaries. The chapters, like those of the other volumes in this collection, are written by experts who have a broad view on the subject. They have an expository character (which fits with the original purpose of this handbook), but some of them also contain original and new material. The Handbook is addressed to researchers and to graduate students.
1
11
2016
978-3-03719-160-6
978-3-03719-660-1
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/160
http://www.ems-ph.org/doi/10.4171/160
IRMA Lectures in Mathematics and Theoretical Physics
26
Introduction to Teichmüller theory, old and new, V
Athanase
Papadopoulos
Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE
General
1
16
1
10.4171/160-1/1
http://www.ems-ph.org/doi/10.4171/160-1/1
Identities on hyperbolic manifolds
Martin
Bridgeman
Boston College, CHESTNUT HILL, UNITED STATES
Ser Peow
Tan
National University of Singapore, SINGAPORE, SINGAPORE
Manifolds and cell complexes
Group theory and generalizations
Functions of a complex variable
General
In this survey, we discuss four classes of identities due principally to Basmajian, McShane, Bridgeman-Kahn and Luo-Tan on hyperbolic manifolds and provide a unified approach for proving them. We also elucidate on the connections between the various identities.
19
53
1
10.4171/160-1/2
http://www.ems-ph.org/doi/10.4171/160-1/2
Problems on the Thurston metric
Weixu
Su
Fudan University, SHANGHAI, CHINA
Several complex variables and analytic spaces
Functions of a complex variable
General
We present a list of problems related to the study of Thurston metric on Teichmüller space. The problems originate in discussions by the participants at the 2012 AIM workshop “Lipschitz metric on Teichmüller space”. Some of the problems were worked on by small groups, and they remain open. Some others were suggested by various participants at the closing problem session of the conference or added after the conference. We have updated the list in order to include it in the present Handbook.
55
72
1
10.4171/160-1/3
http://www.ems-ph.org/doi/10.4171/160-1/3
Meyer functions and the signature of fibered 4-manifolds
Yusuke
Kuno
Tsuda College, TOKYO, JAPAN
Algebraic geometry
Group theory and generalizations
Several complex variables and analytic spaces
Manifolds and cell complexes
We give a survey on Meyer functions, with emphasis on their application to the signature of fibered 4-manifolds.
75
96
1
10.4171/160-1/4
http://www.ems-ph.org/doi/10.4171/160-1/4
The Goldman–Turaev Lie bialgebra and the Johnson homomorphisms
Nariya
Kawazumi
University of Tokyo, TOKYO, JAPAN
Yusuke
Kuno
Tsuda College, TOKYO, JAPAN
Group theory and generalizations
Several complex variables and analytic spaces
Manifolds and cell complexes
General
We survey a geometric approach to the Johnson homomorphisms using the Goldman–Turaev Lie bialgebra.
97
165
1
10.4171/160-1/5
http://www.ems-ph.org/doi/10.4171/160-1/5
A survey of the Johnson homomorphisms of the automorphism groups of free groups and related topics
Takao
Satoh
Tokyo University of Science, TOKYO, JAPAN
Group theory and generalizations
General
This is a survey on the Johnson homomorphisms of the automorphism groups of free groups. We exposit some well known facts and recent developments for the Johnson homomorphisms and its related topics.
167
209
1
10.4171/160-1/6
http://www.ems-ph.org/doi/10.4171/160-1/6
Geometry and dynamics on character varieties
Inkang
Kim
KIAS, SEOUL, SOUTH KOREA
Topological groups, Lie groups
Differential geometry
Manifolds and cell complexes
General
We survey recent progress on character varieties. We give a general definition of a character variety, yet focus on rigidity and flexibility issues of a given representation. On the other hand, we study dynamical aspects on the character variety, the relation to bounded cohomology theory and several topological issues using different branches of mathematics. We try to give as many familiar examples as possible to motivate the readers.
213
235
1
10.4171/160-1/7
http://www.ems-ph.org/doi/10.4171/160-1/7
Compactifications and reduction theory of geometrically finite locally symmetric spaces
Lizhen
Ji
University of Michigan, ANN ARBOR, UNITED STATES
Topological groups, Lie groups
Algebraic geometry
General
There are two closely related classes of groups arising from Fuchsian groups and their actions on the hyperbolic plane $\H^2$: discrete subgroups of semisimple Lie groups acting on symmetric spaces, and mapping class groups and their subgroups acting on Teichmüller space. Convex cocompact Fuchsian groups have been generalized to Anosov subgroups of semisimple Lie groups of higher rank, which have played an important role in higher Teichmüller theory. They have also been generalized to convex cocompact subgroups of mapping class groups. Due to the fact that Teichmüller space shares some similarities with symmetric spaces of rank one, the analogy between convex cocompact Fuchsian groups and convex cocompact subgroups of mapping class groups is rather complete. But less is known about actions of Anosov subgroups on symmetric spaces of higher rank. In this chapter, we discuss three conjectures on compactifications and coarse fundamental domains for locally symmetric spaces associated with Anosov subgroups of noncompact semisimple Lie groups, and describe results, motivations, and evidence for these conjectures. One conjecture deals with the existence of maximal Satake compactifications of Anosov locally symmetric spaces, and other two are concerned with the characterization of Anosov subgroups. By comparing these locally symmetric spaces of infinite volume with symmetric spaces and locally symmetric spaces of finite volume, and by examining applications of the maximal Satake compactification of symmetric spaces and the Borel-Serre compactification of locally symmetric spaces of finite volume, we conclude that the conjectural maximal Satake compactifications of Anosov locally symmetric spaces arising from the maximal Satake compactification of symmetric spaces are the natural compactifications. In general, there is more than one maximal Satake compactification for Anosov locally symmetric spaces. To explain this non-uniqueness, we develop a reduction theory for Anosov subgroups by introducing the crucial notion of anti-Siegel sets. We also explain how geometric boundaries of the maximal Satake compactifications are related to problems in the spectral theory of Laplace operators of the Riemannian manifolds under consideration, and conclude with some questions on the Martin compactification of Anosov locally symmetric spaces, which is related to positive harmonic functions.
237
305
1
10.4171/160-1/8
http://www.ems-ph.org/doi/10.4171/160-1/8
Representations of fundamental groups of 2-manifolds
Lisa
Jeffrey
University of Toronto, TORONTO, ONTARIO, CANADA
Global analysis, analysis on manifolds
Partial differential equations
General
This chapter aims to provide a survey on the subject of representations of fundamental groups of 2-manifolds, or in other guises flat connections on orientable 2-manifolds or moduli spaces parametrizing holomorphic vector bundles on Riemann surfaces. It emphasizes the relationships between the different descriptions of these spaces.
307
317
1
10.4171/160-1/9
http://www.ems-ph.org/doi/10.4171/160-1/9
Extremal quasiconformal mappings and quadratic differentials
Oswald
Teichmüller
, Berlin, GERMANY
General
321
483
1
10.4171/160-1/10
http://www.ems-ph.org/doi/10.4171/160-1/10
A commentary on Teichmüller’s paper Extremale quasikonforme Abbildungen und quadratische Differentiale
Vincent
Alberge
Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE
Athanase
Papadopoulos
Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE
Weixu
Su
Fudan University, SHANGHAI, CHINA
General
We provide a mathematical commentary on Teichmüller’s paper Extremale quasikonforme Abbildungen und quadratische Differentiale (Extremal quasiconformal mappings of closed oriented Riemann surfaces), Abh. Preuss. Akad. Wiss., Math.- Naturw. Kl. 1940, No.22, 1–197 (1940). The paper is quoted in several works, although it was read by very few people. Some of the results it contains were re- discovered later on and published without any reference to Teichmüller. In this commentary, we highlight the main results and the main ideas contained in that paper and we describe some of the important developments they gave rise to.
485
531
1
10.4171/160-1/11
http://www.ems-ph.org/doi/10.4171/160-1/11
Determination of extremal quasiconformal mappings of closed oriented Riemann surfaces
Oswald
Teichmüller
, Berlin, GERMANY
General
533
567
1
10.4171/160-1/12
http://www.ems-ph.org/doi/10.4171/160-1/12
A commentary on Teichmüller’s paper Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Flächen
Annette
A’Campo-Neuen
Universität Basel, BASEL, SWITZERLAND
Norbert
A’Campo
Universität Basel, BASEL, SWITZERLAND
Vincent
Alberge
Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE
Athanase
Papadopoulos
Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE
General
This is a mathematical commentary on Teichmüller’s paper Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Flächen (Determination of extremal quasiconformal maps of closed oriented Riemann surfaces) [24], (1943). This paper is among the last (and may be the last one) that Teichmüller wrote on the theory of moduli. It contains the proof of the so-called Teichmüller existence theorem for a closed surface of genus $g ≥ 2$. For this proof, the author defines a mapping between a space of equivalence classes of marked Riemann surfaces (the Teichmüller space) and a space of equivalence classes of certain Fuchsian groups (the so-called Fricke space). After that, he defines a map between the latter and the Euclidean space of dimension $6g−6$. Using Brouwer’s theorem of invariance of domain, he shows that this map is a home- omorphism. This involves in particular a careful definition of the topologies of Fricke space, the computation of its dimension, and comparison results between hyperbolic distance and quasiconformal dilatation. The use of the invariance of domain theorem is in the spirit of Poincar ́e and Klein’s use of the so-called “continuity principle” in their attempts to prove the uniformization theorem.
569
580
1
10.4171/160-1/13
http://www.ems-ph.org/doi/10.4171/160-1/13
Measure and Integration
Dietmar
Salamon
ETH Zürich, Switzerland
Measure and integration
Primary: 28-01; Secondary: 28C05, 28C10, 28C15, 35J05, 43A05, 44A35, 46B22, 46C05, 46E27, 46E30
sigma-Algebra, Lebesgue monotone convergence, Caratheodory criterion, Lebesgue measure, Borel measure, Dieudonné’s measure, Riesz Representation Theorem, Alexandrov Double Arrow Space, Sorgenfrey Line, separability, Cauchy–Schwarz inequality, Jensen’s inequality, Egoroff’s theorem, Hardy’s inequality, absolutely continuous measure, truly continuous measure, singular measure, signed measure, Radon–Nikodym Theorem, Lebesgue Decomposition Theorem, Hahn Decomposition Theorem, Jordan Decomposition Theorem, Hardy–Littlewood maximal inequality, Vitali’s Covering Lemma, Lebesgue point, Lebesgue Differentiation Theorem, Banach-Zarecki Theorem, Vitali–Caratheodory Theorem, Cantor function, product sigma-algebra, Fubini’s Theorem, convolution, Young’s inequality, mollifier, Marcinciewicz interpolation, Poisson identity, Green’s formula, Calderon–Zygmund inequality, Haar measure, modular character
The book is intended as a companion to a one semester introductory lecture course on measure and integration. After an introduction to abstract measure theory it proceeds to the construction of the Lebesgue measure and of Borel measures on locally compact Hausdorff spaces, $L^p$ spaces and their dual spaces and elementary Hilbert space theory. Special features include the formulation of the Riesz Representation Theorem in terms of both inner and outer regularity, the proofs of the Marcinkiewicz Interpolation Theorem and the Calderon–Zygmund inequality as applications of Fubini’s theorem and Lebesgue differentiation, the treatment of the generalized Radon–Nikodym theorem due to Fremlin, and the existence proof for Haar measures. Three appendices deal with Urysohn’s Lemma, product topologies, and the inverse function theorem. The book assumes familiarity with first year analysis and linear algebra. It is suitable for second year undergraduate students of mathematics or anyone desiring an introduction to the concepts of measure and integration.
3
29
2016
978-3-03719-159-0
978-3-03719-659-5
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/159
http://www.ems-ph.org/doi/10.4171/159
EMS Textbooks in Mathematics
Metric Measure Geometry
Gromov’s Theory of Convergence and Concentration of Metrics and Measures
Takashi
Shioya
Tohoku University, Sendai, Japan
Differential geometry
Measure and integration
Functions of a complex variable
Probability theory and stochastic processes
Primary: 53C23; Secondary: 28A33, 30Lxx, 35P15, 53C20, 54Exx, 58C40, 58J50, 60B10
Metric measure space, concentration of measure phenomenon, observable distance, pyramid, convergence of spaces, curvature-dimension condition, Laplacian, dissipation
This book studies a new theory of metric geometry on metric measure spaces, originally developed by M. Gromov in his book “Metric Structures for Riemannian and Non-Riemannian Spaces” and based on the idea of the concentration of measure phenomenon due to Lévy and Milman. A central theme in this text is the study of the observable distance between metric measure spaces, defined by the difference between 1-Lipschitz functions on one space and those on the other. The topology on the set of metric measure spaces induced by the observable distance function is weaker than the measured Gromov–Hausdorff topology and allows to investigate a sequence of Riemannian manifolds with unbounded dimensions. One of the main parts of this presentation is the discussion of a natural compactification of the completion of the space of metric measure spaces. The stability of the curvature-dimension condition is also discussed. This book makes advanced material accessible to researchers and graduate students interested in metric measure spaces.
1
5
2016
978-3-03719-158-3
978-3-03719-658-8
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/158
http://www.ems-ph.org/doi/10.4171/158
IRMA Lectures in Mathematics and Theoretical Physics
25
Absolute Arithmetic and $\mathbb F_1$-Geometry
Koen
Thas
University of Gent, Belgium
Combinatorics
Number theory
Commutative rings and algebras
Algebraic geometry
Primary: 05E18, 11M26, 13F35, 13K05, 14A15, 14A20, 14A22, 14G15, 14G40, 14H10, 18A05, 19E08, 20B25, 20G05, 20G35, 20M25, 51E24; Secondary: 05E05, 06B10, 11G20, 11G25, 11R18, 11T55, 13A35, 13C60, 14C40, 14F05, 14L15, 14M25, 14M26, 14P10, 15B48, 16G20, 16Y60, 18D50, 18F20, 20E42, 20F36, 20M14, 20M32, 20N20, 51B25, 55N30, 55P42, 55Q45
The field with one element, $\mathbb F_1$-geometry, combinatorial $\mathbb F_1$-geometry, non-additive category, Deitmar scheme, graph, monoid, motive, zeta function, automorphism group, blueprint, Euler characteristic, K-theory, Grassmannian, Witt ring, noncommutative geometry, Witt vector, total positivity, moduli space of curves, operad, torificiation, Absolute Arithmetic, counting function, Weil conjectures, Riemann Hypothesis
It has been known for some time that geometries over finite fields, their automorphism groups and certain counting formulae involving these geometries have interesting guises when one lets the size of the field go to 1. On the other hand, the nonexistent field with one element, $\mathbb F_1$, presents itself as a ghost candidate for an absolute basis in Algebraic Geometry to perform the Deninger–Manin program, which aims at solving the classical Riemann Hypothesis. This book, which is the first of its kind in the $\mathbb F_1$-world, covers several areas in $\mathbb F_1$-theory, and is divided into four main parts – Combinatorial Theory, Homological Algebra, Algebraic Geometry and Absolute Arithmetic. Topics treated include the combinatorial theory and geometry behind $\mathbb F_1$, categorical foundations, the blend of different scheme theories over $\mathbb F_1$ which are presently available, motives and zeta functions, the Habiro topology, Witt vectors and total positivity, moduli operads, and at the end, even some arithmetic. Each chapter is carefully written by experts, and besides elaborating on known results, brand new results, open problems and conjectures are also met along the way. The diversity of the contents, together with the mystery surrounding the field with one element, should attract any mathematician, regardless of speciality.
7
25
2016
978-3-03719-157-6
978-3-03719-657-1
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/157
http://www.ems-ph.org/doi/10.4171/157
The Weyl functor. Introduction to Absolute Arithmetic
Koen
Thas
Universiteit Gent, GENT, BELGIUM
Combinatorics
Number theory
Commutative rings and algebras
Algebraic geometry
3
36
1
10.4171/157-1/1
http://www.ems-ph.org/doi/10.4171/157-1/1
Belian categories
Anton
Deitmar
Universität Tübingen, TÜBINGEN, GERMANY
Combinatorics
Number theory
Commutative rings and algebras
Algebraic geometry
39
80
1
10.4171/157-1/2
http://www.ems-ph.org/doi/10.4171/157-1/2
The combinatorial-motivic nature of $\mathbb F_1$-schemes
Koen
Thas
Universiteit Gent, GENT, BELGIUM
Combinatorics
Number theory
Commutative rings and algebras
Algebraic geometry
83
159
1
10.4171/157-1/3
http://www.ems-ph.org/doi/10.4171/157-1/3
A blueprinted view on $\mathbb F_1$-geometry
Oliver
Lorscheid
, RIO DE JANEIRO, BRAZIL
Combinatorics
Number theory
Commutative rings and algebras
Algebraic geometry
161
219
1
10.4171/157-1/4
http://www.ems-ph.org/doi/10.4171/157-1/4
Absolute geometry and the Habiro topology
Lieven
Le Bruyn
University of Antwerp, ANTWERP, BELGIUM
Combinatorics
Number theory
Commutative rings and algebras
Algebraic geometry
221
271
1
10.4171/157-1/5
http://www.ems-ph.org/doi/10.4171/157-1/5
Witt vectors, semirings, and total positivity
James
Borger
Australian National University, CANBERRA, AUSTRALIA
Combinatorics
Number theory
Commutative rings and algebras
Algebraic geometry
273
329
1
10.4171/157-1/6
http://www.ems-ph.org/doi/10.4171/157-1/6
Moduli operad over $\mathbb F_1$
Yuri
Manin
, BONN, GERMANY
Matilde
Marcolli
California Institute of Technology, PASADENA, UNITED STATES
Combinatorics
Number theory
Commutative rings and algebras
Algebraic geometry
331
361
1
10.4171/157-1/7
http://www.ems-ph.org/doi/10.4171/157-1/7
A taste of Weil theory in characteristic one
Koen
Thas
Universiteit Gent, GENT, BELGIUM
Combinatorics
Number theory
Commutative rings and algebras
Algebraic geometry
365
386
1
10.4171/157-1/8
http://www.ems-ph.org/doi/10.4171/157-1/8
Higher-Dimensional Generalized Manifolds: Surgery and Constructions
Alberto
Cavicchioli
Università degli Studi di Modena e Reggio Emilia, Italy
Friedrich
Hegenbarth
Università degli Studi di Milano, Italy
Dušan
Repovš
University of Ljubljana, Slovenia
Manifolds and cell complexes
Category theory; homological algebra
$K$-theory
Primary: 57P05, 57P10, 57P99, 57R65, 57R67; Secondary: 18F15, 19J25, 57N15, 57N60, 57N65
Homology manifold, Poincaré duality, degree 1 normal map, boundedly controlled surgey, surgery spectrum, assembly map, Quinn index, Euclidean neighborhood retract, cell-like resolution, disjoint disks property, manifold recognition problem
Generalized manifolds are a most fascinating subject to study. They were introduced in the 1930s, when topologists tried to detect topological manifolds among more general spaces (this is nowadays called the manifold recognition problem). As such, generalized manifolds have served to understand the nature of genuine manifolds. However, it soon became more important to study the category of generalized manifolds itself. A breakthrough was made in the 1990s, when several topologists discovered a systematic way of constructing higher-dimensional generalized manifolds, based on advanced surgery techniques. In fact, the development of controlled surgery theory and the study of generalized manifolds developed in parallel. In this process, earlier studies of geometric surgery turned out to be very helpful. Generalized manifolds will continue to be an attractive subject to study, for there remain several unsolved fundamental problems. Moreover, they hold promise for new research, e.g. for finding appropriate structures on these spaces which could bring to light geometric (or even analytic) aspects of higher-dimensional generalized manifolds. This is the first book to systematically collect the most important material on higher-dimensional generalized manifolds and controlled surgery. It is self-contained and its extensive list of references reflects the historic development. The book is based on our graduate courses and seminars, as well as our talks given at various meetings, and is suitable for advanced graduate students and researchers in algebraic and geometric topology.
5
31
2016
978-3-03719-156-9
978-3-03719-656-4
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/156
http://www.ems-ph.org/doi/10.4171/156
EMS Series of Lectures in Mathematics
Tempered Homogeneous Function Spaces
Hans
Triebel
University of Jena, Germany
Functional analysis
Fourier analysis
If one tries to transfer assertions for the inhomogeneous spaces $A^s_{p,q} (\mathbb R^n)$, $A \in \{B,F \}$, appropriately to their homogeneous counterparts ${\overset {\, \ast}{A}}{}^s_{p,q} (\mathbb R^n)$ within the framework of the dual pairing $\big( S(\mathbb R^n), S'(\mathbb R^n) \big)$ then it is hard to make a mistake as long as the parameters $p,q,s$ are restricted by $0 < p,q \le \infty$ and, in particular, $n(\frac {1}{p} – 1) < s < \frac {n}{p}$. It is the main aim of these notes to say what this means. This book is addressed to graduate students and mathematicians having a working knowledge of basic elements of the theory of function spaces, especially of type $B^s_{p,q}$ and $F^s_{p,q}$.
9
30
2015
978-3-03719-155-2
978-3-03719-655-7
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/155
http://www.ems-ph.org/doi/10.4171/155
EMS Series of Lectures in Mathematics
3-Manifold Groups
Matthias
Aschenbrenner
University of California Los Angeles, USA
Stefan
Friedl
Universität Regensburg, Germany
Henry
Wilton
University of Cambridge, UK
Manifolds and cell complexes
Group theory and generalizations
Primary: 57M05, 57M27; Secondary: 20E26
3-manifolds, fundamental groups, Geometrization theorem, hyperbolic 3-manifolds, virtually special groups
The field of 3-manifold topology has made great strides forward since 1982, when Thurston articulated his influential list of questions. Primary among these is Perelman's proof of the Geometrization Conjecture, but other highlights include the Tameness Theorem of Agol and Calegari–Gabai, the Surface Subgroup Theorem of Kahn–Markovic, the work of Wise and others on special cube complexes, and finally Agol's proof of the Virtual Haken Conjecture. This book summarizes all these developments and provides an exhaustive account of the current state of the art of 3-manifold topology, especially focussing on the consequences for fundamental groups of 3-manifolds. As the first book on 3-manifold topology that incorporates the exciting progress of the last two decades, it will be an invaluable resource for researchers in the field who need a reference for these developments. It also gives a fast-paced introduction to this material – although some familiarity with the fundamental group is recommended, little other previous knowledge is assumed, and the book is accessible to graduate students. The book closes with an extensive list of open questions, which will also be of interest to graduate students and established researchers alike.
8
20
2015
978-3-03719-154-5
978-3-03719-654-0
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/154
http://www.ems-ph.org/doi/10.4171/154
EMS Series of Lectures in Mathematics
Free Loop Spaces in Geometry and Topology
Including the monograph Symplectic cohomology and Viterbo’s theorem by Mohammed Abouzaid
Janko
Latschev
Universität Hamburg, Germany
Alexandru
Oancea
Sorbonne Universités, Paris, France
Differential geometry
Commutative rings and algebras
Associative rings and algebras
Algebraic topology
Primary: 53D40, 53D12, 53D25, 53D35, 53D37, 55P35, 55P50, 55P62, 55P92, 13D03, 13D07, 13D09, 13D10, 16E40, 16E45; Secondary: 57R15, 57R19, 57R56, 57R70, 57R91
Loop space, symplectic geometry, symplectic topology, string topology, Morse theory, Hochschild and cyclic homology, operations on Hochschild and cyclic homology, rational homotopy theory, minimal models, Lagrangian embeddings, pseudo-holomorphic curves
In the late 1990s two initially unrelated developments brought free loop spaces into renewed focus. In 1999, Chas and Sullivan introduced a wealth of new algebraic operations on the homology of these spaces under the name of string topology, the full scope of which is still not completely understood. A few years earlier, Viterbo had discovered a first deep link between the symplectic topology of cotangent bundles and the topology of their free loop space. In the past 15 years, many exciting connections between these two viewpoints have been found. Still, researchers working on one side of the story often know quite little about the other. One of the main purposes of this book is to facilitate communication between topologists and symplectic geometers thinking about free loop spaces. It was written by active researchers coming to the topic from both perspectives and provides a concise overview of many of the classical results, while also beginning to explore the new directions of research that have emerged recently. As one highlight, it contains a research monograph by M. Abouzaid which proves a strengthened version of Viterbo’s isomorphism between the homology of the free loop space of a manifold and the symplectic cohomology of its cotangent bundle, following a new strategy. The book grew out of a learning seminar on free loop spaces held at Strasbourg University in 2008–2009, and should be accessible to a graduate student with a general interest in the topic. It focuses on introducing and explaining the most important aspects rather than offering encyclopedic coverage, while providing the interested reader with a broad basis for further studies and research.
9
30
2015
978-3-03719-153-8
978-3-03719-653-3
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/153
http://www.ems-ph.org/doi/10.4171/153
IRMA Lectures in Mathematics and Theoretical Physics
24
Coulomb Gases and Ginzburg–Landau Vortices
Sylvia
Serfaty
Université Pierre et Marie Curie (Paris VI), France
Statistical mechanics, structure of matter
Linear and multilinear algebra; matrix theory
Partial differential equations
82B05, 82B21, 82B26, 15B52, 82D55; 35A15, 35J20, 35J60
Coulomb gas, Log gas, one-component plasma, statistical mechanics, Ginzburg-Landau, superconductivity, vortices, Abrikosov lattice, crystallization, random matrices, renormalized energy, mean-field limit, large deviations
The topic of this book is systems of points in Coulomb interaction, in particular, the classical Coulomb gas, and vortices in the Ginzburg–Landau model of superconductivity. The classical Coulomb and Log gases are classical statistical mechanics models, which have seen important developments in the mathematical literature due to their connection with random matrices and approximation theory. At low temperature, these systems are expected to “cristallize” to so-called Fekete sets, which exhibit microscopically a lattice structure. The Ginzburg–Landau model, on the other hand, describes superconductors. In superconducting materials subjected to an external magnetic field, densely packed point vortices emerge, forming perfect triangular lattice patterns, so-called Abrikosov lattices. This book describes these two systems and explores the similarity between them. It presents the mathematical tools developed to analyze the interaction between the Coulomb particles or the vortices, at the microscopic scale, and describes a “renormalized energy” governing the point patterns. This is believed to measure the disorder of a point configuration, and to be minimized by the Abrikosov lattice in dimension 2. The book gives a self-contained presentation of results on the mean field limit of the Coulomb gas system, with or without temperature, and of the derivation of the renormalized energy. It also provides a streamlined presentation of the similar analysis that can be performed for the Ginzburg–Landau model, including a review of the vortex-specific tools and the derivation of the critical fields, the mean-field limit and the renormalized energy.
3
20
2015
978-3-03719-152-1
978-3-03719-652-6
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/152
http://www.ems-ph.org/doi/10.4171/152
Zurich Lectures in Advanced Mathematics
Spectral Theory in Riemannian Geometry
Olivier
Lablée
Université Joseph Fourier Grenoble 1, Saint Martin d’Hères, France
Global analysis, analysis on manifolds
Partial differential equations
Operator theory
Calculus of variations and optimal control; optimization
58J35, 58J37, 58J50, 58J53, 35P05, 35P15, 35P20, 47A05, 47A10, 47A12, 47A60, 47A75, 49R05, 53C21
Spectral theory, linear operators, spectrum of operators, spectral geometry, eigenvalues, Laplacian, inverse problems, Riemannian geometry, analysis on manifolds
Spectral theory is a diverse area of mathematics that derives its motivations, goals and impetus from several sources. In particular, the spectral theory of the Laplacian on a compact Riemannian manifold is a central object in differential geometry. From a physical point a view, the Laplacian on a compact Riemannian manifold is a fundamental linear operator which describes numerous propagation phenomena: heat propagation, wave propagation, quantum dynamics, etc. Moreover, the spectrum of the Laplacian contains vast information about the geometry of the manifold. This book gives a self-containded introduction to spectral geometry on compact Riemannian manifolds. Starting with an overview of spectral theory on Hilbert spaces, the book proceeds to a description of the basic notions in Riemannian geometry. Then its makes its way to topics of main interests in spectral geometry. The topics presented include direct and inverse problems. Direct problems are concerned with computing or finding properties on the eigenvalues while the main issue in inverse problems is “knowing the spectrum of the Laplacian, can we determine the geometry of the manifold?” Addressed to students or young researchers, the present book is a first introduction in spectral theory applied to geometry. For readers interested in pursuing the subject further, this book will provide a basis for understanding principles, concepts and developments of spectral geometry.
2
18
2015
978-3-03719-151-4
978-3-03719-651-9
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/151
http://www.ems-ph.org/doi/10.4171/151
EMS Textbooks in Mathematics
Hybrid Function Spaces, Heat and Navier-Stokes Equations
Hans
Triebel
University of Jena, Germany
Functional analysis
Partial differential equations
Fourier analysis
Integral equations
46-02, 46E35, 42B35, 42C40, 35K05, 35Q30, 76D03, 76D05
Function spaces, Morrey spaces, heat equations, Navier-Stokes equations
This book is the continuation of Local Function Spaces, Heat and Navier–Stokes Equations (Tracts in Mathematics 20, 2013) by the author. A new approach is presented to exhibit relations between Sobolev spaces, Besov spaces, and Hölder–Zygmund spaces on the one hand and Morrey–Campanato spaces on the other. Morrey–Campanato spaces extend the notion of functions of bounded mean oscillation. These spaces play a crucial role in the theory of linear and nonlinear PDEs. Chapter 1 (Introduction) describes the main motivations and intentions of this book. Chapter 2 is a selfcontained introduction into Morrey spaces. Chapter 3 deals with hybrid smoothness spaces (which are between local and global spaces) in Euclidean n-space based on the Morrey–Campanato refinement of the Lebesgue spaces. The presented approach relies on wavelet decompositions. This is applied in Chapter 4 to linear and nonlinear heat equations in global and hybrid spaces. The obtained assertions about function spaces and nonlinear heat equations are used in the Chapters 5 and 6 to study Navier–Stokes equations in hybrid and global spaces. This book is addressed to graduate students and mathematicians having a working knowledge of basic elements of (global) function spaces, and who are interested in applications to nonlinear PDEs with heat and Navier–Stokes equations as prototypes.
1
15
2015
978-3-03719-150-7
978-3-03719-650-2
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/150
http://www.ems-ph.org/doi/10.4171/150
EMS Tracts in Mathematics
24
Valuation Theory in Interaction
Antonio
Campillo
Universidad de Valladolid, Spain
Franz-Viktor
Kuhlmann
University of Saskatchewan, Saskatoon, Canada
Bernard
Teissier
Institut de Mathématiques de Jussieu, Paris, France
Field theory and polynomials
Order, lattices, ordered algebraic structures
Commutative rings and algebras
Algebraic geometry
Primary: 03CXX, 12JXX, 12E30, 12F10, 13A18, 14H20, 14M25; secondary: 06FXX, 11SXX, 11U09, 12DXX, 12E05, 12F05, 12GXX, 12L12, 13D40, 13F30, 13H05, 13JXX, 13N15, 14BXX, 14C20, 14EXX, 14F10, 14J17, 14HXX, 14PXX, 16W60, 32P05, 32SXX, 37A05, 54F50
Valuation, defect, Abhyankar valuation, divisorial valuation, completion, local uniformization, toric geometry, key polynomial, excellent ring, local ring, valuation centered at a local domain, valuative tree, dicritical divisor, Rees valuation, Izumi’s theorem, plane curve singularity, Newton tree, rational surface singularity, Whitney stratification, jet scheme, embedded Nash problem, higher local field, wild ramification, dynamical system, irreducible polynomial, additive polynomial, Hilbertian field, large field, Galois theory, C-minimality, cell decomposition, imaginary element, formally real field, R-place, Hardy field, exponential-logarithmic series field, asymptotic integration, Hahn field, truncation, quasi-valuation.
Having its classical roots, since more than a century, in algebraic number theory, algebraic geometry and the theory of ordered fields and groups, valuation theory has seen an amazing expansion into many other areas in recent decades. Moreover, having been dormant for a while in algebraic geometry, it has now been reintroduced as a tool to attack the open problem of resolution of singularities in positive characteristic and to analyse the structure of singularities. Driven by this topic, and by its many new applications in other areas, also the research in valuation theory itself has been intensified, with a particular emphasis on the deep open problems in positive characteristic. As important examples for the expansion of valuation theory, it has become extremely useful in the theory of complex dynamical systems, and in the study of non-oscillating trajectories of real analytic vector fields in three dimensions. Analogues of the Riemann-Zariski valuation spaces have been found to be the proper framework for questions of intersection theory in algebraic geometry and in the analysis of singularities of complex plurisubharmonic functions. In a different direction, the relation between Berkovich geometry, tropical geometry and valuation spaces, on the one hand, and the geometry of arc spaces and valuation spaces, on the other, have begun to deepen and clarify. Ever since its beginnings, valuation theory and Galois theory have grown closely together and influenced each other. Arguably, studying and understanding the extensions of valuations in algebraic field extensions is one of the most important questions in valuation theory, whereas using valuation theory is one of he most important tools in studying Galois extensions of fields, as well as constructing field extensions with given properties. The well established topic of the model theory of valued fields is also being transformed, in particular through the study of valued fields with functions and operators, and through the study of types over valued fields. The multifaceted development of valuation theory has been monitored by two International Conferences and Workshops: the first in 1999 in Saskatoon, Canada, and the second in 2011 in Segovia and El Escorial in Spain. This book grew out of the second conference and presents high quality papers on recent research together with survey papers that illustrate the state of the art in several areas and applications of valuation theory. The book is addressed to researchers and graduate students who work in valuation theory or the areas where it is applied, as well as a general mathematical audience interested in the expansion and usefulness of the valuation theoretical approach, which has been called the “most analytic” form of algebraic reasoning. For young mathematicians who want to enter these areas of research, it provides a valuable source of up-to-date information.
9
1
2014
978-3-03719-149-1
978-3-03719-649-6
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/149
http://www.ems-ph.org/doi/10.4171/149
EMS Series of Congress Reports
A study of irreducible polynomials over henselian valued fields via distinguished pairs
Kamal
Aghigh
K.N. Toosi University of Technology, Tehran, IRAN
Anuj
Bishnoi
Panjab University, Chandigarh, INDIA
Sudesh
Khanduja
(IISER, Mohali), SAS Nagar, Punjab, INDIA
Sanjeev
Kumar
Panjab University, Chandigarh, INDIA
Field theory and polynomials
General
In this paper, we give an introduction of the phenomenon of lifting with respect to residually transcendental extensions, the notion of distinguished pairs and complete distinguished chains which lead to the study of certain invariants associated to irreducible polynomials over valued fields. We give an overview of various results regarding these concepts and their applications.
1
10
1
10.4171/149-1/1
http://www.ems-ph.org/doi/10.4171/149-1/1
On fields of totally $\mathfrak{S}$-adic numbers. With an appendix by Florian Pop
Lior
Bary-Soroker
Tel Aviv University, Tel Aviv, ISRAEL
Arno
Fehm
University of Konstanz, Konstanz, GERMANY
Field theory and polynomials
General
Given a finite set $\mathfrak{S}$ of places of a number field, we prove that the field of totally $\mathfrak{S}$-adic algebraic numbers is not Hilbertian.
11
15
1
10.4171/149-1/2
http://www.ems-ph.org/doi/10.4171/149-1/2
Infinite towers of Artin-Schreier defect extensions of rational function fields
Anna
Blaszczok
University of Silesia, Katowice, POLAND
Field theory and polynomials
Associative rings and algebras
General
We consider Artin-Scheier defect extensions of rational function fields in two variables over fields of positive characteristic. We study the problem of constructing infinite towers of such extensions. We classify Artin-Schreier defect extensions into "dependent" and "independent" ones, according to whether they are connected with purely inseparable defect extensions, or not. To understand the meaning of the classification for the issue of local uniformization, we consider various valuations of the rational function field and investigate for which it admits an infinite tower of dependent or independent Artin-Schreier defect extensions. We give also a criterion for a valued field of positive characteristic $p$ with $p$-divisible value group and perfect residue field to admit infinitely many parallel dependent Artin-Schreier defect extensions or an infinite tower of such extensions.
16
54
1
10.4171/149-1/3
http://www.ems-ph.org/doi/10.4171/149-1/3
A refinement of Izumi's Theorem
Sébastien
Boucksom
CNRS-Université Paris 6, PARIS CEDEX 05, FRANCE
Charles
Favre
École Polytechnique, Palaiseau Cedex, FRANCE
Mattias
Jonsson
University of Michigan, Ann Arbor, UNITED STATES
Commutative rings and algebras
Associative rings and algebras
Several complex variables and analytic spaces
General
We improve Izumi's inequality, which states that any divisorial valuation $v$ centered at a closed point $0$ on a normal algebraic variety $Y$ is controlled by the order of vanishing at $0$. More precisely, as $v$ ranges through valuations that are monomial with respect to coordinates in a fixed birational model $X$ dominating $Y$, we show that for any regular function $f$ on $Y$ at $0$, the function $v\mapsto v(f) / {\rm ord}_0(f)$ $d_0$ is uniformly Lipschitz continuous as a function of the weight defining $v$. As a consequence, the volume of $v$ is also a Lipschitz continuous function. Our proof uses toroidal techniques as well as positivity properties of the images of suitable nef divisors under birational morphisms.
55
81
1
10.4171/149-1/4
http://www.ems-ph.org/doi/10.4171/149-1/4
Multivariable Hodge theoretical invariants of germs of plane curves. II
Pierrette
Cassou-Noguès
Université Bordeaux I, Talence Cedex, FRANCE
Anatoly
Libgober
University of Illinois at Chicago, CHICAGO, UNITED STATES
Algebraic geometry
Several complex variables and analytic spaces
General
The paper describes several invariants of plane curve singularities in terms of the data of associated Newton trees. Newton trees of singularities are discussed in detail also. The invariants which we study include the constants and faces of quasi-adjunction, log-canonical walls and Arnold-Steenbrink spectrum. As one of the consequences of these calculations we show the failure of ACC for the set of constants of quasi-adjunction of all plane curve singularities, which contains the set of log-canonical thresholds as a subset.
82
135
1
10.4171/149-1/5
http://www.ems-ph.org/doi/10.4171/149-1/5
Existence des diviseurs dicritiques, d’après S.S. Abhyankar
Vincent
Cossart
Université de Versailles Saint-Quentin, Versailles Cedex, FRANCE
Mickaël
Matusinski
Université Bordeaux 1, Talence Cedex, FRANCE
Guillermo
Moreno-Socías
CNRS/UVSQ, bt. Fermat, VERSAILLES CEDEX, FRANCE
Algebraic geometry
General
In this article, there are new proofs of the existence and unicity of dicritical divisors of a pencil of plane curves of $\langle F,G\rangle$ Incidentally, we prove the equivalence between dicritical divisors and Rees valuations. Furthermore, in the case where $G_{\mathrm{red}}$ is regular at the base points of $\langle F,G\rangle$, we have that $F/G$ is residually a polynomial along any dicritical divisor; this reproves geometrically [2, Theorem (7.1)]. As a corollary of the latter proof, we get a generalization of the connectedness theorem of [8].
136
147
1
10.4171/149-1/6
http://www.ems-ph.org/doi/10.4171/149-1/6
Invariants of the graded algebras associated to divisorial valuations dominating a rational surface singularity
Vincent
Cossart
Université de Versailles Saint-Quentin, Versailles Cedex, FRANCE
Olivier
Piltant
Université de Versailles Saint-Quentin, Versailles Cedex, FRANCE
Ana
Reguera
Universidad de Valladolid, VALLADOLID, SPAIN
Algebraic geometry
General
Let $(R,M)$ be a rational surface singularity and $\nu_E$ be a prime divisor of the second kind for $R$. Then $gr_{\nu_E} R$ is finitely generated over $R/M$. We recover information about the dual graph of the minimal resolution $\widetilde X$ of $\text{Spec } R$ from the set of all $gr_{\nu_E} R$. In particular we characterize those graded algebras corresponding to the exceptional curves in $\widetilde X$.
148
166
1
10.4171/149-1/7
http://www.ems-ph.org/doi/10.4171/149-1/7
An introduction to $C$-minimal structures and their cell decomposition theorem
Pablo
Cubides Kovacsics
Université Paris Diderot, PARIS CEDEX 13, FRANCE
Mathematical logic and foundations
Commutative rings and algebras
General
Developments in valuation theory, especially the study of algebraically closed valued fields, have used the model theory of $C$-minimal structures in different places, e.g., the work of Hrushovski-Kazdhan in [5] and Haskell-Hrushovski-Macpherson in [3]. We intend with this text both to promulgate a basic comprehension of $C$-minimality for mathematicians interested in valuation theory (equipped with a basic knowledge of model theory), and to provide a slightly different presentation of the cell decomposition theorem proved by Haskell and Macpherson in [6].
167
207
1
10.4171/149-1/8
http://www.ems-ph.org/doi/10.4171/149-1/8
Valuation semigroups of Noetherian local domains
Steven
Cutkosky
University of Missouri, COLUMBIA, UNITED STATES
Commutative rings and algebras
Algebraic geometry
General
In this article we consider the problem of determining the valuation semigroup of a valuation dominating a Noetherian local ring. We give some general results and examples.
208
218
1
10.4171/149-1/9
http://www.ems-ph.org/doi/10.4171/149-1/9
Additive polynomials over perfect fields
Salih
Durhan
Middle East Technical University, Mersin 10, TURKEY
Number theory
General
Additive polynomials in one variable over valued fields of positive characteristic are sufficiently well understood in terms of their approximation properties. We extend results in this direction to multi-variable additive polynomials over perfect valued fields.
219
225
1
10.4171/149-1/10
http://www.ems-ph.org/doi/10.4171/149-1/10
On $\mathbb{R}$-places and related topics
Danielle
Gondard-Cozette
Université Pierre et Marie Curie, PARIS CEDEX 05, FRANCE
Commutative rings and algebras
Mathematical logic and foundations
Field theory and polynomials
Algebraic geometry
In this survey $K$ will be a formally real fi eld, which means that $-1$ is not a finite sum of squares of elements of $K,$ hence $K$ has characteristic $0$. As often in the literature, we shall write real field insteadof formally real fieldIt is well known from Artin-Schreier theory that such fields are exactly those admitting at least one total order compatible with the field structure. After some background in Real Algebra, we introduce and study the space of $\mathbb{R}$-places. Thereafter, we present other mathematical notions, such as valuation fans, orderings of higher level and the real holomorphy ring. By use of these tools we obtain an outstanding result in Real Algebraic Geometry. Finally we provide some steps towards an abstract theory of $\mathbb{R}$-places.
226
251
1
10.4171/149-1/11
http://www.ems-ph.org/doi/10.4171/149-1/11
Extending valuations to formal completions
Francisco Javier
Herrera Govantes
Universidad de Sevilla, SEVILLA, SPAIN
Miguel Ángel
Olalla Acosta
Universidad de Sevilla, SEVILLA, SPAIN
Mark
Spivakovsky
Université Paul Sabatier, Toulouse Cedex 9, FRANCE
Bernard
Teissier
UMR 7586 du CNRS, PARIS CEDEX 13, FRANCE
Field theory and polynomials
Commutative rings and algebras
Algebraic geometry
Associative rings and algebras
This paper is an extended version of the talk given by Miguel Angel Olalla Ácosta at the International Conference on Valuation Theory in El Escorial in July 2011. Its purpose is to provide an introduction to our joint paper [5] without grinding through all of its technical details. We refer the reader to [5] for details and proofs; only a few proofs are given in the present paper.
252
265
1
10.4171/149-1/12
http://www.ems-ph.org/doi/10.4171/149-1/12
Extending real valuations to skew polynomial rings
Ángel
Granja
Universidad de León, León, SPAIN
M.
Martínez
Universidad de Valladolid, Valladolid, SPAIN
C.
Rodríguez
Universidad de Léon, Léon, SPAIN
Commutative rings and algebras
Field theory and polynomials
General topology
General
Let $D$ be a division ring, $T$ be a variable over $D$, $\sigma $ be an endomorphism of $D$, $\delta $ be a $\sigma$-derivation on $D$ and $R=D[T; \sigma , \delta]$ the left skew polynomial ring over $D$. We show that the partially ordered set $(Val_\nu(R),\preceq)$ of $\sigma$-compatible real valuations on $R$ extending a fixed proper real valuation $\nu $ on $D$ has a natural structure of complete parameterized non-metric tree.
266
287
1
10.4171/149-1/13
http://www.ems-ph.org/doi/10.4171/149-1/13
Stratifications in valued fields
Immanuel
Halupczok
University of Leeds, Leeds, UNITED KINGDOM
Field theory and polynomials
Several complex variables and analytic spaces
General
In these notes a new, strong notion of stratifications which describe singularities of sets in Henselian valued fields is given. The first part presents the definition, some examples, and the main result about their existence. The second part explains how such a stratification in a valued field induces a classical Whitney stratification in $\mathbb{R}$.
288
296
1
10.4171/149-1/14
http://www.ems-ph.org/doi/10.4171/149-1/14
Imaginaries and definable types in algebraically closed valued fields
Ehud
Hrushovski
Hebrew University, JERUSALEM, ISRAEL
Mathematical logic and foundations
Field theory and polynomials
General
We give an exposition of material from [1], [2] and [3], regarding definable types in the model completion of the theory of valued fields, and the classification of imaginary sorts. The latter is given a new proof, based on definable types rather than invariant types, and on the notion of generic reparametrization. I also try to bring out the relation to the geometry of [3] - stably dominated definable types as the model theoretic incarnation of a Berkovich point.
297
319
1
10.4171/149-1/15
http://www.ems-ph.org/doi/10.4171/149-1/15
Defects of algebraic function fields, completion defects and defect quotients
Franz-Viktor
Kuhlmann
University of Saskatchewan, SASKATOON SASK., CANADA
Asim
Naseem
GC University, Lahore, PAKISTAN
Field theory and polynomials
Algebraic geometry
General
The defect (also called ramification deficiency) of valued field extensions is a major stumbling block in deep open problems of valuation theory in positive characteristic. For a detailed analysis, we define and investigate two finer notions of defect: the completion defect and the defect quotient. We define all three defects for finite valued field extensions as well as for certain valued function fields (those with Abhyankar valuations that are allowed to be nontrivial on the ground field). These defects of valued function fields have played an important role in genus reduction formulas that were presented by several authors. We prove the most general known form of the Finiteness and Independence Theorem for the defect of valued function fields. Further, we investigate the completion defect and the defect quotient in detail and present analogues of the results that hold for the usual defect.
320
349
1
10.4171/149-1/16
http://www.ems-ph.org/doi/10.4171/149-1/16
On generalized series fields and exponential-logarithmic series fields with derivations
Mickaël
Matusinski
Université Bordeaux 1, Talence Cedex, FRANCE
Field theory and polynomials
Commutative rings and algebras
General
We survey some important properties of fields of generalized series and of exponential-logarithmic series, with particular emphasis on their possible di fferential structure, based on a joint work of the author with S. Kuhlmann [40, 39].
350
372
1
10.4171/149-1/17
http://www.ems-ph.org/doi/10.4171/149-1/17
Jet schemes of rational double point singularities
Hussein
Mourtada
Institut Mathématique de Jussieu-Paris Rive Gauche, PARIS CEDEX 13, FRANCE
Algebraic geometry
Commutative rings and algebras
General
We prove that for $m\in \mathbb{N},~m$ large enough, the number of irreducible components of the schemes of $m-$jets centered at a point which is a double point singularity is independent of $m$ and is equal to the number of exceptional curves on the minimal resolution of the singularity. We also relate some irreducible components of the jet schemes of an $E_6$ singularity to its "minimal" embedded resolutions of singularities.
373
388
1
10.4171/149-1/18
http://www.ems-ph.org/doi/10.4171/149-1/18
Valuations centered at a two-dimensional regular local domain: infima and topologies
Josnei
Novacoski
University of Silesia, Katowice, POLAND
Commutative rings and algebras
Algebraic geometry
General
Take a two-dimensional regular local domain $R$. The space of all valuations centered at $R$ has a non-metric tree structure, called the valuative tree of $R$. However, the notion of non-metric tree appearing in the literature does not guarantee the existence of infimum for a non-empty set of valuations. We give a more general definition of a rooted non-metric tree and prove that the valuative tree has this more general property. We also generalize some topological results related to a non-metric tree. For instance, we show that the weak tree topology is always coarser than the metric topology given by any parametrization.
389
403
1
10.4171/149-1/19
http://www.ems-ph.org/doi/10.4171/149-1/19
Reduction of local uniformization to the rank one case
Josnei
Novacoski
University of Silesia, Katowice, POLAND
Mark
Spivakovsky
Université Paul Sabatier, Toulouse Cedex 9, FRANCE
Commutative rings and algebras
Algebraic geometry
General
The main result of this paper is that in order to prove the local uniformization theorem for local domains, it is enough to prove it for rank one valuations. Our proof does not depend on the nature of the class of local domains for which we want to prove local uniformization. We prove also the reductions for diff erent versions of the local uniformization theorem.
404
431
1
10.4171/149-1/20
http://www.ems-ph.org/doi/10.4171/149-1/20
Little survey on large fields - old & new
Florian
Pop
University of Pennsylvania, PHILADELPHIA, UNITED STATES
Field theory and polynomials
General
The large elds were introduced by the author in [60] and subsequently acquired several other names. This little survey includes earlier and new developments, and at the end of each section we mention a few open questions.
432
463
1
10.4171/149-1/21
http://www.ems-ph.org/doi/10.4171/149-1/21
Quasi-valuations -- topology and the weak approximation theorem
Shai
Sarussi
Sce College, Ashdod, ISRAEL
Commutative rings and algebras
General topology
General
Suppose $F$ is a field with a nontrivial valuation $v$ and valuation ring $O_{v}$, $E$ is a finite field extension and $w$ is a quasi-valuation on $E$ extending $v$. We study the topology induced by $w$. We prove that the quasi-valuation ring determines the topology, independent of the choice of its quasi-valuation. Moreover, we prove the weak approximation theorem for quasi-valuations.
464
473
1
10.4171/149-1/22
http://www.ems-ph.org/doi/10.4171/149-1/22
Overweight deformations of affine toric varieties and local uniformization
Bernard
Teissier
UMR 7586 du CNRS, PARIS CEDEX 13, FRANCE
Algebraic geometry
General
Given an equicharacteristic complete noetherian local ring $R$ with algebraically closed residue field $k$, we first present a combinatorial proof of \emph{embedded} local uniformization for zero-dimensional valuations of $R$ whose associated graded ring ${\rm gr}_\nu R$ with respect to the filtration defined by the valuation is a finitely generated $k$-algebra. The main idea here is that some of the birational toric maps which provide embedded pseudo-resolutions for the affine toric variety corresponding to ${\rm gr}_\nu R$ also provide local uniformizations for $\nu$ on $R$. These valuations are necessarily Abhyankar (for zero-dimensional valuations this means that the value group is $\mathbf Z^r$ with $r={\rm dim}R$).\Par In a second part we show that conversely, given an excellent noetherian equicharacteristic local domain $R$ with algebraically closed residue field, if the zero-dimensional valuation $\nu$ of $R$ is Abhyankar, there are local domains $R'$ which are essentially of finite type over $R$ and dominated by the valuation ring $R_\nu$ ($\nu$-modifications of $R$) such that the semigroup of values of $\nu$ on $R'$ is finitely generated, and therefore so is the $k$-algebra ${\rm gr}_\nu R'$. Combining the two results and using the fact that Abhyankar valuations behave well under completion gives a proof of local uniformization for rational Abhyankar valuations and, by a specialization argument, for all Abhyankar valuations. \par As a by-product we obtain a description of the valuation ring of a rational Abhyankar valuation as an inductive limit indexed by $\mathbf N$ of birational toric maps of regular local rings. One of our main tools, the valuative Cohen theorem, is then used to study the extensions of rational monomial Abhyankar valuations of the ring $k[[x_1,\ldots ,x_r]]$ to monogenous integral extensions and the nature of their key polynomials. In the conclusion we place the results in the perspective of local embedded resolution of singularities by a single toric modification after an appropriate re-embedding.
474
565
1
10.4171/149-1/23
http://www.ems-ph.org/doi/10.4171/149-1/23
Detecting valuations using small Galois groups
Adam
Topaz
University of California, BERKELEY, UNITED STATES
Field theory and polynomials
General
In this note we show how to detect valuations using almost-abelian pro-$\ell$ Galois groups of a field. In particular, we show that "commuting-liftable" subgroups of Galois groups arise, in a controlled way, from Kummer-duals of (principal-)units of valuations.
568
578
1
10.4171/149-1/24
http://www.ems-ph.org/doi/10.4171/149-1/24
Truncation in Hahn fields
Lou
van den Dries
University of Illinois at Urbana-Champaign, URBANA, UNITED STATES
Order, lattices, ordered algebraic structures
Field theory and polynomials
General
We consider truncation closed subgroups, subrings, and subfi elds of Hahn fields, and show that the property of being truncation closed is preserved under various natural ways of extending these substructures.
579
595
1
10.4171/149-1/25
http://www.ems-ph.org/doi/10.4171/149-1/25
The ergodicity of 1-Lipschitz transformations on 2-adic spheres
Ekaterina
Yurova
Linnaeus University, Vaxjo, SWEDEN
Dynamical systems and ergodic theory
General
In this paper we present results about ergodicity of dynamical systems on $2$-adic spheres for 1-Lipschitz maps $f:\mathbb Z_2\rightarrow \mathbb Z_2$ announced in [8], and extension of Theorem 3 from [8] for the case of spheres of radii greater than $\frac{1}{8}.$ We propose a new approach to study ergodic properties of 1-Lipschitz transformations of $2$-adic spheres. We use a representation of continuous functions $f$ via its van der Put series. This technique allows us to go beyond the classes of smooth 1-Lipschitz transformations which were studied earlier.
596
599
1
10.4171/149-1/26
http://www.ems-ph.org/doi/10.4171/149-1/26
Ramification of higher local fields approaches and questions
Liang
Xiao
University of California at Irvine, Irvine, UNITED STATES
Igor
Zhukov
St. Petersburg University, Peterhof, St. Petersburg, RUSSIAN FEDERATION
Algebraic geometry
Number theory
General
A survey paper contains facts, ideas and problems related to ramifi cation in fi nite extensions of complete discrete valuation fi elds with arbitrary residue fields. Some new results are included.
600
656
1
10.4171/149-1/27
http://www.ems-ph.org/doi/10.4171/149-1/27
Sophus Lie and Felix Klein: The Erlangen Program and Its Impact in Mathematics and Physics
Lizhen
Ji
University of Michigan, Ann Arbor, USA
Athanase
Papadopoulos
Université de Strasbourg, France
History and biography
Topological groups, Lie groups
Geometry
Differential geometry
01-00, 01-02, 01A05, 01A55, 01A70, 22-00, 22-02, 22-03, 51N15, 51P05, 53A20, 53A35, 53B50, 54H15, 58E40
Sophus Lie, Felix Klein, the Erlangen program, group action, Lie group action, symmetry, projective geometry, non-Euclidean geometry, spherical geometry, hyperbolic geometry, transitional geometry, discrete geometry, transformation group
The Erlangen program expresses a fundamental point of view on the use of groups and transformation groups in mathematics and physics. The present volume is the first modern comprehensive book on that program and its impact in contemporary mathematics and physics. Klein spelled out the program, and Lie, who contributed to its formulation, is the first mathematician who made it effective in his work. The theories that these two authors developed are also linked to their personal history and to their relations with each other and with other mathematicians, incuding Hermann Weyl, Élie Cartan, Henri Poincaré, and many others. All these facets of the Erlangen program appear in the present volume. The book is written by well-known experts in geometry, physics and history of mathematics and physics. It is addressed to mathematicians, to graduate students, and to all those interested in the development of mathematical ideas.
4
30
2015
978-3-03719-148-4
978-3-03719-648-9
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/148
http://www.ems-ph.org/doi/10.4171/148
IRMA Lectures in Mathematics and Theoretical Physics
23
Sophus Lie, a giant in mathematics
Lizhen
Ji
University of Michigan, ANN ARBOR, UNITED STATES
History and biography
General
1
26
1
10.4171/148-1/1
http://www.ems-ph.org/doi/10.4171/148-1/1
Felix Klein: his life and mathematics
Lizhen
Ji
University of Michigan, ANN ARBOR, UNITED STATES
History and biography
General
27
58
1
10.4171/148-1/2
http://www.ems-ph.org/doi/10.4171/148-1/2
Klein and the Erlangen Programme
Jeremy
Gray
The Open University, MILTON KEYNES, UNITED KINGDOM
History and biography
General
59
75
1
10.4171/148-1/3
http://www.ems-ph.org/doi/10.4171/148-1/3
Klein’s “Erlanger Programm”: do traces of it exist in physical theories?
Hubert
Goenner
Universität Göttingen, GÖTTINGEN, GERMANY
History and biography
General
77
90
1
10.4171/148-1/4
http://www.ems-ph.org/doi/10.4171/148-1/4
On Klein’s So-called Non-Euclidean geometry
Norbert
A’Campo
Universität Basel, BASEL, SWITZERLAND
Athanase
Papadopoulos
Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE
History and biography
General
91
136
1
10.4171/148-1/5
http://www.ems-ph.org/doi/10.4171/148-1/5
What are symmetries of PDEs and what are PDEs themselves?
Alexandre
Vinogradov
, LIZZANO IN BELVEDERE (BO), ITALY
History and biography
General
137
190
1
10.4171/148-1/6
http://www.ems-ph.org/doi/10.4171/148-1/6
Transformation groups in non-Riemannian geometry
Charles
Frances
Université Paris Sud, ORSAY CEDEX, FRANCE
History and biography
General
191
216
1
10.4171/148-1/7
http://www.ems-ph.org/doi/10.4171/148-1/7
Transitional geometry
Norbert
A’Campo
Universität Basel, BASEL, SWITZERLAND
Athanase
Papadopoulos
Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE
History and biography
General
217
235
1
10.4171/148-1/8
http://www.ems-ph.org/doi/10.4171/148-1/8
On the projective geometry of constant curvature spaces
Athanase
Papadopoulos
Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE
Sumio
Yamada
Gakushuin University, TOKYO, JAPAN
History and biography
General
237
245
1
10.4171/148-1/9
http://www.ems-ph.org/doi/10.4171/148-1/9
The Erlangen program and discrete differential geometry
Yuri
Suris
Technische Universität Berlin, BERLIN, GERMANY
History and biography
General
247
281
1
10.4171/148-1/10
http://www.ems-ph.org/doi/10.4171/148-1/10
Three-dimensional gravity – an application of Felix Klein’s ideas in physics
Catherine
Meusburger
Universität Erlangen-Nürnberg, ERLANGEN, GERMANY
History and biography
General
283
306
1
10.4171/148-1/11
http://www.ems-ph.org/doi/10.4171/148-1/11
Invariances in physics and group theory
Jean-Bernard
Zuber
Universite Pierre et Marie Curie, PARIS CEDEX 05, FRANCE
History and biography
General
307
324
1
10.4171/148-1/12
http://www.ems-ph.org/doi/10.4171/148-1/12
Handbook of Hilbert Geometry
Athanase
Papadopoulos
Université de Strasbourg, France
Marc
Troyanov
École Polytechnique Fédérale de Lausanne, Switzerland
Differential geometry
Geometry
Convex and discrete geometry
Global analysis, analysis on manifolds
01A55, 01-99, 35Q53, 37D25, 37D20, 37D40, 47H09, 51-00, 51-02, 51-03, 51A05, 51B20, 51F99, 51K05, 51K10, 51K99, 51M10, 52A07, 52A20, 52A99, 53A20, 53A35, 53B40, 53C22, 53C24, 53C60, 53C70, 53B40, 54H20, 57S25, 58-00, 58-02, 58-03, 58B20, 58D05, 58F07.
Hilbert metric, Funk metric, non-symmetric metric, Finsler geometry, Minkowski space, Minkowski functional, convexity, Cayley-Klein-Beltrami model, projective manifold, projective volume, Busemann curvature, Busemann volume, horofunction, geodesic flow, Teichmüller space, Hilbert fourth problem, entropy, geodesic, Perron-Frobenius theory, geometric structure, holonomy homomorphism
This volume presents surveys, written by experts in the field, on various classical and the modern aspects of Hilbert geometry. They are assuming several points of view: Finsler geometry, calculus of variations, projective geometry, dynamical systems, and others. Some fruitful relations between Hilbert geometry and other subjects in mathematics are emphasized, including Teichmüller spaces, convexity theory, Perron–Frobenius theory, representation theory, partial differential equations, coarse geometry, ergodic theory, algebraic groups, Coxeter groups, geometric group theory, Lie groups and discrete group actions. The Handbook is addressed to both students who want to learn the theory and researchers working in the area.
12
1
2014
978-3-03719-147-7
978-3-03719-647-2
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/147
http://www.ems-ph.org/doi/10.4171/147
IRMA Lectures in Mathematics and Theoretical Physics
22
Weak Minkowski spaces
Athanase
Papadopoulos
Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE
Marc
Troyanov
Ecole Polytechnique Fédérale de Lausanne, LAUSANNE, SWITZERLAND
Geometry
Differential geometry
General
We define the notion of weak Minkowski metric and prove some basic properties of such metrics. We also highlight some of the important analogies between Minkowski geometry and the Funk and Hilbert geometries.
11
32
1
10.4171/147-1/1
http://www.ems-ph.org/doi/10.4171/147-1/1
From Funk to Hilbert geometry
Athanase
Papadopoulos
Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE
Marc
Troyanov
Ecole Polytechnique Fédérale de Lausanne, LAUSANNE, SWITZERLAND
Geometry
Differential geometry
General
We survey some basic geometric properties of the Funk metric of a convex set in $\mathbb{R}^n$. In particular, we study its geodesics, its topology, its metric balls, its convexity properties, its perpendicularity theory and its isometries. The Hilbert metric is a symmetrization of the Funk metric, and we show some properties of the Hilbert metric that follow directly from the properties we prove for the Funk metric.
33
67
1
10.4171/147-1/2
http://www.ems-ph.org/doi/10.4171/147-1/2
Funk and Hilbert geometries from the Finslerian viewpoint
Marc
Troyanov
Ecole Polytechnique Fédérale de Lausanne, LAUSANNE, SWITZERLAND
Differential geometry
General
In 1929, Paul Funk and Ludwig Berwald gave a characterization of Hilbert geometries from the Finslerian viewpoint. They showed that a smooth Finsler metric in a strongly convex bounded domain of $\mathbb{R}^n$ is the Hilbert geometry in that domain if and only if it is complete, if its geodesics are straight lines and if its flag curvature is equal to $-1$. The goal of this chapter is to explain these notions in details, to illustrate the relation between Hilbert geometry, Finsler geometry and the calculus of variations, and to prove the Funk–Berwald characterization theorem.
69
110
1
10.4171/147-1/3
http://www.ems-ph.org/doi/10.4171/147-1/3
On the Hilbert geometry of convex polytopes
Constantin
Vernicos
Université Montpellier 2, MONTPELLIER CEDEX, FRANCE
Differential geometry
Geometry
General
We survey the Hilbert geometry of convex polytopes. In particular we present two important characterizations of these geometries, the first one in terms of the volume growth of their metric balls, the second one as a metric bilipschitzly equivalent to the Hilbert geometry in the simplex.
111
125
1
10.4171/147-1/4
http://www.ems-ph.org/doi/10.4171/147-1/4
The horofunction boundary and isometry group of the Hilbert geometry
Cormac
Walsh
Ecole Polytechnique, PALAISEAU CEDEX, FRANCE
Convex and discrete geometry
General
The horofunction boundary is a means of compactifying metric spaces that was introduced by Gromov in the 1970s. We describe explicitly the horofunction boundary of the Hilbert geometry, and sketch how it may be used to study the isometry group of this space.
127
146
1
10.4171/147-1/5
http://www.ems-ph.org/doi/10.4171/147-1/5
Characterizations of hyperbolic geometry among Hilbert geometries
Ren
Guo
Oregon State University, CORVALLIS, UNITED STATES
Geometry
Convex and discrete geometry
Differential geometry
General
This chapter is a survey of different characterizations of hyperbolic geometry among Hilbert geometries.
147
158
1
10.4171/147-1/6
http://www.ems-ph.org/doi/10.4171/147-1/6
Around groups in Hilbert geometry
Ludovic
Marquis
Université de Rennes I, RENNES CEDEX, FRANCE
Global analysis, analysis on manifolds
Partial differential equations
General
In this chapter, we survey groups of projective transformations acting on a Hilbert geometry.
207
261
1
10.4171/147-1/7
http://www.ems-ph.org/doi/10.4171/147-1/7
The geodesic flow of Finsler and Hilbert geometries
Mickaël
Crampon
Universidad de Santiago de Chile, Santiago, CHILE
Dynamical systems and ergodic theory
Differential geometry
General
This is a survey of the dynamics of the geodesic flow of Hilbert geometries. The main idea is to compare this flow with the geodesic flow of negatively curved Finsler or Riemannian manifolds, by making links between various useful objects and by comparing results and questions.
161
206
1
10.4171/147-1/8
http://www.ems-ph.org/doi/10.4171/147-1/8
Dynamics of Hilbert nonexpansive maps
Anders
Karlsson
Université de Genève, GENÈVE, SWITZERLAND
Operator theory
Geometry
General
In his work on the foundations of geometry, David Hilbert observed that a formula which appeared in works by Klein gives rise to a complete metric on any bounded convex domain. Some decades later, Garrett Birkhoff and Hans Samelson noticed that this metric has interesting applications, when considering certain maps of convex cones that contract the metric. Such situations have since arisen in many contexts, pure and applied, and could be called nonlinear Perron–Frobenius theory. This note centers around one dynamical aspect of this theory.
263
273
1
10.4171/147-1/9
http://www.ems-ph.org/doi/10.4171/147-1/9
Birkhoff’s version of Hilbert’s metric and its applications in analysis
Bas
Lemmens
University of Kent, CANTERBURY, UNITED KINGDOM
Roger
Nussbaum
Rutgers University, PISCATAWAY, UNITED STATES
Operator theory
Differential geometry
General topology
General
Birkhoff's version of Hilbert's metric is a distance between pairs of rays in a closed cone, and is closely related to Hilbert's classical cross-ratio metric. The version we discuss here was popularized by Bushell and can be traced back to the work of Garrett Birkhoff and Hans Samelson. It has found numerous applications in mathematical analysis, especially in the analysis of linear, and nonlinear, mappings on cones. Some of these applications are discussed in this chapter. Birkhoff's version of Hilbert's metric provides a different perspective on Hilbert geometries and naturally leads to infinite-dimensional generalizations. We illustrate this by showing some of its uses in the geometric analysis of Hilbert geometries.
275
303
1
10.4171/147-1/10
http://www.ems-ph.org/doi/10.4171/147-1/10
Convex real projective structures and Hilbert metrics
Inkang
Kim
KIAS, SEOUL, SOUTH KOREA
Athanase
Papadopoulos
Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE
Geometry
Manifolds and cell complexes
General
We review some basic concepts related to convex real projective structures from the differential geometry point of view. We start by recalling a Riemannian metric which originates in the study of affine spheres using the Blaschke connection (work of Calabi and of Cheng--Yau) mentioning its relation with the Hilbert metric. We then survey some of the deformation theory of convex real projective structures on surfaces. We describe in particular how the set of (Hilbert) lengths of simple closed curves is used in a parametrization of the deformation space in analogy with the classical Fenchel–Nielsen parameters of Teichmüller space (work of Goldman). We then mention parameters of this deformation space that arise in the work of Hitchin on the character variety of representations of the fundamental group of the surface in $\mathrm{SL}(3,\mathbb{R})$. In this character variety, the component of the character variety that corresponds to projective structures is identified with the vector space of pairs of holomorphic quadratic and cubic differentials over a fixed Riemann surface. Labourie and Loftin (independently) obtained parameter spaces that use the cubic differentials and affine spheres. We then display some similarities and differences between Hilbert geometry and hyperbolic geometry using geodesic currents and topological entropy. Finally, we discuss geodesic flows associated to Hilbert metrics and compactifications of spaces of convex real projective structures on surfaces. This makes another analogy with works done on the Teichmüller space of the surface.
307
338
1
10.4171/147-1/11
http://www.ems-ph.org/doi/10.4171/147-1/11
Weil–Petersson Funk metric on Teichmüller space
Hideki
Miyachi
Osaka University, OSAKA, JAPAN
Ken’ichi
Ohshika
Osaka University Graduate School of Science, OSAKA, JAPAN
Sumio
Yamada
Gakushuin University, TOKYO, JAPAN
Global analysis, analysis on manifolds
Partial differential equations
General
As a deformation space of hyperbolic metrics defined on a closed surface of genus $g \geq 2$, Teichmüller space can be regarded as a convex set with respect to the Weil--Petersson geometry. The convexity is used to construct a new distance function, called the Weil–Petersson Funk metric, which is a weak metric, lacking the symmetry and non-degeneracy conditions. The distance between two points is defined as a supremum of functions on the pair of points indexed by the set of the simple closed curves of the given surface. This set acts as the index set of the supporting hyperplanes of Teichmüller space regarded as a Weil–Petersson convex body. This Funk-type construction also appears in defining the two well-known distance functions on Teichmüller space: the Teichmüller metric and the Thurston metric. We emphasize in this chapter that the underlying idea for the three distance functions is to treat the Teichmüller space as a convex body.
339
352
1
10.4171/147-1/12
http://www.ems-ph.org/doi/10.4171/147-1/12
Funk and Hilbert geometries in spaces of constant curvature
Athanase
Papadopoulos
Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE
Sumio
Yamada
Gakushuin University, TOKYO, JAPAN
Global analysis, analysis on manifolds
Geometry
Differential geometry
General
We survey the Funk and Hilbert geometries of open convex sets in the sphere $S^n$ and in the hyperbolic space $\mathbb{H}^n$. The theories are developed in analogy with the classical theory in Euclidean space. It is rather unexpected that the Funk geometry, whose definition and development use the affine structure of Euclidean space, has analogues in the non-linear spaces $S^n$ and $\mathbb{H}^n$, where there are no analogies. As we shall see, the existence of a Funk geometry in these non-linear spaces is based on some non-Euclidean trigonometric formulae which display some kind of similarity between (the hyperbolic sine of the lengths of) sides of right triangles. The Hilbert metric in each of the constant curvature settings is a symmetrization of the Funk metric. We show that the Hilbert metric of a convex subset in a space of constant curvature can also be defined using a notion of a cross ratio which is proper to that space.
353
379
1
10.4171/147-1/13
http://www.ems-ph.org/doi/10.4171/147-1/13
On the origin of Hilbert geometry
Marc
Troyanov
Ecole Polytechnique Fédérale de Lausanne, LAUSANNE, SWITZERLAND
Geometry
History and biography
General
In this brief chapter we succinctly comment on the historical origin of Hilbert geometry. In particular, we give a summary of the letter in which David Hilbert informs his friend and colleague Felix Klein about his discovery of this geometry.
383
389
1
10.4171/147-1/14
http://www.ems-ph.org/doi/10.4171/147-1/14
Hilbert’s fourth problem
Athanase
Papadopoulos
Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE
History and biography
Geometry
Differential geometry
Global analysis, analysis on manifolds
Hilbert's fourth problem asks for the construction and the study of metrics on subsets of projective space for which the projective line segments are geodesics. Several solutions of the problem were given so far, depending on more precise interpretations of this problem, with various additional conditions satisfied. The most interesting solutions are probably those inspired by an integral formula that was first introduced in this theory by Herbert Busemann. Besides that, Busemann and his school made a thorough investigation of metrics defined on subsets of projective space for which the projective lines are geodesics and they obtained several results, characterizing several classes of such metrics. We review some of the developments and important results related to Hilbert's problem, especially those that arose from Busemann's work, mentioning recent results and connections with several branches of mathematics, including Riemannian geometry, the foundations of mathematics, the calculus of variations, metric geometry and Finsler geometry. Hilbert metrics – the subject of this handbook – constitute a basic class of metrics that satisfy the requirements of Hilbert's problem.
391
431
1
10.4171/147-1/15
http://www.ems-ph.org/doi/10.4171/147-1/15
Open problems
General
433
442
1
10.4171/147-1/16
http://www.ems-ph.org/doi/10.4171/147-1/16
Emil Artin and Beyond – Class Field Theory and $L$-Functions
Della
Dumbaugh
University of Richmond, USA
Joachim
Schwermer
University of Vienna, Austria
History and biography
Number theory
01A60, 01A70, 11R37, 11R39, 11S37, 11S39, 11Fxx, 11Mxx
Number theory, class field theory, L-functions, automorphic L-functions; history of mathematics, Emil Artin
This book explores the development of number theory, and class field theory in particular, as it passed through the hands of Emil Artin, Claude Chevalley and Robert Langlands in the middle of the twentieth century. Claude Chevalley’s presence in Artin’s 1931 Hamburg lectures on class field theory serves as the starting point for this volume. From there, it is traced how class field theory advanced in the 1930s and how Artin’s contributions influenced other mathematicians at the time and in subsequent years. Given the difficult political climate and his forced emigration as it were, the question of how Artin created a life in America within the existing institutional framework, and especially of how he continued his education of and close connection with graduate students, is considered. In particular, Artin’s collaboration in algebraic number theory with George Whaples and his student Margaret Matchett’s thesis work “On the zeta-function for ideles” in the 1940s are investigated. A (first) study of the influence of Artin on present day work on a non-abelian class field theory finishes the book. The volume consists of individual essays by the authors and two contributors, James Cogdell and Robert Langlands, and contains relevant archival material. Among these, the letter from Chevalley to Helmut Hasse in 1935 is included, where he introduces the notion of ideles and explores their significance, along with the previously unpublished thesis by Matchett and the seminal letter of Langlands to André Weil of 1967 where he lays out his ideas regarding a non-abelian class field theory. Taken together, these chapters offer a view of both the life of Artin in the 1930s and 1940s and the development of class field theory at that time. They also provide insight into the transmission of mathematical ideas, the careful steps required to preserve a life in mathematics at a difficult moment in history, and the interplay between mathematics and politics (in more ways than one). Some of the technical points in this volume require a sophisticated understanding of algebra and number theory. The broader topics, however, will appeal to a wider audience that extends beyond mathematicians and historians of mathematics to include historically minded individuals, particularly those with an interest in the time period.
3
31
2015
978-3-03719-146-0
978-3-03719-646-5
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/146
http://www.ems-ph.org/doi/10.4171/146
Heritage of European Mathematics
Uniformization of Riemann Surfaces
Revisiting a hundred-year-old theoremTranslated from the French by Robert G. Burns
Henri Paul
de Saint-Gervais
Paris
History and biography
General
(primary; secondary): 30F10; 01A55, 14H55, 30-01, 30-03, 30Fxx
Riemann surfaces, uniformization, Fuchsian groups, continuity method, Gauss, Riemann, Schwarz, Poincaré, Klein, Koebe
In 1907 Paul Koebe and Henri Poincar é almost simultaneously proved the uniformization theorem: Every simply connected Riemann surface is isomorphic to the plane, the open unit disc, or the sphere. It took a whole century to get to the point of stating this theorem and providing a convincing proof of it, relying as it did on prior work of Gauss, Riemann, Schwarz, Klein, Poincar é, and Koebe, among others. The present book o ffers an overview of the maturation process of this theorem. The evolution of the uniformization theorem took place in parallel with the emergence of modern algebraic geometry, the creation of complex analysis, the fi rst stirrings of functional analysis, and with the flowering of the theory of di fferential equations and the birth of topology. The uniformization theorem was thus one of the lightning rods of 19th century mathematics. Rather than describe the history of a single theorem, our aim is to return to the original proofs, to look at these through the eyes of modern mathematicians, to enquire as to their correctness, and to attempt to make them rigorous while respecting insofar as possible the state of mathematical knowledge at the time, or, if this should prove impossible, then using modern mathematical tools not available to their authors. This book will be useful to today's mathematicians wishing to cast a glance back at the history of their discipline. It should also provide graduate students with a non-standard approach to concepts of great importance for modern research.
1
31
2016
978-3-03719-145-3
978-3-03719-645-8
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/145
http://www.ems-ph.org/doi/10.4171/145
Heritage of European Mathematics
Karl Löwner and His Student Lipman Bers – Pre-war Prague Mathematicians
Martina
Bečvářová
Czech Technical University, Prague, Czech Republic
Ivan
Netuka
Charles University, Prague, Czech Republic
History and biography
Number theory
01A60, 01A70, 11R37, 11R39, 11S37, 11S39, 11Fxx, 11Mxx
Mathematical analysis, matrix functions, geometric function theory, potential theory, 20th century history of mathematics
This monograph is devoted to two distinguished mathematicians, Karel Löwner (1893–1968) and Lipman Bers (1914–1993), whose lives are dramatically interlinked with key historical events of the 20th century. K. Löwner, Professor of Mathematics at the German University in Prague (Czechoslovakia), was dismissed from his position because he was a Jew, and emigrated to the USA in 1939 (where he changed his name to Charles Loewner). Earlier, he had published several outstanding papers in complex analysis and a masterpiece on matrix functions. In particular, his ground-breaking parametric method in geometric function theory from 1923, which led to Löwner’s celebrated differential equation, brought him world-wide fame and turned out to be a cornerstone in de Branges’ proof of the Bieberbach conjecture. Unexpectedly, Löwner’s differential equation has gained recent prominence with the introduction of a conformally invariant stochastic process called stochastic Loewner evolution (SLE) by O. Schramm in 2000. SLE features in two Fields Medal citations from 2006 and 2010. L. Bers was the final Prague Ph.D. student of K. Löwner. His dissertation on potential theory (1938), completed shortly before his emigration and long thought to be irretrievably lost, was found in 2006. It is here made accessible for the first time, with an extensive commentary, to the mathematical community. This monograph presents an in-depth account of the lives of both mathematicians, with special emphasis on the pre-war period. Löwner’s teaching activities and professional achievements are presented in the context of the prevailing complex political situation and against the background of the wider development of mathematics in Europe. Each of his publications is accompanied by an extensive commentary, tracing the origin and motivation of the problem studied, and describing the state-of-art at the time of the corresponding mathematical field. Special attention is paid to the impact of the results obtained and to the later development of the underlying ideas, thus connecting Löwner’s achievements to current research activity. The text is based on an extensive archival search, and most of the archival findings appear here for the first time. Anyone with an interest in mathematics and the history of mathematics will enjoy reading this book about two famous mathematicians of the 20th century.
4
10
2015
978-3-03719-144-6
978-3-03719-644-1
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/144
http://www.ems-ph.org/doi/10.4171/144
Heritage of European Mathematics
Faà di Bruno Hopf Algebras, Dyson–Schwinger Equations, and Lie–Butcher Series
Kurusch
Ebrahimi-Fard
Universidad Autónoma de Madrid, Spain
Frédéric
Fauvet
Université de Strasbourg, France
Combinatorics
Order, lattices, ordered algebraic structures
Associative rings and algebras
Approximations and expansions
Primary: 05E15, 06A07, 16T05, 41A58, 58D05, 93C10; Secondary: 05C05, 81T18, 34A25, 34M25, 47H20, 65L05, 81T15, 81T16
Faà di Bruno formula, Dyson–Schwinger equations, geometric numerical integration, Butcher series, Lie–Butcher series, nonlinear control systems, nonlinear operators, combinatorial Hopf algebras, pre-Lie algebras, Lie algebras, trees, Ecalle’s mould calcul
Since the early works of G.-C. Rota and his school, Hopf algebras have been instrumental in algebraic combinatorics. In a seminal 1998 paper, A. Connes and D. Kreimer presented a Hopf algebraic approach to renormalization in perturbative Quantum Field Theory (QFT). This work triggered an abundance of new research on applications of Hopf algebraic techniques in QFT as well as other areas of theoretical physics. Furthermore, these new developments were complemented by progress made in other domains of applications, such as control theory, dynamical systems, and numerical integration methods. Especially in the latter context, it became clear that J. Butcher’s work from the early 1970s was well ahead of its time. The present volume emanated from a conference hosted in June 2011 by IRMA at Strasbourg University in France. Researchers from different scientific communities who share similar techniques and objectives gathered at this meeting to discuss new ideas and results on Faà di Bruno algebras, Dyson–Schwinger equations, and Butcher series. The purpose of this book is to present a coherent set of lectures reflecting the state of the art of research on combinatorial Hopf algebras relevant to high energy physics, control theory, dynamical systems, and numerical integration methods. More specifically, connections between Dyson–Schwinger equations, Faà di Bruno algebras, and Butcher series are examined in great detail. This volume is aimed at researchers and graduate students interested in combinatorial and algebraic aspects of QFT, control theory, dynamical systems and numerical analysis of integration methods. It contains introductory lectures on the various constructions that are emerging and developing in these domains.
6
30
2015
978-3-03719-143-9
978-3-03719-643-4
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/143
http://www.ems-ph.org/doi/10.4171/143
IRMA Lectures in Mathematics and Theoretical Physics
21
Foreword
José
Gracia-Bondía
Universidad Complutense, ZARAGOZA, SPAIN
Combinatorics
General
1
8
1
10.4171/143-1/1
http://www.ems-ph.org/doi/10.4171/143-1/1
Pre-Lie algebras and systems of Dyson–Schwinger equations
Loïc
Foissy
Centre Universitaire de la Mi-Voix, CALAIS CEDEX, FRANCE
General
These lecture notes contain a review of the results of [15], [16], [17], and [19] about combinatorial Dyson–Schwinger equations and systems. Such an equation or system generates a subalgebra of a Connes–Kreimer Hopf algebra of decorated trees, and we shall say that the equation or the system is Hopf if the associated subalgebra is Hopf. We first give a classi cation of the Hopf combinatorial Dyson–Schwinger equations. The proof of the existence of the Hopf subalgebra uses pre-Lie structures and is different from the proof of [15] and [17]. We consider afterwards systems of Dyson-Schwinger equations. We give a description of Hopf systems, with the help of two families of special systems (quasi-cyclic and fundamental) and four operations on systems (change of variables, dilatation, extension, concatenation). We also give a few result on the dual Lie algebras. Again, the proof of the existence of these Hopf subalgebras uses pre-Lie structures and is different from the proof of [16].
9
89
1
10.4171/143-1/2
http://www.ems-ph.org/doi/10.4171/143-1/2
Five interpretations of Faà di Bruno’s formula
Alessandra
Frabetti
Université Claude Bernard Lyon 1, VILLEURBANNE CEDEX, FRANCE
Dominique
Manchon
Université Blaise Pascal, AUBIÈRE CEDEX, FRANCE
Associative rings and algebras
Combinatorics
Group theory and generalizations
Topological groups, Lie groups
In these lectures we present five interpretations of the Faà di Bruno formula which computes the $n$-th derivative of the composition of two functions of one variable: in terms of groups, Lie algebras and Hopf algebras, in combinatorics and within operads.
91
147
1
10.4171/143-1/3
http://www.ems-ph.org/doi/10.4171/143-1/3
A Faà di Bruno Hopf algebra for analytic nonlinear feedback control systems
W. Steven
Gray
Old Dominion University, NORFOLK, UNITED STATES
Luis
Duffaut Espinosa
University of NSW at the Australian Defence Force Academy, CANBERRA, AUSTRALIA
Systems theory; control
Associative rings and algebras
Operator theory
General
In many applications, nonlinear input-output systems are interconnected in various ways to model complex systems. If a component system is analytic, meaning it can be described in terms of a Chen–Fliess functional series expansion, then it can be represented uniquely by a formal power series over a noncommutative alphabet. System interconnections are then characterized in terms of operations on formal power series. This paper provides an introduction to this methodology with an emphasis on feedback systems, which are ubiquitous in modern technology. In this case, a Faà di Bruno type Hopf algebra is de ned for a group of integral operators, where operator composition is the group product. Using a series expansion for the antipode, an explicit formula for the generating series of the compositional inverse operator is derived. This result produces an explicit formula for the generating series of a feedback system, which had been an open problem until recently.
149
217
1
10.4171/143-1/4
http://www.ems-ph.org/doi/10.4171/143-1/4
On algebraic structures of numerical integration on vector spaces and manifolds
Alexander
Lundervold
Bergen University College, BERGEN, NORWAY
Hans
Munthe-Kaas
University of Bergen, BERGEN, NORWAY
Global analysis, analysis on manifolds
Partial differential equations
General
Numerical analysis of time-integration algorithms has applied advanced algebraic techniques for more than fourty years. An explicit description of the group of characters in the Butcher–Connes–Kreimer Hopf algebra fi rst appeared in Butcher’s work on composition of integration methods in 1972. In more recent years, the analysis of structure preserving algorithms, geometric integration techniques and integration algorithms on manifolds have motivated the incorporation of other algebraic structures in numerical analysis. In this paper we will survey algebraic structures that have found applications within these areas. This includes pre-Lie structures for the geometry of flat and torsion free connections appearing in the analysis of numerical fl ows on vector spaces. The much more recent post-Lie and D-algebras appear in the analysis of flows on manifolds with flat connections with constant torsion. Dynkin and Eulerian idempotents appear in the analysis of non-autonomous flows and in backward error analysis. Non-commutative Bell polynomials and a non-commutative Faà di Bruno Hopf algebra are other examples of structures appearing naturally in the numerical analysis of integration on manifolds.
219
263
1
10.4171/143-1/5
http://www.ems-ph.org/doi/10.4171/143-1/5
Simple and contracting arborification
Emmanuel
Vieillard-Baron
Université de Bourgogne, DIJON CEDEX, FRANCE
Combinatorics
Order, lattices, ordered algebraic structures
Ordinary differential equations
General
We present a complete exposition of Ecalle’s arbori fication–coarborifi cation formalism, which is an essential component of his Mould Calculus, and we include in particular original results on the composition of arbori fied moulds. The connections with recent works regarding combinatorial Hopf algebras on trees are made but we give all the proofs in a self contained way, and we treat numerous examples.
265
353
1
10.4171/143-1/6
http://www.ems-ph.org/doi/10.4171/143-1/6
Strong QCD and Dyson–Schwinger equations
Craig
Roberts
Argonne National Laboratory, ARGONNE, UNITED STATES
Quantum theory
General
The real-world properties of quantum chromodynamics (QCD) – the strongly interacting piece of the Standard Model – are dominated by two emergent phenomena: con finement; namely, the theory’s elementary degrees-of-freedom – quarks and gluons – have never been detected in isolation; and dynamical chiral symmetry breaking (DCSB), which is a remarkably effective mass generating mechanism, responsible for the mass of more than 98% of visible matter in the Universe. These phenomena are not apparent in the formulae that de fine QCD, yet they play a principal role in determining Nature’s observable characteristics. Much remains to be learnt before con finement can properly be understood. On the other hand, the last decade has seen important progress in the use of relativistic quantum field theory, so that we can now explain the origin of DCSB and are beginning to demonstrate its far-reaching consequences. Dyson–Schwinger equations have played a critical role in these advances. These lecture notes provide an introduction to Dyson–Schwinger equations (DSEs), QCD and hadron physics, and illustrate the use of DSEs to predict observable phenomena.
355
458
1
10.4171/143-1/7
http://www.ems-ph.org/doi/10.4171/143-1/7
Four Faces of Number Theory
Kathrin
Bringmann
Universität zu Köln, Germany
Yann
Bugeaud
IRMA Strasbourg, France
Titus
Hilberdink
University of Reading, UK
Jürgen
Sander
Universität Hildesheim, Germany
Number theory
Computer science
Primary: 11-02, 11J81; Secondary 11A63, 11J04, 11J13, 11J68, 11J70, 11J87, 68R15
Transcendence, algebraic number, Schmidt Subspace Theorem, continued fraction, digital expansion, Diophantine approximation
This book arose from courses given at the International Summer School organized in August 2012 by the number theory group of the Department of Mathematics at the University of Würzburg. It consists of four essentially self-contained chapters and presents recent research results highlighting the strong interplay between number theory and other fields of mathematics, such as combinatorics, functional analysis and graph theory. The book is addressed to (under)graduate students who wish to discover various aspects of number theory. Remarkably, it demonstrates how easily one can approach frontiers of current research in number theory by elementary and basic analytic methods. Kathrin Bringmann gives an introduction to the theory of modular forms and, in particular, so-called Mock theta-functions, a topic which had been untouched for decades but has obtained much attention in the last years. Yann Bugeaud is concerned with expansions of algebraic numbers. Here combinatorics on words and transcendence theory are combined to derive new information on the sequence of decimals of algebraic numbers and on their continued fraction expansions. Titus Hilberdink reports on a recent and rather unexpected approach to extreme values of the Riemann zeta-function by use of (multiplicative) Toeplitz matrices and functional analysis. Finally, Jürgen Sander gives an introduction to algebraic graph theory and the impact of number theoretical methods on fundamental questions about the spectra of graphs and the analogue of the Riemann hypothesis.
11
23
2015
978-3-03719-142-2
978-3-03719-642-7
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/142
http://www.ems-ph.org/doi/10.4171/142
EMS Series of Lectures in Mathematics
Lectures on Universal Teichmüller Space
Armen
Sergeev
Steklov Mathematical Institute, Moscow, Russia
Global analysis, analysis on manifolds
Differential geometry
Primary: 58B20, 58B25, 58B34; Secondary: 53C55, 53D50
Teichmüller spaces, conformal maps, quasisymmetric homeomorphisms, Kähler manifolds, geometric quantization, noncommutative geometry
This book is based on a lecture course given by the author at the Educational Center of Steklov Mathematical Institute in 2011. It is designed for a one semester course for undergraduate students, familiar with basic differential geometry, complex and functional analysis. The universal Teichmüller space $\mathcal T$ is the quotient of the space of quasisymmetric homeomorphisms of the unit circle modulo Möbius transformations. The first part of the book is devoted to the study of geometric and analytic properties of $\mathcal T$. It is an infinite-dimensional Kähler manifold which contains all classical Teichmüller spaces of compact Riemann surfaces as complex submanifolds which explains the name “universal Teichmüller space”. Apart from classical Teichmüller spaces, $\mathcal T$ contains the space $\mathcal S$ of diffeomorphisms of the circle modulo Möbius transformations. The latter space plays an important role in the quantization of the theory of smooth strings. The quantization of $\mathcal T$ is presented in the second part of the book. In contrast with the case of diffeomorphism space $\mathcal S$, which can be quantized in frames of the conventional Dirac scheme, the quantization of $\mathcal T$ requires an absolutely different approach based on the noncommutative geometry methods. The book concludes with a list of 24 problems and exercises which can be used during the examinations.
8
12
2014
978-3-03719-141-5
978-3-03719-641-0
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/141
http://www.ems-ph.org/doi/10.4171/141
EMS Series of Lectures in Mathematics
Elliptic PDEs, Measures and Capacities
From the Poisson Equation to Nonlinear Thomas–Fermi Problems
Augusto
Ponce
Université catholique de Louvain, Belgium
Measure and integration
Potential theory
Primary: 28-02, 31-01, 35-02, 35R06; Secondary: 26B20, 26B35, 26D10, 28A12, 28A25, 28A33, 28A78, 28C05, 31B05, 31B10, 31B15, 31B20, 31B35, 35A01, 35A02, 35A08, 35A15, 35A23, 35A35, 35B05, 35B33, 35B45, 35B50, 35B51, 35B60, 35B65, 35C15, 35D30, 35J05, 35J10, 35J15, 35J20, 35J25, 35J60, 35J61, 35J86, 35J91, 35Q40, 35Q75, 35R05, 46E27, 46E30, 46E35, 49J40, 49J45, 46N20, 49S05
Balayage method, Borel measure, Chern–Simons equation, continuous potential, diffuse measure, Dirichlet problem, elliptic PDE, Euler–Lagrange equation, extremum solution, fractional Sobolev inequality, Frostman’s lemma, Hausdorff measure, Hausdorff content, Kato’s inequality, Laplacian, Lebesgue set, Lebesgue space, Marcinkiewicz space, maximum principle, minimization problem, Morrey’s imbedding, obstacle problem, Perron’s method, Poisson equation, potential theory, precise representative, reduced measure, regularity theory, removable singularity, Riesz representation theorem, Schrödinger operator, semilinear equation, Sobolev capacity, Sobolev space, subharmonic, superharmonic, sweeping-out method, Thomas–Fermi equation, trace inequality, Weyl’s lemma
Partial differential equations (PDEs) and geometric measure theory (GMT) are branches of analysis whose connections are usually not emphasized in introductory graduate courses. Yet, one cannot dissociate the notions of mass or electric charge, naturally described in terms of measures, from the physical potential they generate. Having such a principle in mind, this book illustrates the beautiful interplay between tools from PDEs and GMT in a simple and elegant way by investigating properties like existence and regularity of solutions of linear and nonlinear elliptic PDEs. Inspired by a variety of sources, from the pioneer balayage scheme of Poincaré to more recent results related to the Thomas–Fermi and the Chern–Simons models, the problems covered in this book follow an original presentation, intended to emphasize the main ideas in the proofs. Classical techniques like regularity theory, maximum principles and the method of sub- and supersolutions are adapted to the setting where merely integrability or density assumptions on the data are available. The distinguished role played by capacities and precise representatives is also explained. Other special features are: • the remarkable equivalence between Sobolev capacities and Hausdorff contents in terms of trace inequalities; • the strong approximation of measures in terms of capacities or densities, normally absent from GMT books; • the rescue of the strong maximum principle for the Schrödinger operator involving singular potentials. This book invites the reader to a trip through modern techniques in the frontier of elliptic PDEs and GMT, and is addressed to graduate students and researchers having some deep interest in analysis. Most of the chapters can be read independently, and only basic knowledge of measure theory, functional analysis and Sobolev spaces is required.
10
14
2016
978-3-03719-140-8
978-3-03719-640-3
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/140
http://www.ems-ph.org/doi/10.4171/140
EMS Tracts in Mathematics
23
Foundations of Garside Theory
Patrick
Dehornoy
Université de Caen, France
François
Digne
Université de Picardie Jules-Verne, Amiens, France
Eddy
Godelle
Université de Caen, France
Daan
Krammer
University of Warwick, Coventry, UK
Jean
Michel
Université Denis Diderot Paris 7, France
Group theory and generalizations
Combinatorics
Category theory; homological algebra
Computer science
Primary: 20Fxx, 20F05, 20F10, 20F36, 20F60, 20F65, 20M05, 20M10; Secondary: 05A05, 18B40, 18G35, 20B30, 20F55, 20L05, 20M50, 20N02, 68Q17
Group, monoid, category, greedy decomposition, normal decomposition, symmetric normal decomposition, Garside family, Garside map, Garside element, Garside monoid, Garside group, word problem, conjugacy problem, braid group, Artin–Tits group, Deligne–Luzstig variety, self-distributivity, ordered group, Yang–Baxter equation, cell decomposition
Winner of the 2014 EMS Monograph Award! This text is a monograph in algebra, with connections toward geometry and low-dimensional topology. It mainly involves groups, monoids, and categories, and aims at providing a unified treatment for those situations in which one can find distinguished decompositions by iteratively extracting a maximal fragment lying in a prescribed family. Initiated in 1969 by F. A. Garside in the case of Artin’s braid groups, this approach turned out to lead to interesting results in a number of cases, the central notion being what the authors call a Garside family. At the moment, the study is far from complete, and the purpose of this book is both to present the current state of the theory and to be an invitation for further research. There are two parts: the bases of a general theory, including many easy examples, are developed in Part A, whereas various more sophisticated examples are specifically addressed in Part B. In order to make the content accessible to a wide audience of nonspecialists, exposition is essentially self-contained and very few prerequisites are needed. In particular, it should be easy to use the current text as a textbook both for Garside theory and for the more specialized topics investigated in Part B: Artin–Tits groups, Deligne-Lusztig varieties, groups of algebraic laws, ordered groups, structure groups of set-theoretic solutions of the Yang–Baxter equation. The first part of the book can be used as the basis for a graduate or advanced undergraduate course.
6
1
2015
978-3-03719-139-2
978-3-03719-639-7
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/139
http://www.ems-ph.org/doi/10.4171/139
EMS Tracts in Mathematics
22
Analytic Projective Geometry
Eduardo
Casas-Alvero
Universitat de Barcelona, Spain
Geometry
51-01, 51N15; 51N10, 51N20
Projective geometry, affine geometry, Euclidean geometry, linear varieties, cross ratio, projectivities, quadrics, pencils of quadrics, correlations
Projective geometry is concerned with the properties of figures that are invariant by projecting and taking sections. It is considered one of the most beautiful parts of geometry and plays a central role because its specializations cover the whole of the affine, Euclidean and non-Euclidean geometries. The natural extension of projective geometry is projective algebraic geometry, a rich and active field of research. Regarding its applications, results and techniques of projective geometry are today intensively used in computer vision. This book contains a comprehensive presentation of projective geometry, over the real and complex number fields, and its applications to affine and Euclidean geometries. It covers central topics such as linear varieties, cross ratio, duality, projective transformations, quadrics and their classifications – projective, affine and metric –, as well as the more advanced and less usual spaces of quadrics, rational normal curves, line complexes and the classifications of collineations, pencils of quadrics and correlations. Two appendices are devoted to the projective foundations of perspective and to the projective models of plane non-Euclidean geometries. The presentation uses modern language, is based on linear algebra and provides complete proofs. Exercises are proposed at the end of each chapter; many of them are beautiful classical results. The material in this book is suitable for courses on projective geometry for undergraduate students, with a working knowledge of a standard first course on linear algebra. The text is a valuable guide to graduate students and researchers working in areas using or related to projective geometry, such as algebraic geometry and computer vision, and to anyone wishing to gain an advanced view on geometry as a whole.
5
10
2014
978-3-03719-138-5
978-3-03719-638-0
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/138
http://www.ems-ph.org/doi/10.4171/138
EMS Textbooks in Mathematics
MATHEON – Mathematics for Key Technologies
Peter
Deuflhard
Konrad-Zuse-Zentrum, Berlin, Germany
Martin
Grötschel
Konrad-Zuse-Zentrum; Berlin, Germany
Dietmar
Hömberg
Technische Universität Berlin, Germany
Ulrich
Horst
Humboldt-Universität zu Berlin, Germany
Jürg
Kramer
Humboldt-Universität zu Berlin, Germany
Volker
Mehrmann
Technische Universität Berlin, Germany
Konrad
Polthier
Freie Universität Berlin, Germany
Frank
Schmidt
Konrad-Zuse-Zentrum, Berlin, Germany
Christof
Schütte
Freie Universität Berlin, Germany
Martin
Skutella
Technische Universität Berlin, Germany
Jürgen
Sprekels
Weierstraß Institut für Angewandte Analysis und Stochastik, Berlin, Germany
History and biography
General
00-02, 01-02
Mathematics for life sciences, mathematics for networks, mathematics for production, mathematics for electronic and optical devices, mathematics for finance, mathematics for visualization, mathematical education
Mathematics: intellectual endeavor, production factor, key technology, key to key technologies? Mathematics is all of these! The last three of its facets have been the focus of the research and development in the Berlin-based DFG Research Center MATHEON in the last twelve years. Through these activities MATHEON has become an international trademark for carrying out creative, application-driven research in mathematics and for cooperating with industrial partners in the solution of complex problems in key technologies. Modern key technologies have become highly sophisticated, integrating aspects of engineering, computer, business and other sciences. Flexible mathematical models, as well as fast and accurate methods for numerical simulation and optimization open new possibilities to handle the indicated complexities, to react quickly, and to explore new options. Researchers in mathematical fields such as Optimization, Discrete Mathematics, Numerical Analysis, Scientific Computing, Applied Analysis and Stochastic Analysis have to work hand in hand with scientists and engineers to fully exploit this potential and to strengthen the transversal role of mathematics in the solution of the challenging problems in key technologies. This book presents in seven chapters the highlights of the research work carried out in the MATHEON application areas: Life Sciences, Networks, Production, Electronic and Photonic Devices, Finance, Visualization, and Education. The chapters summarize many of the contributions, put them in the context of current mathematical research activities and outline their impact in various key technologies. To make some of the results more easily accessible to the general public, a large number of “showcases” are presented that illustrate a few success stories.
4
28
2014
978-3-03719-137-8
978-3-03719-637-3
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/137
http://www.ems-ph.org/doi/10.4171/137
EMS Series in Industrial and Applied Mathematics
1
A Spinorial Approach to Riemannian and Conformal Geometry
Jean-Pierre
Bourguignon
IHÉS, Bures-sur-Yvette, France
Oussama
Hijazi
Université de Lorraine, Nancy, France
Jean-Louis
Milhorat
Université de Nantes, France
Andrei
Moroianu
Université de Versailles-St Quentin, France
Sergiu
Moroianu
Institutul de Matematică al Academiei Române, București, Romania
Differential geometry
Nonassociative rings and algebras
Ordinary differential equations
Partial differential equations
Primary: 53C27, 53A30, Secondary: 53C26, 53C55, 53C80, 17B10, 34L40, 35S05
Dirac operator, Penrose operator, Spin geometry, Spinc geometry, conformal geometry, Kähler manifolds, Quaternion-Kähler manifolds, Weyl geometry, representation theory, Killing spinors, eigenvalues
The book gives an elementary and comprehensive introduction to Spin Geometry, with particular emphasis on the Dirac operator which plays a fundamental role in differential geometry and mathematical physics. After a self-contained presentation of the basic algebraic, geometrical, analytical and topological ingredients, a systematic study of the spectral properties of the Dirac operator on compact spin manifolds is carried out. The classical estimates on eigenvalues and their limiting cases are discussed next, highlighting the subtle interplay of spinors and special geometric structures. Several applications of these ideas are presented, including spinorial proofs of the Positive Mass Theorem or the classification of positive Kähler–Einstein contact manifolds. Representation theory is used to explicitly compute the Dirac spectrum of compact symmetric spaces. The special features of the book include a unified treatment of Spin$^\mathrm c$ and conformal spin geometry (with special emphasis on the conformal covariance of the Dirac operator), an overview with proofs of the theory of elliptic differential operators on compact manifolds based on pseudodifferential calculus, a spinorial characterization of special geometries, and a self-contained presentation of the representation-theoretical tools needed in order to apprehend spinors. This book will help advanced graduate students and researchers to get more familiar with this beautiful, though not sufficiently known, domain of mathematics with great relevance to both theoretical physics and geometry.
6
29
2015
978-3-03719-136-1
978-3-03719-636-6
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/136
http://www.ems-ph.org/doi/10.4171/136
EMS Monographs in Mathematics
Compactness and Stability for Nonlinear Elliptic Equations
Emmanuel
Hebey
Université de Cergy-Pontoise, France
Global analysis, analysis on manifolds
Partial differential equations
58J05, 35J15
Blow-up theory, compactness, critical nonlinear elliptic equations, stability
The book offers an expanded version of lectures given at ETH Zürich in the framework of a Nachdiplomvorlesung. Compactness and stability for nonlinear elliptic equations in the inhomogeneous context of closed Riemannian manifolds are investigated, a field presently undergoing great development. The author describes blow-up phenomena and presents the progress made over the past years on the subject, giving an up-to-date description of the new ideas, concepts, methods, and theories in the field. Special attention is devoted to the nonlinear stationary Schrödinger equation and to its critical formulation. Intended to be as self-contained as possible, the book is accessible to a broad audience of readers, including graduate students and researchers.
7
1
2014
978-3-03719-134-7
978-3-03719-634-2
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/134
http://www.ems-ph.org/doi/10.4171/134
Zurich Lectures in Advanced Mathematics
Diffusion Processes and Stochastic Calculus
Fabrice
Baudoin
Purdue University, West Lafayette, USA
Probability theory and stochastic processes
60-01, 60G07, 60J60, 60J65, 60H05, 60H07
Brownian motion, diffusion processes, Malliavin calculus, rough paths theory, semigroup theory, stochastic calculus, stochastic processes
The main purpose of the book is to present at a graduate level and in a self-contained way the most important aspects of the theory of continuous stochastic processes in continuous time and to introduce to some of its ramifications like the theory of semigroups, the Malliavin calculus and the Lyons’ rough paths. It is intended for students, or even researchers, who wish to learn the basics in a concise but complete and rigorous manner. Several exercises are distributed throughout the text to test the understanding of the reader and each chapter ends up with bibliographic comments aimed to those interested in exploring further the materials. The stochastic calculus has been developed in the 1950s and the range of its applications is huge and still growing today. Besides being a fundamental component of modern probability theory, domains of applications include but are not limited to: mathematical finance, biology, physics, and engineering sciences. The first part of the text is devoted the general theory of stochastic processes, we focus on existence and regularity results for processes and on the theory of martingales. This allows to quickly introduce the Brownian motion and to study its most fundamental properties. The second part deals with the study of Markov processes, in particular diffusions. Our goal is to stress the connections between these processes and the theory of evolution semigroups. The third part deals with stochastic integrals, stochastic differential equations and Malliavin calculus. Finally, in the fourth and final part we present an introduction to the very new theory of rough paths by Terry Lyons.
7
7
2014
978-3-03719-133-0
978-3-03719-633-5
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/133
http://www.ems-ph.org/doi/10.4171/133
EMS Textbooks in Mathematics
Metric Spaces, Convexity and Nonpositive Curvature
Second edition
Athanase
Papadopoulos
IRMA, Strasbourg, France
Real functions
Functions of a complex variable
Several complex variables and analytic spaces
Geometry
26-01, 30F25, 30F45, 30F60, 32G15, 32Q45, 51-01, 51K05, 51K10, 51M09, 51M10, 51F99, 52-01, 52A07, 52A41, 53-01, 53C70, 54-01, 54E35
This book is about metric spaces of nonpositive curvature in the sense of Busemann, that is, metric spaces whose distance function satisfies a convexity condition. It also contains a systematic introduction to metric geometry, as well as a detailed presentation of some facets of convexity theory that are useful in the study of nonpositive curvature. The concepts and the techniques are illustrated by many examples, in particular from hyperbolic geometry, Hilbert geometry and Teichmüller theory. For the second edition, some corrections and a few additions have been made, and the bibliography has been updated.
1
18
2014
978-3-03719-132-3
978-3-03719-632-8
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/132
http://www.ems-ph.org/doi/10.4171/132
IRMA Lectures in Mathematics and Theoretical Physics
6
The Defocusing NLS Equation and Its Normal Form
Benoît
Grébert
Université de Nantes, France
Thomas
Kappeler
University of Zürich, Switzerland
Partial differential equations
Ordinary differential equations
Dynamical systems and ergodic theory
35Q55, 37K15, 37K10, 34L40, 34L20
Defocusing NLS equation, integrable PDEs, normal forms, action and angle variables, Zakharov–Shabat operators
The theme of this monograph is the nonlinear Schrödinger equation. This equation models slowly varying wave envelopes in dispersive media and arises in various physical systems such as water waves, plasma physics, solid state physics and nonlinear optics. More specifically, this book treats the defocusing nonlinear Schrödinger (dNLS) equation on the circle with a dynamical systems viewpoint. By developing the normal form theory it is shown that this equation is an integrable partial differential equation in the strongest possible sense. In particular, all solutions of the dNLS equation on the circle are periodic, quasi-periodic or almost-periodic in time and Hamiltonian perturbations of this equation can be studied near solutions far away from the equilibrium. The book is not only intended for specialists working at the intersection of integrable PDEs and dynamical systems, but also for researchers farther away from these fields as well as for graduate students. It is written in a modular fashion, each of its chapters and appendices can be read independently of each other.
3
15
2014
978-3-03719-131-6
978-3-03719-631-1
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/131
http://www.ems-ph.org/doi/10.4171/131
EMS Series of Lectures in Mathematics
Lecture Notes on Cluster Algebras
Robert
Marsh
University of Leeds, UK
Commutative rings and algebras
Combinatorics
Algebraic geometry
Nonassociative rings and algebras
13F60; 05E40, 14M15, 17B22, 17B63, 18E30, 20F55, 51F15, 52B05, 52B11, 57Q15
Associahedron, cluster algebra, cluster complex, Dynkin diagram, finite mutation type, Grassmannian, Laurent phenomenon, reflection group, periodicity, polytope, quiver mutation, root system, Somos sequence, surface
Cluster algebras are combinatorially defined commutative algebras which were introduced by S. Fomin and A. Zelevinsky as a tool for studying the dual canonical basis of a quantized enveloping algebra and totally positive matrices. The aim of these notes is to give an introduction to cluster algebras which is accessible to graduate students or researchers interested in learning more about the field, while giving a taste of the wide connections between cluster algebras and other areas of mathematics. The approach taken emphasizes combinatorial and geometric aspects of cluster algebras. Cluster algebras of finite type are classified by the Dynkin diagrams, so a short introduction to reflection groups is given in order to describe this and the corresponding generalized associahedra. A discussion of cluster algebra periodicity, which has a close relationship with discrete integrable systems, is included. The book ends with a description of the cluster algebras of finite mutation type and the cluster structure of the homogeneous coordinate ring of the Grassmannian, both of which have a beautiful description in terms of combinatorial geometry.
1
18
2014
978-3-03719-130-9
978-3-03719-630-4
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/130
http://www.ems-ph.org/doi/10.4171/130
Zurich Lectures in Advanced Mathematics
From Newton to Boltzmann: Hard Spheres and Short-range Potentials
Isabelle
Gallagher
Université Paris-Diderot France
Laure
Saint-Raymond
Ecole Normale Superieure, France
Benjamin
Texier
Université Paris Diderot - Paris 7, France
Partial differential equations
35Q20; 35Q70
Boltzmann equation, particle systems, propagation of chaos, BBGKY hierarchy, hard spheres, clusters, collision trees
The question addressed in this monograph is the relationship between the time-reversible Newton dynamics for a system of particles interacting via elastic collisions, and the irreversible Boltzmann dynamics which gives a statistical description of the collision mechanism. Two types of elastic collisions are considered: hard spheres, and compactly supported potentials.. Following the steps suggested by Lanford in 1974, we describe the transition from Newton to Boltzmann by proving a rigorous convergence result in short time, as the number of particles tends to infinity and their size simultaneously goes to zero, in the Boltzmann-Grad scaling. Boltzmann’s kinetic theory rests on the assumption that particle independence is propagated by the dynamics. This assumption is central to the issue of appearance of irreversibility. For finite numbers of particles, correlations are generated by collisions. The convergence proof establishes that for initially independent configurations, independence is statistically recovered in the limit. This book is intended for mathematicians working in the fields of partial differential equations and mathematical physics, and is accessible to graduate students with a background in analysis.
1
18
2014
978-3-03719-129-3
978-3-03719-629-8
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/129
http://www.ems-ph.org/doi/10.4171/129
Zurich Lectures in Advanced Mathematics
Basic Noncommutative Geometry
Second edition
Masoud
Khalkhali
The University of Western Ontario, London, Canada
Global analysis, analysis on manifolds
58-02; 58B34
Noncommutative space, noncommutative quotient, groupoid, cyclic cohomology, Connes–Chern character, index formula
This text provides an introduction to noncommutative geometry and some of its applications. It can be used either as a textbook for a graduate course or for self-study. It will be useful for graduate students and researchers in mathematics and theoretical physics and all those who are interested in gaining an understanding of the subject. One feature of this book is the wealth of examples and exercises that help the reader to navigate through the subject. While background material is provided in the text and in several appendices, some familiarity with basic notions of functional analysis, algebraic topology, differential geometry and homological algebra at a first year graduate level is helpful. Developed by Alain Connes since the late 1970s, noncommutative geometry has found many applications to long-standing conjectures in topology and geometry and has recently made headways in theoretical physics and number theory. The book starts with a detailed description of some of the most pertinent algebra-geometry correspondences by casting geometric notions in algebraic terms, then proceeds in the second chapter to the idea of a noncommutative space and how it is constructed. The last two chapters deal with homological tools: cyclic cohomology and Connes–Chern characters in K-theory and K-homology, culminating in one commutative diagram expressing the equality of topological and analytic index in a noncommutative setting. Applications to integrality of noncommutative topological invariants are given as well. Two new sections have been added to this second edition: one concerns the Gauss–Bonnet theorem and the definition and computation of the scalar curvature of the curved noncommutative two torus, and the second is a brief introduction to Hopf cyclic cohomology. The bibliography has been extended and some new examples are presented.
12
13
2013
978-3-03719-128-6
978-3-03719-628-1
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/128
http://www.ems-ph.org/doi/10.4171/128
EMS Series of Lectures in Mathematics
Lectures on Representations of Surface Groups
François
Labourie
Université Paris Sud, Orsay, France
Differential geometry
Several complex variables and analytic spaces
Global analysis, analysis on manifolds
53D30, 53C10; 58D27, 32G15, 58J28
Surface groups, symplectic geometry, Lie groups, character variety
The subject of these notes is the character variety of representations of a surface group in a Lie group. We emphasize the various points of view (combinatorial, differential, algebraic) and are interested in the description of its smooth points, symplectic structure, volume and connected components. We also show how a three manifold bounded by the surface leaves a trace in this character variety. These notes were originally designed for students with only elementary knowledge of differential geometry and topology. In the first chapters, we do not insist in the details of the differential geometric constructions and refer to classical textbooks, while in the more advanced chapters proofs occasionally are provided only for special cases where they convey the flavor of the general arguments. These notes could also be used by researchers entering this fast expanding field as motivation for further studies proposed in a concluding paragraph of every chapter.
12
13
2013
978-3-03719-127-9
978-3-03719-627-4
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/127
http://www.ems-ph.org/doi/10.4171/127
Zurich Lectures in Advanced Mathematics
Jacques Tits, Œuvres – Collected Works Volumes I–IV
Francis
Buekenhout
Université libre de Bruxelles, Belgium
Bernhard
Mühlherr
Justus Liebig University Giessen, Germany
Jean-Pierre
Tignol
Université catholique de Louvain, Belgium
Hendrik
Van Maldeghem
Ghent University, Belgium
General
Combinatorics
Nonassociative rings and algebras
Group theory and generalizations
00B60, 05-XX, 17-XX, 20-XX, 22-XX, 51-XX
Jacques Tits was awarded the Wolf Prize in 1993 and the Abel Prize (jointly with John Thompson) in 2008. The impact of his contributions in algebra, group theory and geometry made over a span of more than five decades is incalculable. Many fundamental developments in several fields of mathematics have their origin in ideas of Tits. A number of Tits’ papers mark the starting point of completely new directions of research. Outstanding examples are papers on quadratic forms, on Kac–Moody groups and on what subsequently became known as the Tits-alternative. These volumes contain an almost complete collection of Tits’ mathematical writings. They include, in particular, a number of published and unpublished manuscripts which have not been easily accessible until now. This collection of Tits’ contributions in one place makes the evolution of his mathematical thinking visible. The development of his theory of buildings and BN-pairs and its bearing on the theory of algebraic groups, for example, reveal a fascinating story. Along with Tits’ mathematical writings, these volumes contain biographical data, survey articles on aspects of Tits’ work and comments by the editors on the content of some of his papers. With the publication of these volumes, a major piece of 20th century mathematics is being made available to a wider audience.
11
15
2013
978-3-03719-126-2
978-3-03719-626-7
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/126
http://www.ems-ph.org/doi/10.4171/126
Heritage of European Mathematics
Advances in Representation Theory of Algebras
David
Benson
University of Aberdeen, UK
Henning
Krause
University of Bielefeld, Germany
Andrzej
Skowroński
Nicolaus Copernicus University, Toruń, Poland
Associative rings and algebras
16Gxx; 13Dxx, 18Exx, 20Cxx
Representation theory, finite dimensional algebras, Lie algebras, cluster algebras, homological algebra, derived categories, triangulated categories, rings and modules
This volume presents a collection of articles devoted to representations of algebras and related topics. Dististinguished experts in this field presented their work at the International Conference on Representations of Algebras which took place 2012 in Bielefeld. Many of the expository surveys are included here. Researchers of representation theory will find in this volume interesting and stimulating contributions to the development of the subject.
1
3
2014
978-3-03719-125-5
978-3-03719-625-0
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/125
http://www.ems-ph.org/doi/10.4171/125
EMS Series of Congress Reports
Infinite dimensional tilting theory
Lidia
Angeleri Hügel
Università degli Studi di Verona, VERONA, ITALY
Associative rings and algebras
Commutative rings and algebras
General
Infinite dimensional tilting modules are abundant in representation theory. They occur when studying torsion pairs in module categories, when looking for complements to partial tilting modules, or in connection with the Homological Conjectures. They share many properties with classical tilting modules, but they also give rise to interesting new phenomena as they are intimately related with localization, both at the level of module categories and of derived categories. In these notes, we review the main features of infinite dimensional tilting modules. We discuss the relationship with approximation theory and with localization. Finally, we focus on some classification results and we give a geometric interpretation of tilting.
1
37
1
10.4171/125-1/1
http://www.ems-ph.org/doi/10.4171/125-1/1
A survey of modules of constant Jordan type and vector bundles on projective space
David
Benson
University of Aberdeen, ABERDEEN, UNITED KINGDOM
Group theory and generalizations
Algebraic geometry
General
This is a transcription of a series of four lectures given at the Seattle Summer $\pi$-School on Cohomology and Support in Representation Theory and Related Topics, 27–30 July 2012. These were repeated in a mildly compressed form as a series of three lectures at the Workshop on Representations of Algebras in Bielefeld, 8–11 August 2012. Most of the material here also appears in greatly expanded form in a set of notes available online [2], which I eventually hope to publish as a book. This document may serve as an introduction to that longer work. The sections here correspond to the lectures given in Seattle. The first lecture introduces the concept of modules of constant Jordan type, and develops some of their properties, especially the question of what Jordan types can occur. To get much further with this subject it is necessary to work with vector bundles on projective space. So the second lecture gives an algebraic introduction to this subject, and explains how vector bundles arise from modules of constant Jordan type. We give an outline of the realisation theorem, which states that (up to a Frobenius twist if $p$ is odd) every vector bundle comes from a module of constant Jordan type. The third lecture introduces the theory of Chern classes, and uses these to provide restrictions on the behaviour of modules of constant Jordan types. In particular, this explains the appearance of the Frobenius twist for $p$ odd in the realisation theorem. Finally in the fourth lecture we give a short algebraic proof of the Hirzebruch–Riemann–Roch theorem for projective space. Schwarzenberger’s conditions on Chern roots are usually regarded as a consequence of Hirzebruch–Riemann–Roch, but in our development the logical order is reversed. Hirzebruch–Riemann–Roch is then used to provide further restrictions on possible Jordan types of modules.
39
63
1
10.4171/125-1/2
http://www.ems-ph.org/doi/10.4171/125-1/2
On representation-finite algebras and beyond
Klaus
Bongartz
Bergische Universität Wuppertal, WUPPERTAL, GERMANY
Associative rings and algebras
General
We give a survey on the theory of representation-finite and certain minimal representation-infinite algebras. The main goals are the existence of multiplicative bases and of coverings with good properties. Both are attained using ray-categories. As applications we include a proof of a sharper version of the second Brauer–Thrall conjecture and of the fact that there are no gaps in the lengths of the indecomposables.
65
101
1
10.4171/125-1/3
http://www.ems-ph.org/doi/10.4171/125-1/3
Quiver Hecke algebras and categorification
Jonathan
Brundan
University of Oregon, EUGENE, UNITED STATES
Associative rings and algebras
Nonassociative rings and algebras
General
This is a brief introduction to the quiver Hecke algebras of Khovanov, Lauda and Rouquier, emphasizing their application to the categorification of quantum groups. The text is based on lectures given by the author at the ICRA workshop in Bielefeld in August, 2012.
103
133
1
10.4171/125-1/4
http://www.ems-ph.org/doi/10.4171/125-1/4
Ordered exchange graphs
Thomas
Brüstle
Bishop's University, SHERBROOKE, CANADA
Dong
Yang
Nanjing University, NANJING, CHINA
General
The exchange graph of a cluster algebra encodes the combinatorics of mutations of clusters. Through the recent “categorifications” of cluster algebras using representation theory one obtains a whole variety of exchange graphs associated with objects such as a finite-dimensional algebra or a differential graded algebra concentrated in non-positive degrees. These constructions often come from variations of the concept of tilting, the vertices of the exchange graph being torsion pairs, t-structures, silting objects, support $\tau$-tilting modules and so on. All these exchange graphs stemming from representation theory have the additional feature that they are the Hasse quiver of a partial order which is naturally defined for the objects. In this sense, the exchange graphs studied in this article can be considered as a generalization or as a completion of the poset of tilting modules which has been studied by Happel and Unger. The goal of this article is to axiomatize the thus obtained structure of an ordered exchange graph, to present the various constructions of ordered exchange graphs and to relate them among each other.
135
193
1
10.4171/125-1/5
http://www.ems-ph.org/doi/10.4171/125-1/5
Introduction to Donaldson–Thomas invariants
Sergey
Mozgovoy
Trinity College, DUBLIN, IRELAND
Commutative rings and algebras
Associative rings and algebras
Category theory; homological algebra
General
These notes represent three lectures given at the International Conference on Representations of Algebras (ICRA) held at Bielefeld in August 2012. The goal of the lectures was to introduce refined Donaldson–Thomas invariants for the categories of modules over Jacobian algebras associated to quivers with potentials. We define refined Donaldson–Thomas invariants and compute them in some simple cases and then discuss their basic properties, including integrality, positivity, and wall-crossing phenomena.
195
210
1
10.4171/125-1/6
http://www.ems-ph.org/doi/10.4171/125-1/6
Cluster algebras and singular supports of perverse sheaves
Hiraku
Nakajima
Kyoto University, KYOTO, JAPAN
Commutative rings and algebras
Nonassociative rings and algebras
Partial differential equations
General
We propose an approach to Geiss–Leclerc–Schroer’s conjecture on the cluster algebra structure on the coordinate ring of a unipotent subgroup and the dual canonical base. It is based on singular supports of perverse sheaves on the space of representations of a quiver, which give the canonical base.
211
230
1
10.4171/125-1/7
http://www.ems-ph.org/doi/10.4171/125-1/7
Representations and cohomology of finite group schemes
Julia
Pevtsova
University of Washington, SEATTLE, UNITED STATES
Group theory and generalizations
Associative rings and algebras
General
This is a survey article covering developments in representation theory of finite group schemes over the last fifteen years. We start with the finite generation of cohomology of a finite group scheme and proceed to discuss various consequences and theories that ultimately grew out of that result. This includes the theory of one-parameter subgroups and rank varieties for infinitesimal group schemes; the $\pi$-points and $\Pi$-support spaces for finite group schemes, modules of constant rank and constant Jordan type, and construction of bundles on varieties closely related to $\operatorname{Proj}\operatorname{H}^{\bullet}(G,k)$ for an infinitesimal group scheme $G$. The material is mostly complementary to the article of D. Benson on elementary abelian $p$-groups in the same volume; we concentrate on the aspects of the theory which either hold generally for any finite group scheme or are specific to finite group schemes which are not finite groups. In the last section we discuss varieties of elementary subalgebras of modular Lie algebras, generalizations of modules of constant Jordan type, and new constructions of bundles on projective varieties associated to a modular Lie algebra.
231
261
1
10.4171/125-1/8
http://www.ems-ph.org/doi/10.4171/125-1/8
Superdecomposable pure-injective modules
Mike
Prest
University of Manchester, MANCHESTER, UNITED KINGDOM
Associative rings and algebras
Mathematical logic and foundations
General
Existence of superdecomposable pure-injective modules reflects complexity in the category of finite-dimensional representations. We describe the relation in terms of pointed modules. We present methods for producing superdecomposable pure-injectives and give some details of recent work of Harland doing this in the context of tubular algebras.
263
296
1
10.4171/125-1/9
http://www.ems-ph.org/doi/10.4171/125-1/9
Exact model categories, approximation theory, and cohomology of quasi-coherent sheaves
Jan
Šťovíček
Charles University, PRAHA 8, CZECH REPUBLIC
Algebraic topology
Category theory; homological algebra
General
Our aim is to give a fairly complete account on the construction of compatible model structures on exact categories and symmetric monoidal exact categories, in some cases generalizing previously known results. We describe the close connection of this theory to approximation theory and cotorsion pairs. We also discuss the motivating applications with the emphasis on constructing monoidal model structures for the derived category of quasi-coherent sheaves of modules over a scheme.
297
367
1
10.4171/125-1/10
http://www.ems-ph.org/doi/10.4171/125-1/10
Invariant Manifolds in Discrete and Continuous Dynamical Systems
Kaspar
Nipp
ETH Zürich, Switzerland
Daniel
Stoffer
ETH Zürich, Switzerland
Dynamical systems and ergodic theory
Ordinary differential equations
Numerical analysis
37-02; 37Cxx, 37Dxx, 34Cxx, 34Dxx, 65Lxx, 65P10
Discrete and continuous dynamical systems, invariant manifolds, foliation, singular perturbations, geometric numerical integration, differential algebraic equations
In this book dynamical systems are investigated from a geometric viewpoint. Admitting an invariant manifold is a strong geometric property of a dynamical system. This text presents rigorous results on invariant manifolds and gives examples of possible applications. In the first part discrete dynamical systems in Banach spaces are considered. Results on the existence and smoothness of attractive and repulsive invariant manifolds are derived. In addition, perturbations and approximations of the manifolds and the foliation of the adjacent space are treated. In the second part analogous results for continuous dynamical systems in finite dimensions are established. In the third part the theory developed is applied to problems in numerical analysis and to singularly perturbed systems of ordinary differential equations. The mathematical approach is based on the so-called graph transform, already used by Hadamard in 1901. The aim is to establish invariant manifold results in a simple setting providing quantitative estimates. The book is targeted at researchers in the field of dynamical systems interested in precise theorems easy to apply. The application part might also serve as an underlying text for a student seminar in mathematics.
8
7
2013
978-3-03719-124-8
978-3-03719-624-3
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/124
http://www.ems-ph.org/doi/10.4171/124
EMS Tracts in Mathematics
21
Local Function Spaces, Heat and Navier–Stokes Equations
Hans
Triebel
University of Jena, Germany
Functional analysis
Partial differential equations
Fourier analysis
46-02, 46E35, 42B35, 42C40, 35K05, 35Q30, 76D03, 76D05
Function spaces, Morrey–Campanato spaces, Gagliardo–Nirenberg inequalities, heat equations, Navier–Stokes equations
In this book a new approach is presented to exhibit relations between Sobolev spaces, Besov spaces, and Hölder–Zygmund spaces on the one hand and Morrey–Campanato spaces on the other. Morrey–Campanato spaces extend the notion of functions of bounded mean oscillation. These spaces play an important role in the theory of linear and nonlinear PDEs. Chapters 1–3 deal with local smoothness spaces in Euclidean n-space based on the Morrey–Campanato refinement of the Lebesgue spaces. The presented approach relies on wavelet decompositions. This is applied in Chapter 4 to Gagliardo–Nirenberg inequalities. Chapter 5 deals with linear and nonlinear heat equations in global and local function spaces. The obtained assertions about function spaces and nonlinear heat equations are used in Chapter 6 to study Navier–Stokes equations. The book is addressed to graduate students and mathematicians having a working knowledge of basic elements of (global) function spaces, and who are interested in applications to nonlinear PDEs with heat and Navier–Stokes equations as prototypes.
5
29
2013
978-3-03719-123-1
978-3-03719-623-6
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/123
http://www.ems-ph.org/doi/10.4171/123
EMS Tracts in Mathematics
20
Infinitesimal Geometry of Quasiconformal and Bi-Lipschitz Mappings in the Plane
Bogdan
Bojarski
IM PAN, Warsaw, Poland
Vladimir
Gutlyanskii
National Academy of Science of Ukraine, Donetsk, Ukraine
Olli
Martio
Finnish Academy of Science and Letters, Helsinki, Finland
Vladimir
Ryazanov
National Academy of Science of Ukraine, Donetsk, Ukraine
Functions of a complex variable
Partial differential equations
30C65, 30C75, 35J46, 35J50, 35J56, 35J70, 35Q35, 35Q60, 37F30, 37F40, 37F45, 57R99
Quasiconformal mappings, bi-Lipschitz mappings, Beltrami equations, local and boundary behavior, infinitesimal space, convergence and compactness theory, asymptotic linearity, rotation problems, conformal differentiability
This book is intended for researchers interested in new aspects of local behavior of plane mappings and their applications. The presentation is self-contained, but the reader is assumed to know basic complex and real analysis. The study of the local and boundary behavior of quasiconformal and bi-Lipschitz mappings in the plane forms the core of the book. The concept of the infinitesimal space is used to investigate the behavior of a mapping at points without differentiability. This concept, based on compactness properties, is applied to regularity problems of quasiconformal mappings and quasiconformal curves, boundary behavior, weak and asymptotic conformality, local winding properties, variation of quasiconformal mappings, and criteria of univalence. Quasiconformal and bi-Lipschitz mappings are instrumental for understanding elasticity, control theory and tomography and the book also offers a new look at the classical areas such as the boundary regularity of a conformal map. Complicated local behavior is illustrated by many examples. The text offers a detailed development of the background for graduate students and researchers. Starting with the classical methods to study quasiconformal mappings, this treatment advances to the concept of the infinitesimal space and then relates it to other regularity properties of mappings in Part II. The new unexpected connections between quasiconformal and bi-Lipschitz mappings are treated in Part III. There is an extensive bibliography.
5
29
2013
978-3-03719-122-4
978-3-03719-622-9
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/122
http://www.ems-ph.org/doi/10.4171/122
EMS Tracts in Mathematics
19
Erwin Schrödinger – 50 Years After
Wolfgang
Reiter
University of Vienna, Austria
Jakob
Yngvason
University of Vienna, Austria
History and biography
Quantum theory
01-02, 81-02, 81-03, 81P05, 81P15
Erwin Schrödinger (1887–1961) was an Austrian physicist famous for the equation named after him and which earned him the Nobel Prize in 1933. This book contains lectures presented at the international symposium Erwin Schrödinger – 50 Years After held at the Erwin Schrödinger International Institute for Mathematical Physics in January 2011 to commemorate the 50th anniversary of Schrödinger’s death. The text covers a broad spectrum of topics ranging from personal reminiscences to foundational questions of quantum mechanics and historical accounts of Schrödinger’s work. Besides the lectures presented at the symposium the volume also contains articles specially written for this occasion. The contributions give an overview of Schrödinger’s legacy to the sciences from the standpoint of some of present day’s leading scholars in the field. The book addresses students and researchers in mathematics, physics and the history of science.
4
11
2013
978-3-03719-121-7
978-3-03719-621-2
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/121
http://www.ems-ph.org/doi/10.4171/121
ESI Lectures in Mathematics and Physics
Erwin Schrödinger – personal reminiscences
Walter
Thirring
, WIEN, AUSTRIA
History and biography
Quantum theory
General
In this article the author recollects some memories of his encounters with Schrödinger, beginning in 1937 when he first met him in Vienna up to the final years after Schrödinger’s return to Vienna.
1
7
1
10.4171/121-1/1
http://www.ems-ph.org/doi/10.4171/121-1/1
Schrödinger and the genesis of wave mechanics
Jürgen
Renn
, BERLIN, GERMANY
History and biography
Quantum theory
General
In the context of a new analysis of the notebooks of Erwin Schrödinger, the paper deals with the question of the relation between Schrödinger's creation of wave mechanics and the contemporary efforts by Werner Heisenberg and his colleagues to establish a new quantum mechanics. How can one explain, from a broader historical and epistemological perspective, the astonishing simultaneity and complementarity of these discoveries? The paper argues that neither the physical problems with which both approaches deal nor what ultimately turned out to be their common mathematical ground are sufficient to explain their complementarity. Instead, their closeness is explained by analyzing their common roots in classical mechanics and its transformation in the light of the most fundamental new quantum law, the relation between energy and frequency found by Planck. It is shown, in particular, that for both approaches a bridge between quantum and classical aspects involving this relation was crucial. In the case of Heisenberg, this bridge was given by Bohr’s correspondence principle. In the case of Schrödinger it was constituted by Hamilton’s optical-mechanical analogy.
9
36
1
10.4171/121-1/2
http://www.ems-ph.org/doi/10.4171/121-1/2
Do we understand quantum mechanics – finally?
Jürg
Fröhlich
ETH Zürich, ZÜRICH, SWITZERLAND
Baptiste
Schubnel
ETH Zürich, ZÜRICH, SWITZERLAND
Quantum theory
General
This paper reviews some of our understanding of general quantum mechanics. It starts with the exposition of an abstract algebraic formalism useful to formulate classical and quantum-mechanical models of physical systems. It then highlights the essential differences between classical models (commutative algebra) and quantum-mechanical models (non-commutative algebra) of physical systems. It is explained in which sense classical models are “realistic” and deterministic, while quantum-mechanical models are intrinsically probabilistic – in spite of the fact that the Heisenberg time-evolution of operators representing physical quantities is “deterministic”. The quantum theory of time-ordered sequences of measurements is developed in some detail, and the crucial role of “decoherence” in the emergence of facts – or “(almost) consistent histories” – is explained. Some technical matters (Bell inequalities, quantum marginal problem) are discussed in appendices.
37
84
1
10.4171/121-1/3
http://www.ems-ph.org/doi/10.4171/121-1/3
Schrödinger’s cat and her laboratory cousins
Anthony
Leggett
University of Illinois at Urbana-Champaign, URBANA, UNITED STATES
Quantum theory
General
In a famous 1935 paper, Erwin Schroedinger pointed out the apparently bizarre consequences which flow from the assumption that the formalism of quantum mechanics can be extrapolated from the world of electrons and atoms to the level of our own direct consciousness (the “Cat” paradox). In this paper I first discuss some alleged resolutions of the paradox, then point out that the premise is not self-evident, and define a class of alternative (“macrorealistic”) scenarios. I then review the extent to which the experimental predictions of quantum mechanics are currently tested in the direction of the everyday world, and explain a crucial test, yet to be performed, of these predictions vis-a-vis those of macrorealism.
87
108
1
10.4171/121-1/4
http://www.ems-ph.org/doi/10.4171/121-1/4
Digital and open system quantum simulation with trapped ions
Markus
Müller
Universität Innsbruck, INNSBRUCK, AUSTRIA
Peter
Zoller
Universität Innsbruck, INNSBRUCK, AUSTRIA
Quantum theory
General
We summarize some of the recent developments in building a quantum simulator with quantum optical systems, in particular laser cooled trapped ions. Our discussion focuses on digital and open system quantum simulation as realized recently for the first time.
109
121
1
10.4171/121-1/5
http://www.ems-ph.org/doi/10.4171/121-1/5
Optomechanical Schrödinger cats – a case for space
Rainer
Kaltenbaek
Universität Wien, WIEN, AUSTRIA
Markus
Aspelmeyer
Universität Wien, WIEN, AUSTRIA
Quantum theory
General
Quantum optomechanics exploits radiation pressure effects inside optical cavities. It can be used to generate quantum states of the center-of-mass motion of massive mechanical objects, thereby opening up a new parameter regime for macroscopic quantum experiments. The challenging experimental conditions to maintain and observe quantum coherence for increasingly large objects may require a space environment rather than an earth-bound laboratory. We introduce a possible space experiment to study the wave-packet expansion of massive objects. This forms the basis for Schrödinger cat states of unprecedented size and mass.
123
132
1
10.4171/121-1/6
http://www.ems-ph.org/doi/10.4171/121-1/6
A quantum discontinuity: The Schrödinger–Bohr dialogue
Helge
Kragh
University of Aarhus, AARHUS, DENMARK
Quantum theory
General
Parts of Erwin Schrödinger’s career and views of science can be illuminated through his interaction with another pioneer of quantum theory, Niels Bohr. Both physicists endeavoured to understand nature on a basic level, where physics cannot be easily separated from natural philosophy in a broad sense of the term. In spite of having much in common, they fundamentally disagreed about the meaning and aim of quantum mechanics. The essay describes the Schrödinger–Bohr dialogue as it unfolded from their first contact in 1921 to their last exchange of letters in 1952.
135
151
1
10.4171/121-1/7
http://www.ems-ph.org/doi/10.4171/121-1/7
The debate between Hendrik A. Lorentz and Schrödinger on wave mechanics
Anne
Kox
University of Amsterdam, AMSTERDAM, NETHERLANDS
Quantum theory
General
In 1923 the Austrian Erwin Schrödinger, professor of physics in Zurich, and his Dutch colleague Hendrik Lorentz, grand old man of classical physics, engaged in a fascinating correspondence on the foundations of quantum physics. In this lecture I will go back to that year, give a sketch of the personalities of Lorentz and Schrödinger and of their respective careers, and explain the importance for the history of physics of their remarkable exchange of ideas.
153
162
1
10.4171/121-1/8
http://www.ems-ph.org/doi/10.4171/121-1/8
A few reasons why Louis de Broglie discovered matter waves and yet did not discover Schrödinger’s equation
Olivier
Darrigol
Université Paris 7 Denis Diderot, PARIS, FRANCE
Quantum theory
General
In 1923 Louis de Broglie arrived at a shocking notion of matter waves through reasoning so forceful that Albert Einstein and a few quantum experts took it seriously. The purpose of this paper is to analyze de Broglie's discovery and to show how some of his heuristics prevented him to introduce the equation that now bears Schrödinger’s name.
165
174
1
10.4171/121-1/9
http://www.ems-ph.org/doi/10.4171/121-1/9
European Congress of Mathematics Kraków, 2 – 7 July, 2012
Rafał
Latała
University of Warsaw, Poland
Andrzej
Ruciński
A. Mickiewicz University, Poznań, Poland
Paweł
Strzelecki
University of Warsaw, Poland
Jacek
Świątkowski
University of Wrocław, Poland
Dariusz
Wrzosek
University of Warsaw, Poland
Piotr
Zakrzewski
University of Warsaw, Poland
General
00Bxx
The European Congress of Mathematics, held every four years, has become a well-established major international mathematical event. Following those in Paris (1992), Budapest (1996), Barcelona (2000), Stockholm (2004) and Amsterdam (2008), the Sixth European Congress of Mathematics (6ECM) took place in Kraków, Poland, July 2–7, 2012, with about 1000 participants from all over the world. Ten plenary, thirty-three invited lectures and three special lectures formed the core of the program. As at all the previous EMS congresses, ten outstanding young mathematicians received the EMS prizes in recognition of their research achievements. In addition, two more prizes were awarded: the Felix Klein Prize for a remarkable solution of an industrial problem, and – for the first time – the Otto Neugebauer Prize for a highly original and influential piece of work in the history of mathematics. The program was complemented by twenty-four minisymposia with nearly 100 talks, spread over all areas of mathematics. Six panel discussions were organized, covering a variety of issues ranging from the financing of mathematical research to gender imbalance in mathematics. These proceedings present extended versions of most of the invited talks which were delivered during the congress, providing a permanent record of the best what mathematics offers today.
1
3
2014
978-3-03719-120-0
978-3-03719-620-5
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/120
http://www.ems-ph.org/doi/10.4171/120
Some mathematical aspects of water waves
Adrian
Constantin
King's College London, LONDON, UNITED KINGDOM
Mathematical logic and foundations
Combinatorics
Functional analysis
General
We discuss some recent mathematical investigations that o er insight and open up promising perspectives on the fundamental aspect of uid mechanics concerned with the motion beneath a surface water wave.
1
12
1
10.4171/120-1/1
http://www.ems-ph.org/doi/10.4171/120-1/1
Continuous dissipative Euler flows and a conjecture of Onsager
Camillo
De Lellis
Universität Zürich, ZÜRICH, SWITZERLAND
László
Székelyhidi Jr.
Universität Leipzig, LEIPZIG, GERMANY
Partial differential equations
Ordinary differential equations
Differential geometry
Fluid mechanics
It is known since the pioneering works of Sche er and Shnirelman that there are nontrivial distributional solutions to the Euler equations which are compactly supported in space and time. Obviously these solutions do not respect the classical conservation law for the total kinetic energy and they are therefore very irregular. In recent joint works we have proved the existence of continuous and even Holder continuous solutions which dissipate the kinetic energy. Our theorem might be regarded as a rst step towards a conjecture of Lars Onsager, which in 1949 asserted the existence of dissipative Hölder solutions for any Hölder exponent smaller than 1/3.
13
29
1
10.4171/120-1/2
http://www.ems-ph.org/doi/10.4171/120-1/2
Persistent Homology: Theory and Practice
Herbert
Edelsbrunner
Institute of Science and Technology Austria, KLOSTERNEUBURG, AUSTRIA
Dmitriy
Morozov
Lawrence National Laboratory, BERKELEY, UNITED STATES
Algebraic topology
Computer science
General
Persistent homology is a recent grandchild of homology that has found use in science and engineering as well as in mathematics. This paper surveys the method as well as the applications, neglecting completeness in favor of highlighting ideas and directions.
31
50
1
10.4171/120-1/3
http://www.ems-ph.org/doi/10.4171/120-1/3
In a Search for a Structure, Part 1: On Entropy
Misha
Gromov
Le Bois-Marie, BURES-SUR-YVETTE, FRANCE
Measure and integration
Dynamical systems and ergodic theory
Statistical mechanics, structure of matter
General
Mathematics is about \interesting structures". What make a structure interesting is an abundance of interesting problems; we study a structure by solving these problems. The worlds of science, as well as of mathematics itself, is abundant with gems (germs?) of simple beautiful ideas. When and how may such an idea direct you toward beautiful mathematics? I try to present in this talk a few suggestive examples.
51
78
1
10.4171/120-1/4
http://www.ems-ph.org/doi/10.4171/120-1/4
Classification of Algebraic Varieties
Christopher
Hacon
University of Utah, SALT LAKE CITY, UNITED STATES
General
We discuss recent results in the Minimal Model Program that have lead to several breakthroughs in the classi cation of algebraic varieties
79
90
1
10.4171/120-1/5
http://www.ems-ph.org/doi/10.4171/120-1/5
Representations of affine Kac–Moody groups over local and global fields: a survey of some recent results
Alexander
Braverman
Brown University, PROVIDENCE, UNITED STATES
David
Kazhdan
The Hebrew University of Jerusalem, JERUSALEM, ISRAEL
General
91
117
1
10.4171/120-1/6
http://www.ems-ph.org/doi/10.4171/120-1/6
Emergence of the Abrikosov lattice in several models with two dimensional Coulomb interaction
Sylvia
Serfaty
Université Pierre et Marie Curie (Paris VI), PARIS CEDEX 05, FRANCE
Partial differential equations
Convex and discrete geometry
Statistical mechanics, structure of matter
General
We consider three diff erent models coming from physics and involving the Coulomb interaction of points in the plane. The first is the classical Coulomb gas in an external potential, the second is the Ginzburg–Landau model of superconductivity, the third is the Ohta–Kawasaki model of polymers. In superconductivity, one observes in certain regimes the emergence of densely packed point vortices forming perfect triangular lattices named "Abrikosov lattices" in physics. We show how these Abrikosov lattices are expected to emerge in the three systems, via the minimization of a "Coulombian renormalized energy" that we defi ned with Etienne Sandier. We also present applications to the statistical mechanics of Coulomb gases, which are related to some random matrix models, and show how the previous results lead to expecting crystallisation in the low temperature limit.
119
135
1
10.4171/120-1/7
http://www.ems-ph.org/doi/10.4171/120-1/7
Dependent Classes E72
Saharon
Shelah
The Hebrew University of Jerusalem, JERUSALEM, ISRAEL
Mathematical logic and foundations
General
Model theory deals with general classes of structures = models, so the class of rings, the class of algebraically closed elds and other such speci c classes. Counting of so called complete types over models in the class has an important role in the development of model theory in general and so called stability theory in particular. In particular understanding the stable classes: those with relatively few complete types over structures from the class, has been central in model theory and its applications. Lately, we have a parallel recounting theorem under reasonable restrictions, counting types up to conjugacy. The classes for which we have few complete types up to conjugacy are proved to be so called dependent. This is a strong indication that there is much to be said on the classes whose models in the relevant cases has few complete types over them, so called dependent classes.
137
157
1
10.4171/120-1/8
http://www.ems-ph.org/doi/10.4171/120-1/8
Chaining and the Geometry of Stochastic Processes
Michel
Talagrand
Université Pierre et Marie Curie-Paris 6, PARIS CEDEX 6, FRANCE
Probability theory and stochastic processes
Algebraic geometry
General
Kolmogorov introduced a very powerful method called chaining in order to provide bounds for the supremum of a stochastic process. Even in the case of Gaussian processes, this method has reached its seemingly nal form only about 10 years ago. When used optimally in the case of Gaussian processes, it then yields optimal bounds. We explain why we believe that this is also true in many more cases.
159
178
1
10.4171/120-1/9
http://www.ems-ph.org/doi/10.4171/120-1/9
Duflo isomorphism, the Kashiwara–Vergne conjecture and Drinfeld associators
Anton
Alekseev
Université de Genève, GENÈVE, SWITZERLAND
Nonassociative rings and algebras
General
179
193
1
10.4171/120-1/10
http://www.ems-ph.org/doi/10.4171/120-1/10
Coagulation with limited aggregations
Jean
Bertoin
Universität Zürich, ZÜRICH, SWITZERLAND
Statistical mechanics, structure of matter
Probability theory and stochastic processes
General
Smoluchowski's coagulation equations can be used as elementary mathematical models for the formation of polymers. We review here some recent contributions on a variation of this model in which the number of aggregations for each atom is a priori limited. Macroscopic results in the deterministic setting can be explained at the microscopic level by considering a version of stochastic coalescence with limited aggregations, which can be related to the so-called random con guration model of random graph theory
195
209
1
10.4171/120-1/11
http://www.ems-ph.org/doi/10.4171/120-1/11
The Cremona group in two variables
Serge
Cantat
Ecole Normale Supérieure, PARIS CEDEX 05, FRANCE
Algebraic geometry
Group theory and generalizations
Several complex variables and analytic spaces
Dynamical systems and ergodic theory
We survey a few results concerning the Cremona group in two variables.
211
225
1
10.4171/120-1/12
http://www.ems-ph.org/doi/10.4171/120-1/12
Variational models for image inpainting
Vicent
Caselles
Universitat Pompeu-Fabra, BARCELONA, SPAIN
Computer science
Partial differential equations
Numerical analysis
General
In this paper we study some variational models for exemplar-based image inpainting, also referred to as nonlocal methods. Nonlocal methods for denoising and inpainting have gained considerable attention due to their good performance on textured images, a known weakness of classical local methods which are performant in recovering the geometric structure of the image. We rst review a general variational framework for the problem of nonlocal inpainting that exploits the self-similarity of natural images to copy information in a consistent way from the known parts of the image. We single out some particular methods and we review the main properties of the corresponding energies and their minima. We discuss the basic algorithms to minimize the energies and we display some numerical experiments illustrating the main properties of the proposed models.
227
242
1
10.4171/120-1/13
http://www.ems-ph.org/doi/10.4171/120-1/13
KAM theory and its applications: from conservative to dissipative systems
Alessandra
Celletti
Università di Roma 'Tor Vergata', ROMA, ITALY
Mechanics of particles and systems
General
We present a review of KAM theory for conservative and dissipative systems with applications to several models. The theory can be developed under general assumptions, namely a Diophantine condition on the frequency and a non-degeneracy requirement. In particular, we present a KAM theory for conformally symplectic systems, which provides also a very e cient numerical technique to determine the breakdown value of invariant attractors. Applications to model problems, most notably in Celestial Mechanics, are considered. Explicit estimates have been developed in the conservative case, where a computer-assisted implementation of KAM theory gives results in good agreement with the experimental values. Some KAM results for the dissipative standard map and for the dissipative spin-orbit problem in Celestial Mechanics are also discussed.
243
257
1
10.4171/120-1/14
http://www.ems-ph.org/doi/10.4171/120-1/14
Le programme de Langlands $p$-adique
Pierre
Colmez
Université Pierre et Marie Curie, PARIS, FRANCE
Number theory
General
We give a loose introduction to the circle of ideas around the $p$-adic local Langlands correspondence for GL$_2$(Q$_p$).
259
284
1
10.4171/120-1/15
http://www.ems-ph.org/doi/10.4171/120-1/15
Mirror Symmetry and Fano Manifolds
Tom
Coates
Imperial College London, LONDON, UNITED KINGDOM
Alessio
Corti
Imperial College London, LONDON, UNITED KINGDOM
Sergey
Galkin
The University of Tokyo, CHIBA, JAPAN
Vasily
Golyshev
Russian Academy of Sciences, MOSCOW, RUSSIAN FEDERATION
Alexander
Kasprzyk
Imperial College London, LONDON, UNITED KINGDOM
Algebraic geometry
General
285
300
1
10.4171/120-1/16
http://www.ems-ph.org/doi/10.4171/120-1/16
On flat bundles in characteristic 0 and $p > 0$
Hélène
Esnault
FU Berlin, BERLIN, GERMANY
Algebraic geometry
General
We discuss analogies between the fundamental groups of flat bundles in characteristic 0 and $p > 0$.
301
313
1
10.4171/120-1/17
http://www.ems-ph.org/doi/10.4171/120-1/17
Combinatorial realisation of cycles and small covers
Alexander
Gaifullin
Steklov Mathematical Institute, MOSCOW, RUSSIAN FEDERATION
Manifolds and cell complexes
Several complex variables and analytic spaces
Convex and discrete geometry
Differential geometry
In 1940s Steenrod asked if every homology class $z \in H_n(X; \mathbb Z)$ of every topological space $X$ can be realised by an image of the fundamental class of an oriented closed smooth manifold. Thom found a non-realisable 7-dimensional class and proved that for every $n$, there is a positive integer $k(n)$ such that the class $k(n)z$ is always realisable. The proof was by methods of algebraic topology and gave no information on the topology of the manifold which realises the homology class. We give a purely combinatorial construction of a manifold that realises a multiple of a given homology class. For every $n$, this construction yields a manifold $M^n_0$ with the following universality property: For any $X$ and $z \in H_n(X; \mathbb Z)$, a multiple of $z$ can be realised by an image of a (non-rami ed) finite-sheeted covering of $M^n_0$ . Manifolds satisfying this property are called URC-manifolds. The manifold $M^n_0$ is a so-called small cover of the permutahedron, i. e., a manifold glued in a special way out of $2n$ permutahedra. (The permutahedron is a special convex polytope with $(n+1)$! vertices.) Among small covers over other simple polytopes, we find a broad class of examples of URC-manifolds. In particular, in dimension 4, we fi nd a hyperbolic URC-manifold. Thus we prove a conjecture of Kotschick and Loh claiming that a multiple of every homology class can be realised by an image of a hyperbolic manifold. We also obtain some new results on the relationship of URC-manifolds with theory of simplicial volume.
315
330
1
10.4171/120-1/18
http://www.ems-ph.org/doi/10.4171/120-1/18
Remarks on the global regularity for solutions to the incompressible Navier–Stokes equations
Isabelle
Gallagher
Université Paris 7, PARIS CEDEX 05, FRANCE
Partial differential equations
Fourier analysis
General
We review some recent results concerning the Cauchy problem for the three dimensional, homogeneous, incompressible Navier{Stokes equations. We show in particular that the set of initial data generating a global smooth solution is open and connected, and we discuss under which conditions it is open for weak topology.
331
345
1
10.4171/120-1/19
http://www.ems-ph.org/doi/10.4171/120-1/19
Why the empirical sciences need statistics so desperately
Olle
Häggström
Chalmers University of Technology, GÖTEBORG, SWEDEN
Statistics
General
Science can be described as a systematic attempt to extract reliable information about the world. The cognitive capacities of homo sapiens come with various biases, such as our tendencies (a) to detect patterns in what is actually just noise, and (b) to be overly con dent in our conclusions. Thus, the scienti c method needs to involve safeguards against drawing incorrect conclusions due to such biases. A crucial part of the necessary toolbox is the theory of statistical inference. There exists a large and well-developed (but of course incomplete) body of such theory, which, however, researchers across practically all of the empirical sciences do not have su cient access to. The lack of statistical knowledge therefore forms a serious bottleneck in the quest for reliable scienti c advances. As has been observed by several authors in recent years, statistical malpractice is widespread across a broad spectrum of disciplines, including (but not limited to) medicine, cognitive sciences, Earth sciences and social sciences. Here I will rst try to describe the overall situation and provide some concrete examples, and then move on to discuss the more di cult issue of what can and needs to be done.
347
360
1
10.4171/120-1/20
http://www.ems-ph.org/doi/10.4171/120-1/20
Computing the Schrodinger equation with no fear of commutators
Arieh
Iserles
University of Cambridge, CAMBRIDGE, UNITED KINGDOM
Numerical analysis
Nonassociative rings and algebras
General
The discretisation of a linear Schrodinger equation is di cult due to the presence of a small parameter which induces high oscillations. A standard approach consists of a spectral semidiscretisation, followed by an exponential splitting. This, however, is sub-optimal, because, unless we are willing to compute an exceedingly large number of matrix exponentials per step, the very high precision of the space discretisation is marred by low order of the time solver. The underlying reason is that no commutators are allowed in the arguments of the exponentials, to avoid costly computations. It turns out, however, that once we split rst, discretise later, commutators are benign. Thus, we consider formally objects in the free Lie algebra spanned by the Laplacian and by a multiplication with the interaction potential. We demonstrate that it is embedded in a larger Lie algebra where all commutators can be resolved easily. This is the starting point for our symmetric Zassenhaus splitting, expressing the solution operator as a product of simple exponentials of asymptotically decreasing arguments. Once the underlying di erential operator is discretised and the exponentials approximated by suitable means, the outcome is an exceedingly accurate scheme bearing the price tag of $\mathcal O(M$ log$M$) operations per step, where $M$ is the number of degrees of freedom in the discretisation. This is joint work with Philipp Bader (Valencia), Karolina Kropielnicka (Cambridge and Gda nsk) and Pranav Singh (Cambridge).
361
374
1
10.4171/120-1/21
http://www.ems-ph.org/doi/10.4171/120-1/21
Dynamics of non-archimedean Polish groups
Alexander
Kechris
California Institute of Technology, PASADENA, UNITED STATES
Mathematical logic and foundations
Topological groups, Lie groups
Dynamical systems and ergodic theory
General topology
A topological group $G$ is Polish if its topology admits a compatible separable complete metric. Such a group is non-archimedean if it has a basis at the identity that consists of open subgroups. This class of Polish groups includes the pro finite groups and ($\mathbb Q_p; +$) but our main interest here will be on non-locally compact groups. In recent years there has been considerable activity in the study of the dynamics of Polish non-archimedean groups and this has led to interesting interactions between logic, fi nite combinatorics, group theory, topological dynamics, ergodic theory and representation theory. In this paper I will give a survey of some of the main directions in this area of research.
375
397
1
10.4171/120-1/22
http://www.ems-ph.org/doi/10.4171/120-1/22
Cluster algebras and cluster monomials
Bernhard
Keller
Université Paris Diderot - Paris 7, PARIS CEDEX 13, FRANCE
Commutative rings and algebras
Nonassociative rings and algebras
General
Cluster algebras were invented by Sergey Fomin and Andrei Zelevinsky at the beginning of the year 2000. Their motivations came from Lie theory and more precisely from the study of the so-called canonical bases in quantum groups and that of total positivity in algebraic groups. Since then, cluster algebras have been linked to many other subjects ranging from higher Teichmuller theory through discrete dynamical systems to combinatorics, algebraic geometry and representation theory. According to Fomin–Zelevinsky's philosophy, each cluster algebra should admit a `canonical' basis, which should contain the cluster monomials. This led them to formulate, about ten years ago, the conjecture on the linear independence of the cluster monomials. In these notes, we give a concise introduction to cluster algebras and sketch the ingredients of a proof of the conjecture. The proof is valid for all cluster algebras associated with quivers and was obtained in recent joint work with G. Cerulli Irelli, D. Labardini-Fragoso and P.-G. Plamondon.
399
413
1
10.4171/120-1/23
http://www.ems-ph.org/doi/10.4171/120-1/23
Weak solutions to the complex Monge–Ampère equation
Sławomir
Kołodziej
Jagellonian University, KRAKOW, POLAND
Several complex variables and analytic spaces
General
Canonical Kahler metrics, such as Ricci-flat or Käahler–Einstein, are obtained via solving the complex Monge-Ampère equation. The famous Calabi–Yau theorem asserts the existence and regularity of solutions to this equation on compact Käahler manifolds for smooth data. In this note we shall present methods, based on pluripotential theory, which yield the results on the existence and stability of the weak solutions of the Monge–Amp ère equation for possibly degenerate, non-smooth right hand side. Those weak solutions have also interesting applications in geometry. They lead to canonical metrics with singularities, which may occur as the limits of the Kähler–Ricci flow or the limits of families of Calabi–Yau metrics when the Kähler class hits the boundary of the Kähler cone.
415
428
1
10.4171/120-1/24
http://www.ems-ph.org/doi/10.4171/120-1/24
Reinforced random walk
Gady
Kozma
Weizmann Institute of Science, REHOVOT, ISRAEL
Probability theory and stochastic processes
General
We show that linearly reinforced random walk has a recurrent phase in every graph. The proof does not use the magic formula.
429
443
1
10.4171/120-1/25
http://www.ems-ph.org/doi/10.4171/120-1/25
On blow-up curves for semilinear wave equations
Frank
Merle
Université de Cergy-Pontoise, CERGY-PONTOISE CEDEX, FRANCE
Partial differential equations
General
In a joint work with H. Zaag, we consider the semilinear wave equation with focusing power nonlinearity in one space dimension. Blow-up solutions are known to exist, and the solution can be de ned on some domain of de nition under the blow-up curve {$t = T(x)$}. Considering an arbitrary blow-up solution, our goal is to describe its behavior near the blow-up curve, and the geometry of the blow-up curve itself. Such properties are linked to the notion of non-characteristic points on the curve. First, we nd criteria on initial data to ensure the existence or the non-existence of characteristic points. Then, we prove the regularity of the blow-up curve away from characteristic points, and show a surprising isolatedness property for characteristic points, together with the classi cation of the behavior of the solution near them. In order to do this, we introduce for this problem a notion of critical space. Furthermore, we link the geometrical properties of the blow-up curve with the problem of decomposing a general solution into a sum of solitons.
445
457
1
10.4171/120-1/26
http://www.ems-ph.org/doi/10.4171/120-1/26
Commuting higher rank ordinary differential operators
Andrey
Mironov
Siberian Branch of the Russian Academy of Sciences, NOVOSIBIRSK, RUSSIAN FEDERATION
Dynamical systems and ergodic theory
General
The theory of commuting ordinary di erential operators was developed from the beginning of the XX century on. The problem of nding commuting di fferential operators is solved for the case of operators of rank one. For operators of rank greater than one it is still open. In this paper we discuss some results related to operators of rank greater than one.
459
473
1
10.4171/120-1/27
http://www.ems-ph.org/doi/10.4171/120-1/27
Stochastic calculus with respect to the fractional Brownian motion
David
Nualart
University of Kansas, LAWRENCE, UNITED STATES
Probability theory and stochastic processes
General
In this note we survey some recent results on the stochastic calculus with respect to a fractional Brownian motion using Riemann sums approximations.
475
488
1
10.4171/120-1/28
http://www.ems-ph.org/doi/10.4171/120-1/28
Sampling, Interpolation, Translates
Alexander
Olevskii
Tel Aviv University, RAMAT AVIV, TEL AVIV, ISRAEL
Fourier analysis
Functions of a complex variable
Information and communication, circuits
General
The concepts in the title are closely related and belong to the central objects in Fourier Analysis, partly inspired by the signal theory. I’ll present some recent results in these topics. Sec. 2–6 are devoted to the sampling and interpolation problems, sec. 7–8 to the completeness of translates.
489
502
1
10.4171/120-1/29
http://www.ems-ph.org/doi/10.4171/120-1/29
Multidimensional periodic and almost-periodic spectral problems
Leonid
Parnovski
University College London, LONDON, UNITED KINGDOM
Partial differential equations
Operator theory
Quantum theory
General
I discuss various recent results on the spectral properties of multidimensional periodic and, to a smaller extent, almost-periodic operators, including proving the Bethe–Sommerfeld conjecture and obtaining the asymptotic expansion of the integrated density of states for large energies.
503
513
1
10.4171/120-1/30
http://www.ems-ph.org/doi/10.4171/120-1/30
Effective equations for quantum dynamics
Benjamin
Schlein
Universität Bonn, BONN, GERMANY
Quantum theory
Probability theory and stochastic processes
Statistical mechanics, structure of matter
General
We report on recent results concerning the derivation of e ective evolution equations starting from many body quantum dynamics. In particular, we obtain rigorous derivations of nonlinear Hartree equations in the bosonic mean eld limit, with precise bounds on the rate of convergence. Moreover, we present a central limit theorem for the fluctuations around the Hartree dynamics.
515
529
1
10.4171/120-1/31
http://www.ems-ph.org/doi/10.4171/120-1/31
Combinatorics of asymptotic representation theory
Piotr
Śniady
Polish Academy of Sciences, WARSAW 10, POLAND
Group theory and generalizations
Combinatorics
Functional analysis
General
The representation theory of the symmetric groups S(n) is intimately related to combinatorics: combinatorial objects such as Young tableaux and combinatorial algorithms such as Murnaghan–Nakayama rule. In the limit as n tends to infinity, the structure of these combinatorial objects and algorithms becomes complicated and it is hard to extract from them some meaningful answers to asymptotic questions. In order to overcome these difficulties, a kind of dual combinatorics of the representation theory of the symmetric groups was initiated in 1990s. We will concentrate on one of its highlights: Kerov polynomials which express characters in terms of, so called, free cumulants.
531
545
1
10.4171/120-1/32
http://www.ems-ph.org/doi/10.4171/120-1/32
On scale-invariant solutions of the Navier–Stokes equations
Hao
Jia
University of Minnesota, MINNEAPOLIS, UNITED STATES
Vladimír
Šverák
University of Minnesota, MINNEAPOLIS, UNITED STATES
Partial differential equations
Fluid mechanics
General
We discuss the forward self-similar solutions of the Navier{Stokes equations. It appears these solutions may provide an interesting window into non-perturbative regimes of the solutions of the equations.
547
553
1
10.4171/120-1/33
http://www.ems-ph.org/doi/10.4171/120-1/33
Ramsey-theoretic analysis of the conditional structure of weakly-null sequences
Stevo
Todorčević
University of Toronto, TORONTO, ONTARIO, CANADA
Mathematical logic and foundations
Combinatorics
Functional analysis
General
Understanding the possible conditional structure in a given weakly-null sequence $(x_i)$ in some normed space $X$ lies in the heart of several classical problems of this area of mathematics. We will expose the set-theoretic and Ramsey-theoretic methods relevant to both the lack and the existence of this conditional structure. We will concentrate on more recent results and will point out problems for further study.
555
572
1
10.4171/120-1/34
http://www.ems-ph.org/doi/10.4171/120-1/34
Uniqueness results for minimal surfaces and constant mean curvature surfaces in Riemannian manifolds
Simon
Brendle
Stanford University, STANFORD, UNITED STATES
Differential geometry
Relativity and gravitational theory
General
A classical theorem due to Alexandrov asserts that the only embedded surfaces in Euclidean space with constant mean curvature are the round spheres. Alexandrov's theorem is closely related to the classical isoperimetric inequality, which asserts that round spheres have minimal surface area among all surfaces that enclose a given amount of volume. It is an interesting question whether Alexandrov's theorem can be generalized to surfaces of constant mean curvatures in a Riemannian manifold. In particular, if the ambient Riemannian manifold is rotationally symmetric, can one conclude that every embedded surface of constant mean curvature is a sphere of symmetry? This question turns out to have interesting links with general relativity. We also discuss minimal surfaces in the sphere $S^3$ and our recent solution of Lawson's conjecture.
573
583
1
10.4171/120-1/35
http://www.ems-ph.org/doi/10.4171/120-1/35
Stability in geometric and functional inequalities
Alessio
Figalli
University of Texas at Austin, AUSTIN, UNITED STATES
Calculus of variations and optimal control; optimization
Partial differential equations
General
The aim of this note is to review recent stability results for some geometric and functional inequalities, and to describe applications to the long-time asymptotic of evolution equations
585
599
1
10.4171/120-1/36
http://www.ems-ph.org/doi/10.4171/120-1/36
Classification and rigidity for von Neumann algebras
Adrian
Ioana
University of California, San Diego, LA JOLLA, UNITED STATES
Functional analysis
Measure and integration
Dynamical systems and ergodic theory
General
We survey some recent progress on the classi cation of von Neumann algebras arising from countable groups and their measure preserving actions on probability spaces. In particular, we present results which provide classes of (W -superrigid) groups and actions that can be entirely reconstructed from their von Neumann algebras. We also discuss the recent finding of several large families of II$_1$ factors that have a unique group measure space decomposition.
601
625
1
10.4171/120-1/37
http://www.ems-ph.org/doi/10.4171/120-1/37
A nonlinear variational problem in relativistic quantum mechanics
Mathieu
Lewin
Université de Cergy-Pontoise, CERGY-PONTOISE, FRANCE
Partial differential equations
Quantum theory
General
We describe several recent results obtained in collaboration with P. Gravejat, C. Hainzl, E. S er e and J.P. Solovej, concerning a nonlinear model for the relativistic quantum vacuum in interaction with a classical electromagnetic field.
627
641
1
10.4171/120-1/38
http://www.ems-ph.org/doi/10.4171/120-1/38
Grid Diagrams in Heegaard Floer Theory
Ciprian
Manolescu
UCLA, LOS ANGELES, UNITED STATES
Manifolds and cell complexes
General
We review the use of grid diagrams in the development of Heegaard Floer theory. We describe the construction of the combinatorial link Floer complex, and the resulting algorithm for unknot detection. We also explain how grid diagrams can be used to show that the Heegaard Floer invariants of 3-manifolds and 4-manifolds are algorithmically computable (mod 2).
643
657
1
10.4171/120-1/39
http://www.ems-ph.org/doi/10.4171/120-1/39
Random maps and continuum random 2-dimensional geometries
Grégory
Miermont
Université Paris-Sud 11, ORSAY CEDEX, FRANCE
Probability theory and stochastic processes
Combinatorics
Statistical mechanics, structure of matter
General
In the recent years, much progress has been made in the mathematical understanding of the scaling limit of random maps, making precise the sense in which random embedded graphs approach a model of continuum surface. In particular, it is now known that many natural models of random plane maps, for which the faces degrees remain small, admit a universal scaling limit, the Brownian map. Other models, favoring large faces, also admit a one-parameter family of scaling limits, called stable maps. The latter are believed to describe the asymptotic geometry of random maps carrying statistical physics models, as has now been established in some important cases (including the so-called rigid $O(n)$ model on quadrangulations).
659
673
1
10.4171/120-1/40
http://www.ems-ph.org/doi/10.4171/120-1/40
Approximate (Abelian) groups
Tom
Sanders
Oxford University, OXFORD, UNITED KINGDOM
Group theory and generalizations
Number theory
General
Our aim is to discuss the structure of subsets of Abelian groups which behave 'a bit like' cosets (of subgroups). One version of 'a bit like' can be arrived at by relaxing the usual characterisation of cosets: a subset $S$ of an Abelian group is a coset if for every three elements $x, y, z \in S$ we have $x+y–z \in S$. What happens if this is not true 100% of the time but is true, say, 1% of the time? It turns out that this is a situation which comes up quite a lot, and one possible answer is called Frei man's theorem. We shall discuss it and some recent related quantitative advances.
675
689
1
10.4171/120-1/41
http://www.ems-ph.org/doi/10.4171/120-1/41
Shearing and mixing in parabolic flows
Corinna
Ulcigrai
University of Bristol, BRISTOL, UNITED KINGDOM
Dynamical systems and ergodic theory
General
Parabolic flows are dynamical systems in which nearby trajectories diverge with polynomial speed. A classical example is the horocycle flow on a surface of constant negative curvature. Other important classes of examples are smooth area-preserving flows on surfaces, whose study is connected with Teichmueller dynamics, and Heisenberg nilflows. We survey some of the chaotic properties of these flows and some recent results on time changes of the above mentioned classes of examples. We focus in particular on mixing and we explain the shearing mechanism which is responsible for mixing in parabolic dynamics.
691
705
1
10.4171/120-1/42
http://www.ems-ph.org/doi/10.4171/120-1/42
Optimal control theory and some applications to aerospace problems
Emmanuel
Trélat
Université Pierre et Marie Curie (Paris 6), PARIS CEDEX 05, FRANCE
Calculus of variations and optimal control; optimization
Dynamical systems and ergodic theory
Numerical analysis
Systems theory; control
In this proceedings article we rst shortly report on some classical techniques of nonlinear optimal control such as the Pontryagin Maximum Principle and the conjugate point theory, and on their numerical implementation. We illustrate these issues with problems coming from aerospace applications such as the orbit transfer problem which is taken as a motivating example. Such problems are encountered in a longstanding collaboration with the european space industry EADS Astrium. On this kind of nonacademic problem it is shown that the knowledge resulting from the maximum principle is insu fficient for solving adequately the problem, in particular due to the di culty of initializing the shooting method, which is an approach for solving the boundary value problem resulting from the application of the maximum principle. On the orbit transfer problem we show how the shooting method can be successfully combined with a numerical continuation method in order to improve signi cantly its performances. We comment on assumptions ensuring the feasibility of continuation or homotopy methods, which consist of deforming continuously a problem towards a simpler one, and then of solving a series of parametrized problems to end up with the solution of the initial problem. Finally, in view of designing low cost interplanetary space missions, we show how optimal control can be also combined with dynamical system theory, which allows to put in evidence nice properties of the celestial dynamics around Lagrange points that are of great interest for mission design.
707
726
1
10.4171/120-1/43
http://www.ems-ph.org/doi/10.4171/120-1/43
Mathematics and geometric ornamentation in the medieval Islamic world
Jan
Hogendijk
University of Utrecht, UTRECHT, NETHERLANDS
History and biography
Geometry
General
We discuss medieval Arabic and Persian sources on the design and construction of geometric ornaments in Islamic civilization.
727
741
1
10.4171/120-1/44
http://www.ems-ph.org/doi/10.4171/120-1/44
Some mathematical aspects of the planet Earth
José Francisco
Rodrigues
FC Universidade de Lisboa, LISBOA, PORTUGAL
General
The Planet Earth System is composed of several sub-systems: the atmosphere, the liquid oceans, the internal structure and the icecaps and the biosphere. In all of them Mathematics, enhanced by the supercomputers, has currently a key role through the \universal method" for their study, which consists of mathematical modeling, analysis, simulation and control, as it was re-stated by Jacques-Louis Lions in [41]. Much before the advent of computers, the representation of the Earth, navigation and cartography have contributed in a decisive form to the mathematical sciences. Nowadays the International Geosphere-Biosphere Program, sponsored by the International Council of Scienti c Unions, may contribute to stimulate several mathematical research topics. In this article, we present a brief historical introduction to some of the essential mathematics for understanding the Planet Earth, stressing the importance of Mathematical Geography and its role in the Scienti c Revolution(s), the modeling e orts of Winds, Heating, Earthquakes, Climate and their in uence on basic aspects of the theory of Partial Di erential Equations. As a special topic to illustrate the wide scope of these (Geo)physical problems we describe brie y some examples from History and from current research and advances in Free Boundary Problems arising in the Planet Earth. Finally we conclude by referring the potential impact of the international initiative Mathematics of Planet Earth (http://www.mpe2013.org) in Raising Public Awareness of Mathematics, in Research and in the Communication of the Mathematical Sciences to the new generations.
743
762
1
10.4171/120-1/45
http://www.ems-ph.org/doi/10.4171/120-1/45
Turing's Mathematical Work
P.D.
Welch
University of Bristol, BRISTOL, UNITED KINGDOM
History and biography
Mathematical logic and foundations
General
We sketch a brief outline of the mathematical, and in particular the logical, achievements of Turing in this, his centenary year.
763
777
1
10.4171/120-1/46
http://www.ems-ph.org/doi/10.4171/120-1/46
Counting Berg partitions via Sturmian words and substitution tilings
Artur
Siemaszko
University of Warmia and Mazury in Olsztyn, OLSZTYN, POLAND
Maciej
Wojtkowski
University of Warmia and Mazury in Olsztyn, OLSZTYN, POLAND
Dynamical systems and ergodic theory
General
We develop the connection of Berg partitions with special substitution tilings of two tiles. We obtain a new proof that the number of Berg partitions with a fi xed connectivity matrix is equal to half of the sum of its entries, [12]. This approach together with the formula of S e ebold [10], for the number of substitutions preserving a given Sturmian sequence, shows that all of the combinatorial substitutions can be realized geometrically as Berg partitions. We treat Sturmian tilings as intersection tilings of bi-partitions. Using the symmetries of bi-partitions we obtain geometrically the palindromic properties of Sturmian sequences (Theorem 3) established combinatorially by de Luca and Mignosi, [6].
779
790
1
10.4171/120-1/47
http://www.ems-ph.org/doi/10.4171/120-1/47
Geometry and Arithmetic
Carel
Faber
Royal Institute of Technology, Stockholm, Sweden
Gavril
Farkas
Humboldt-Universität zu Berlin, Berlin, Germany
Robin
de Jong
University of Leiden, The Netherlands
Algebraic geometry
Number theory
14; 11
Algebraic geometry, arithmetic geometry
This volume contains 21 articles written by leading experts in the fields of algebraic and arithmetic geometry. The treated topics range over a variety of themes, including moduli spaces of curves and abelian varieties, algebraic cycles, vector bundles and coherent sheaves, curves over finite fields, and algebraic surfaces, among others. The volume originates from the conference “Geometry and Arithmetic”, which was held on the island of Schiermonnikoog in The Netherlands in September 2010.
10
19
2012
978-3-03719-119-4
978-3-03719-619-9
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/119
http://www.ems-ph.org/doi/10.4171/119
EMS Series of Congress Reports
Nef divisors on $\M_{0,n}$ from GIT
Valery
Alexeev
University of Georgia, ATHENS, UNITED STATES
David
Swinarski
Fordham University, BRONX, UNITED STATES
Algebraic geometry
General
We introduce and study the GIT cone of $\overline{M}_{0,n}$, which is generated by the pullbacks of the natural ample line bundles on the GIT quotients $(\mathbb P^1)^n/\!/SL(2)$. We give an explicit formula for these line bundles and prove a number of basic results about the GIT cone. As one application, we prove unconditionally that the log canonical models of $\overline{M}_{0,n}$ with a symmetric boundary divisor coincide with the moduli spaces of weighted curves or with the symmetric GIT quotient, extending the result of Matt Simpson.
1
21
1
10.4171/119-1/1
http://www.ems-ph.org/doi/10.4171/119-1/1
Inoue type manifolds and Inoue surfaces: a connected component of the moduli space of surfaces with $K^2=7, p_g = 0$
Ingrid
Bauer
Universität Bayreuth, BAYREUTH, GERMANY
Fabrizio
Catanese
Universität Bayreuth, BAYREUTH, GERMANY
Algebraic geometry
General
We show that a family of minimal surfaces of general type with $p_g = 0, K^2=7$, constructed by Inoue in 1994, is indeed a connected component of the moduli space: indeed that any surface which is homotopically equivalent to an Inoue surface belongs to the Inoue family. The ideas used in order to show this result motivate us to give a new definition of varieties, which we propose to call Inoue type manifolds: these are obtained as quotients $ \hat{X} / G$, where $ \hat{X} $ is an ample divisor in a $K(\Gamma, 1)$ projective manifold $Z$, and $G$ is a finite group acting freely on $ \hat{X} $. For these types of manifolds we prove a similar theorem to the above, even if weaker, that manifolds homotopically equivalent to Inoue type manifolds are again Inoue type manifolds.
23
56
1
10.4171/119-1/2
http://www.ems-ph.org/doi/10.4171/119-1/2
Non-rationality of the symmetric sextic Fano threefold
Arnaud
Beauville
Université de Nice, NICE CEDEX 2, FRANCE
Algebraic geometry
General
We prove that the symmetric sextic Fano threefold, defined by the equations $\sum X_i=\sum X_i^2=\sum X_i^3=0$ in $\mathbb{P}^6$, is not rational. In view of the work of Prokhorov [P], our result implies that the alternating group $\mathfrak{A}_7$ admits only one embedding into the Cremona group $\mathrm{Cr}_3$ up to conjugacy.
57
60
1
10.4171/119-1/3
http://www.ems-ph.org/doi/10.4171/119-1/3
Brill–Noether loci of stable rank-two vector bundles on a general curve
Ciro
Ciliberto
Università di Roma, ROMA, ITALY
Flaminio
Flamini
Università di Roma Tor Vergata, ROMA, ITALY
Algebraic geometry
General
In this note we give an easy proof of the existence of generically smooth components of the expected dimension of certain Brill–Noether loci of stable rank 2 vector bundles on a curve with general moduli, with related applications to the Hilbert scheme of scrolls.
61
74
1
10.4171/119-1/4
http://www.ems-ph.org/doi/10.4171/119-1/4
Mordell–Weil groups and Zariski triples
José Ignacio
Cogolludo-Agustín
Universidad de Zaragoza, ZARAGOZA, SPAIN
Remke
Kloosterman
Humboldt-Universität zu Berlin, BERLIN, GERMANY
Algebraic geometry
Number theory
General
We prove the existence of three irreducible curves $C_{12,m}$ of degree 12 with the same number of cusps and different Alexander polynomials. This exhibits a Zariski triple. Moreover we provide a set of generators for the elliptic threefold with constant $j$-invariant 0 and discriminant curve $C_{12,m}$. Finally we consider a general degree $d$ base change of $C_{12d,m}$ and calculate the dimension of the equisingular deformation space.
75
89
1
10.4171/119-1/5
http://www.ems-ph.org/doi/10.4171/119-1/5
Approximate computations with modular curves
Jean-Marc
Couveignes
Université Bordeaux 1, TALENCE CEDEX, FRANCE
Bas
Edixhoven
Universiteit Leiden, LEIDEN, NETHERLANDS
Algebraic geometry
Number theory
General
This article gives an introduction for mathematicians interested in numerical computations in algebraic geometry and number theory to some recent progress in algorithmic number theory, emphasising the key role of approximate computations with modular curves and their Jacobians. These approximations are done in polynomial time in the dimension and the required number of significant digits. We explain the main ideas of how the approximations are done, illustrating them with examples, and we sketch some applications in number theory.
91
112
1
10.4171/119-1/6
http://www.ems-ph.org/doi/10.4171/119-1/6
A remark on a conjecture of Paranjape and Ramanan
Friedrich
Eusen
, DÜSSELDORF, GERMANY
Frank-Olaf
Schreyer
Universität des Saarlandes, SAARBRÜCKEN, GERMANY
Algebraic geometry
Number theory
General
In this note, we show that the spaces of global sections of exterior powers of a globally generated line bundle on a curve are not necessarily spanned by locally decomposable sections. The examples are based on the study of generic syzygy varieties. An application of these varieties is a short proof of Mukai's theorem that every smooth curve of genus 7 and Clifford index 3 arises as the intersection of the spinor variety $S \subset \mathbb P^{15}$ with a transversal $\mathbb P^6$.
113
123
1
10.4171/119-1/7
http://www.ems-ph.org/doi/10.4171/119-1/7
On extensions of the Torelli map
Angela
Gibney
University of Georgia, ATHENS, UNITED STATES
Algebraic geometry
Number theory
General
The divisors on $\overline{\operatorname {M}}_g$ that arise as the pullbacks of ample divisors along any extension of the Torelli map to any toroidal compactification of $\operatorname{A}_g$ form a 2-dimensional extremal face of the nef cone of $\overline{\operatorname {M}}_g$, which is explicitly described.
125
136
1
10.4171/119-1/8
http://www.ems-ph.org/doi/10.4171/119-1/8
The classes of singular moduli in the generalized Jacobian
Benedict
Gross
Harvard University, CAMBRIDGE, UNITED STATES
Algebraic geometry
Number theory
General
We reinterpret some results of D. Zagier on the traces of singular moduli, in terms of the generalized Jacobian of the modular curve of level 1, with respect to the divisor 2$(\infty)$.
137
141
1
10.4171/119-1/9
http://www.ems-ph.org/doi/10.4171/119-1/9
The Eisenstein motive for the cohomology of GSp2(ℤ)
Günter
Harder
, BONN, GERMANY
Algebraic geometry
Number theory
General
In his paper [4], Gerard van der Geer discusses the Eisenstein cohomology with coefficients in a sheaf $\tilde M$, which is obtained from a representation for the group $\tilde\Gamma=\GSp_g(\mathbb Z)$. Since we have an arithmetic interpretation of this sheaf, we can endow these cohomology groups with the structure of a mixed motive. A certain part of this cohomology is the compactly supported Eisenstein cohomology and van der Geer determines the structure of this compactly supported Eisenstein motive in the case $g=2$ and a regular coefficient system [4], Cor.~10.2). At the end of this note we compute this part of the cohomology for an arbitrary coefficient system, again in the case $g=2$.
143
164
1
10.4171/119-1/10
http://www.ems-ph.org/doi/10.4171/119-1/10
Cohomology of the moduli stack of coherent sheaves on a curve
Jochen
Heinloth
Universität Duisburg-Essen, ESSEN, GERMANY
Algebraic geometry
Number theory
General
We compute the cohomology of the moduli stack of coherent sheaves on a curve and find that it is a free graded algebra on infinitely many generators.
165
171
1
10.4171/119-1/11
http://www.ems-ph.org/doi/10.4171/119-1/11
New methods for bounding the number of points on curves over finite fields
Everett
Howe
Center for Communications Research, SAN DIEGO, UNITED STATES
Kristin
Lauter
Theory Group, REDMOND, UNITED STATES
Algebraic geometry
Number theory
General
We provide new upper bounds on $N_q(g)$, the maximum number of rational points on a smooth absolutely irreducible genus-$g$ curve over $\mathbb {F}_q$, for many values of $q$ and $g$. Among other results, we find that $N_4(7) = 21$ and $N_8(5) = 29$, and we show that a genus-12 curve over $\mathbb {F}_2$ having 15 rational points must have characteristic polynomial of Frobenius equal to one of three explicitly given possibilities. We also provide sharp upper bounds for the lengths of the shortest vectors in Hermitian lattices of small rank and determinant over the maximal orders of small imaginary quadratic fields of class number 1. Some of our intermediate results can be interpreted in terms of Mordell–Weil lattices of constant elliptic curves over one-dimensional function fields over finite fields. Using the Birch and Swinnerton-Dyer conjecture for such elliptic curves, we deduce lower bounds on the orders of certain Shafarevich–Tate groups.
173
212
1
10.4171/119-1/12
http://www.ems-ph.org/doi/10.4171/119-1/12
Wildly ramified actions and surfaces of general type arising from Artin–Schreier curves
Hiroyuki
Ito
Tokyo University of Science, CHIBA-KEN, JAPAN
Stefan
Schröer
Heinrich-Heine-Universität, DÜSSELDORF, GERMANY
Algebraic geometry
Number theory
General
We analyse the diagonal quotient for the product of certain Artin--Schreier curves. The smooth models are almost always surfaces of general type, with Chern slopes tending asymptotically to 1. The calculation of numerical invariants relies on a close examination of the relevant wild quotient singularity in characteristic $p$. It turns out that the canonical model has $q-1$ rational double points of type $A_{q-1}$, and embeds as a divisor of degree $q$ in $\mathbb P^3$, which is in some sense reminiscent of the classical Kummer quartic.
213
241
1
10.4171/119-1/13
http://www.ems-ph.org/doi/10.4171/119-1/13
A note on a supersingular K3 surface in characteristic 2
Toshiyuki
Katsura
Hosei University, TOKYO, JAPAN
Shigeyuki
Kondō
Nagoya University, NAGOYA, JAPAN
Algebraic geometry
Number theory
General
We construct, on a supersingular K3 surface with Artin invariant 1 in characteristic 2, a set of 21 disjoint smooth rational curves and another set of 21 disjoint smooth rational curves such that each curve in one set intersects exactly 5 curves from the other set with multiplicity 1 by using the structure of a generalized Kummer surface. As a corollary we have a concrete construction of a K3 surface with 21 rational double points of type A1 in characteristic 2.
243
255
1
10.4171/119-1/14
http://www.ems-ph.org/doi/10.4171/119-1/14
The intuitive definition of Du Bois singularities
Sándor
Kovács
University of Washington, SEATTLE, UNITED STATES
Algebraic geometry
Number theory
General
It is proved that for projective varieties having Du Bois singularities is equivalent to the condition that the coherent cohomology groups of the structure sheaf coincide with the appropriate Hodge components of the singular cohomology groups
257
266
1
10.4171/119-1/15
http://www.ems-ph.org/doi/10.4171/119-1/15
Bundles of rank 2 with small Clifford index on algebraic curves
H.
Lange
Universität Erlangen-Nünberg, ERLANGEN, GERMANY
Peter
Newstead
University of Liverpool, LIVERPOOL, UNITED KINGDOM
Algebraic geometry
Number theory
General
In this paper, we construct stable bundles $E$ of rank 2 on suitably chosen curves of any genus $g\ge12$ with maximal Clifford index such that the Clifford index of $E$ takes the minimum possible value for curves with this property.
267
281
1
10.4171/119-1/16
http://www.ems-ph.org/doi/10.4171/119-1/16
Descendents on local curves: Stationary theory
Rahul
Pandharipande
ETH Zürich, ZÜRICH, SWITZERLAND
A.
Pixton
Princeton University, PRINCETON, UNITED STATES
Algebraic geometry
Number theory
General
The stable pairs theory of local curves in 3-folds (equivariant with respect to the scaling 2-torus) is studied with stationary descendent insertions. Reduction rules are found to lower descendents when higher than the degree. Factorization then yields a simple proof of rationality in the stationary case and a proof of the functional equation related to inverting $q$. The method yields an effective determination of stationary descendent integrals. The series $\mathsf{Z}^{\mathsf{cap}}_{d,(d)}( \tau_d(\mathsf{p}))$ plays a special role and is calculated exactly using the stable pairs vertex and an analysis of the solution of the quantum differential equation for the Hilbert scheme of points of the plane.
283
307
1
10.4171/119-1/17
http://www.ems-ph.org/doi/10.4171/119-1/17
A remark on Getzler’s semi-classical approximation
Dan
Petersen
Royal Institute of Technology, STOCKHOLM, SWEDEN
Algebraic geometry
Number theory
General
Ezra Getzler notes in the proof of the main theorem of [3] that “A proof of the theorem could no doubt be given using [a combinatorial interpretation in terms of a sum over necklaces]; however, we prefer to derive it directly from Theorem 2.2”. In this note we give such a direct combinatorial proof using wreath product symmetric functions.
309
316
1
10.4171/119-1/18
http://www.ems-ph.org/doi/10.4171/119-1/18
On the modular curve X0(23)
René
Schoof
Università di Roma Tor Vergata, ROMA, ITALY
Algebraic geometry
Number theory
General
The Jacobian $J_0(23)$ of the modular curve $X_0(23)$ is a semi-stable abelian variety over $\mathbb Q$ with good reduction outside 23. It is simple. We prove that every simple semi-stable abelian variety over $\mathbb Q$ with good reduction outside 23 is isogenous over $\mathbb Q$ to $J_0(23)$.
317
345
1
10.4171/119-1/19
http://www.ems-ph.org/doi/10.4171/119-1/19
Degree 4 unramified cohomology with finite coefficients and torsion codimension 3 cycles
Claire
Voisin
Institut de Mathématiques de Jussieu, PARIS, FRANCE
Algebraic geometry
Number theory
General
We study in this paper degree $4$ unramified cohomology with torsion coefficients of a smooth projective variety $X$. We show that if $CH_0(X)$ is small, it is isomorphic to the group of torsion codimension $3$ cycles with trivial Deligne cycle class, modulo algebraic equivalence.
347
368
1
10.4171/119-1/20
http://www.ems-ph.org/doi/10.4171/119-1/20
Poincaré duality and unimodularity
Yuri
Zarhin
Pennsylvania State University, UNIVERSITY PARK, UNITED STATES
Algebraic geometry
Number theory
General
It is well known that the cup-product pairing on the complementary integral cohomology groups (modulo torsion) of a compact oriented manifold is unimodular. We prove a similar result for the $\ell$-adic cohomology groups of smooth algebraic varieties.
369
376
1
10.4171/119-1/21
http://www.ems-ph.org/doi/10.4171/119-1/21
Singularities in Geometry and Topology
Strasbourg 2009
Vincent
Blanlœil
IRMA, Strasbourg, France
Toru
Ohmoto
Hokkaido University, Sapporo, Japan
Algebraic geometry
Several complex variables and analytic spaces
Manifolds and cell complexes
Global analysis, analysis on manifolds
13A35, 14A22, 14B05, 14B07, 14B15, 14C17, 14D06, 14E15, 14E18, 14F99, 14H25, 14J17, 14M25, 14N99, 18F30, 19K10, 32A27, 32C37, 32F75, 32G15, 32SXX, 35Q75, 52C35, 53C05, 55N35, 55R40, 57N05, 57R18, 57R20, 58A30, 58K10, 58K40, 58K60, 60D05, 83C57
singularity theory, singularities, characteristic classes, Milnor fiber, jet schemes, equisingularity, intersection homology, knot theory, Hodge theory, Fulton–MacPherson bivariant theory, mixed weighted homogeneous, nearby cycles, vanishing cycles
This volume arises from 5th Franco-Japanese Symposium on Singularities, held in Strasbourg in August 2009. The conference brought together an international group of researchers working on singularities in algebraic geometry, analytic geometry and topology, mainly from France and Japan. Besides, it also organized a special session, JSPS Forum on Singularities and Applications, which was aimed to introduce some recent applications of singularity theory to physics and statistics. This book comprises research papers and short lecture notes on advanced topics on singularities. Some surveys on applications that were presented in the Forum are also added. Topics covered include splice surface singularities, b-functions, equisingularity, degenerating families of Riemann surfaces, hyperplane arrangements, mixed singularities, jet schemes, noncommutative blow-ups, characteristic classes of singular spaces, and applications to geometric optics, cosmology and learning theory. Graduate students who wish to learn about various approaches to singularities, as well as experts in the field and researchers in other areas of mathematics and science will find the contributions to this volume a rich source for further study and research.
12
18
2012
978-3-03719-118-7
978-3-03719-618-2
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/118
http://www.ems-ph.org/doi/10.4171/118
IRMA Lectures in Mathematics and Theoretical Physics
20
Optical caustics and their modelling as singularities
Alain
Joets
Université Paris Sud-XI, ORSAY CEDEX, FRANCE
General
Optical caustics are bright patterns, formed by the local focalization of light rays. They are caused, for instance, by the reflection or the refraction of the sun rays through a wavy water surface. In the absence of an appropriate mathematical frame, their main characteristics have remained unrecognized for a long time and the caustics appeared in the literature under different names: evolutes, envelopes, focals, etc. The creation of the singularity theory in the middle of the 20th century radically changed the situation. Caustics are now understood as physical realizations of Lagrangian singularities. In this modelling, one predicts their local classification into five stable types (R. Thom, V. Arnold): folds, cusps, swallowtails, elliptic umbilics and hyperbolic umbilics. This local classification is indeed observed in experiments. However the global properties of the caustics are only partially taken into account by the Lagrangian model. In fact, it has been proved by Yu. Chekanov that the special form of the eikonal equation governing the propagation of the optical wave fronts imposes the existence of a topological constraint on the singular set (representing the caustic in the phase space) and restricts the number of possible bifurcations. Our experiments on caustics produced by bi-periodic structures in liquid crystals confirm the existence of the topological constraint, and validate the modelling of the caustics as special types of Lagrangian singularities.
1
17
1
10.4171/118-1/1
http://www.ems-ph.org/doi/10.4171/118-1/1
On local equisingularity
Helmut
Hamm
Universität Münster, MÜNSTER, GERMANY
Several complex variables and analytic spaces
Algebraic geometry
Global analysis, analysis on manifolds
General
We will prove some generalization of the theorems of Lê and Ramanujam resp. of Timourian for the case where the ambient space is no longer $\mathbb{C}^m$. Furthermore we will derive some weaker result in the case of a family of non-isolated singularities.
19
37
1
10.4171/118-1/2
http://www.ems-ph.org/doi/10.4171/118-1/2
Jet schemes of homogeneous hypersurfaces
Shihoko
Ishii
University of Tokyo, TOKYO, JAPAN
Akiyoshi
Sannai
Nagoya University, NAGOYA, JAPAN
Kei-ichi
Watanabe
College of Humanities and Sciences, TOKYO, JAPAN
Algebraic geometry
General
This paper studies the singularities of jet schemes of homogeneous hypersurfaces of general type. We obtain the condition of the degree and the dimension for the singularities of the jet schemes to be of dense $F$-regular type. This provides us with examples of singular varieties whose $m$-jet schemes have rational singularities for every $m$.
39
49
1
10.4171/118-1/3
http://www.ems-ph.org/doi/10.4171/118-1/3
Singularities in relativity
Tatsuhiko
Koike
Keio University, YOKOHAMA, JAPAN
Relativity and gravitational theory
Partial differential equations
Global analysis, analysis on manifolds
General
Many phenomena of importance in general relativity theory are related to singularities in mathematics. For a simple example, the spacetime regions of extreme gravitational field such as the beginning of the Universe and the fate of a massive star are described by singularities in the differential-geometric sense, i.e., curvature singularities of pseudo-Riemannian manifolds. This type of singularity is one of the main objects of interest in general relativity. A less trivial example is that the formation of a black hole horizon can be described as a blow-up solution of some partial differential equations in a certain coordinate system, which is a singularity in the analytic sense. Another is that the “shape” of the black hole horizon is fully characterised by the set of its nondifferential points which are singularities in the sense of singularity theory. I will explain these connections between singularity and relativity with some comments on my related works.
51
69
1
10.4171/118-1/4
http://www.ems-ph.org/doi/10.4171/118-1/4
On the universal degenerating family of Riemann surfaces
Yukio
Matsumoto
Gakushuin University, TOKYO, JAPAN
Algebraic geometry
Several complex variables and analytic spaces
Manifolds and cell complexes
General
Let $\Sigma_g$ be a closed oriented (topological) surface of genus $g$ ($\geqq 2$). Over the Teichmüller space $T(\Sigma_g)$ of $\Sigma_g$, Bers constructed a universal family $V(\Sigma_g)$ of curves of genus $g$, which would be well called “the tautological family of Riemann surfaces”. The mapping class group $\Gamma_g$ of $\Sigma_g$ acts on $V(\Sigma_g) \to T(\Sigma_g)$ in a fibration preserving manner. Dividing the fiber space by this action, we obtain an “orbifold fiber space” $Y(\Sigma_g) \to M(\Sigma_g)$, where $Y(\Sigma_g)$ and $M(\Sigma_g)$ denote $V(\Sigma_g)/\Gamma_g$ and $T(\Sigma_g)/\Gamma_g$, respectively. The latter quotient $M(\Sigma_g)$ is called the moduli space of $\Sigma_g$. The fiber space $Y(\Sigma_g) \to M(\Sigma_g)$ can be naturally compactified to another orbifold fiber space $\overline{Y(\Sigma_g)} \to \overline{M(\Sigma_g)}$. The base space $\overline{M(\Sigma_g)}$ is called the Deligne–Mumford compactification. Since this compactification is constructed by adding “stable curves” at infinity, it is usually accepted that the compactified moduli space $\overline{M(\Sigma_g)}$ is the coarse moduli space of stable curves of genus $g$. In this paper, we will sketch our argument which leads to a conclusion, somewhat contradictory to the above general acceptance, that the compactified family $\overline{Y(\Sigma_g)} \to \overline{M(\Sigma_g)}$ is the universal degenerating family of Riemann surfaces, i.e., it virtually parametrizes not only stable curves but also all types of degenerate and non-degenerate curves. Our argument is a combination of the Bers–Kra theory and Ashikaga’s precise stable reduction theorem.
71
102
1
10.4171/118-1/5
http://www.ems-ph.org/doi/10.4171/118-1/5
Algebraic local cohomologies and local $b$-functions attached to semiquasihomogeneous singularities with $L(f)=2$
Yayoi
Nakamura
Kinki University, OSAKA, JAPAN
Shinichi
Tajima
University of Tsukuba, IBARAKI, JAPAN
Algebraic geometry
Several complex variables and analytic spaces
General
The role of the weighted degree of algebraic local cohomology classes in the computation of $b$-function is discussed. The result which is similar to quasihomogeneous cases is observed for semiquasihomogeneous isolated singularities with $L(f)=2$.
103
115
1
10.4171/118-1/6
http://www.ems-ph.org/doi/10.4171/118-1/6
A note on the Chern–Schwartz–MacPherson class
Toru
Ohmoto
Hokkaido University, SAPPORO, JAPAN
Algebraic geometry
General
This is a note about the Chern–Schwartz–MacPherson class for certain algebraic stacks which has been introduced in [17]. We also discuss other singular Riemann–Roch type formulas in the same manner.
117
131
1
10.4171/118-1/7
http://www.ems-ph.org/doi/10.4171/118-1/7
On mixed projective curves
Mutsuo
Oka
Tokyo University of Science, TOKYO, JAPAN
Algebraic geometry
General
Let $f(\mathbf{z},\bar{\mathbf{z}})$ be a strongly polar homogeneous polynomial of $n$ variables $\mathbf{z} =(z_1,\dots, z_n)$. This polynomial defines a projective real algebraic variety $V = \{[\mathbf{z}] \in \mathbf{CP}^{n-1}\,|\,f(\mathbf{z},\bar{\mathbf{z}})=0 \}$ in the projective space $ \mathbf{CP}^{n-1}$. The behavior is different from that of the projective hypersurface. The topology is not uniquely determined by the degree of the variety even if $V$ is non-singular. We study a basic property of such a variety.
133
147
1
10.4171/118-1/8
http://www.ems-ph.org/doi/10.4171/118-1/8
Invariants of splice quotient singularities
Tomohiro
Okuma
Yamagata University, YAMAGATA, JAPAN
Algebraic geometry
Several complex variables and analytic spaces
General
This article is a survey of results on analytic invariants of splice quotient singularities induced by Neumann and Wahl. These singularities are natural and broad generalization of quasihomogeneous surface singularities with rational homology sphere links. The “leading terms” of the equations are constructed from the resolution graph. Some analytic invariants of splice quotients can explicitly be computed from their graph.
149
159
1
10.4171/118-1/9
http://www.ems-ph.org/doi/10.4171/118-1/9
A note on the toric duality between the cyclic quotient surface singularities $A_{n,q}$ and $A_{n,n - q}$
Oswald
Riemenschneider
Universität Hamburg, HAMBURG, GERMANY
Algebraic geometry
General
In my lecture at the Franco-Japanese Symposium on Singularities I gave an introduction to the work of Martin Hamm [3] concerning the explicit construction of the versal deformation of cyclic surface singularities. Since that part of his dissertation is already documented in a survey article (cf. [8]), I concentrate in the present note on some other aspect of [3]: the toric duality of the total spaces of the deformations over the monodromy coverings of the Artin components for the singularities $A_{n,q}$ and $A_{n,n - q}$ which themselves are toric duals of each other. Our exhibition is based – as in Hamm’s dissertation – on the algebraic aspects, i.e., the algebras and their generators of these total spaces. We prove Hamm’s remarkable duality result in this note first in detail for the hypersurface case $q = n - 1$ in which the interplay between algebra and geometry of the underlying polyhedral cones is rather obvious, especially when bringing also the “complementarity” of $A_{n,q}$ and $A_{n,n - q}$ into the game. We then treat the dual case $q = 1$ of cones over the rational normal curves once more in order to develop the necessary ideas for transforming the generators in such a way that it becomes transparent how to compute the dual, even in the general situation (which we explain in the last section by an example).
161
179
1
10.4171/118-1/10
http://www.ems-ph.org/doi/10.4171/118-1/10
Nearby cycles and characteristic classes of singular spaces
Jörg
Schürmann
Universität Münster, MÜNSTER, GERMANY
Algebraic geometry
Several complex variables and analytic spaces
Global analysis, analysis on manifolds
General
In this paper we give an introduction to our recent work on characteristic classes of complex hypersurfaces based on some talks given at conferences in Strasbourg, Oberwolfach, and Kagoshima. We explain the relation between nearby cycles for constructible functions or sheaves as well as for (relative) Grothendieck groups of algebraic varieties and mixed Hodge modules, and the specialization of characteristic classes of singular spaces like the Chern-, Todd-, Hirzebruch-, and motivic Chern-classes. As an application we get a description of the differences between the corresponding virtual and functorial characteristic classes of complex hypersurfaces in terms of vanishing cycles related to the singularities of the hypersurface.
181
205
1
10.4171/118-1/11
http://www.ems-ph.org/doi/10.4171/118-1/11
Residues of singular holomorphic distributions
Tatsuo
Suwa
Hokkaido University, SAPPORO, JAPAN
Algebraic geometry
Several complex variables and analytic spaces
Manifolds and cell complexes
Global analysis, analysis on manifolds
We present two types of residue theories for singular holomorphic distributions. The first one is for certain Chern polynomials of the normal sheaf of a distribution and the residues arise from the vanishing, by rank reason, of the relevant characteristic classes on the non-singular part. The second one is for certain Atiyah polynomials of vector bundles admitting an action of a distribution and the residues arise from the Bott type vanishing theorem on the non-singular part.
207
247
1
10.4171/118-1/12
http://www.ems-ph.org/doi/10.4171/118-1/12
Two birational invariants in statistical learning theory
Sumio
Watanabe
Tokyo Institute of Technology, YOKOHAMA, JAPAN
Probability theory and stochastic processes
General
This paper introduces a recent advance in the research between algebraic geometry and statistical learning theory. A lot of statistical models used in information science contain singularities in their parameter spaces, to which the conventional theory can not be applied. The statistical foundation of singular models was been left unknown, because no mathematical base could be found. However, recently new theory was constructed based on algebraic geometry and algebraic analysis. In this paper, we show that statistical estimation process is determined by two birational invariants, the real log canonical threshold and the singular fluctuation. As a result, a new formula is derived, which enables us to estimate the generalization error without any knowledge of the information source. In the discussion, a relation between mathematics and the real world is introduced to pure mathematicians.
249
268
1
10.4171/118-1/13
http://www.ems-ph.org/doi/10.4171/118-1/13
Frobenius morphisms of noncommutative blowups
Takehiko
Yasuda
Osaka University, OSAKA, JAPAN
Algebraic geometry
Commutative rings and algebras
General
We define the Frobenius morphism of certain class of noncommutative blowups in positive characteristic. Thanks to a nice property of the class, the defined morphism is flat. Therefore we say that the noncommutative blowups in this class are Kunz regular. One of such blowups is the one associated to a regular Galois alteration. As a consequence of de Jong's theorem, we see that for every variety over an algebraically closed field of positive characteristic, there exists a noncommutative blowup which is Kunz regular. We also see that a variety with F-pure and FFRT (finite F-representation type) singularities has a Kunz regular noncommutative blowup which is associated to an iteration of the Frobenius morphism of the variety.
269
283
1
10.4171/118-1/14
http://www.ems-ph.org/doi/10.4171/118-1/14
Bivariant motivic Hirzebruch class and a zeta function of motivic Hirzebruch class
Shoji
Yokura
Kagoshima University, KAGOSHIMA, JAPAN
Algebraic geometry
Several complex variables and analytic spaces
Algebraic topology
Manifolds and cell complexes
The Euler–Poincaré characteristic is a generalization of the cardinality (or counting) and its higher homological extension for singular varieties as a natural transformation (what could be put in as its “categorification”) is MacPherson’s Chern class transformation. This transformation furthermore has two main developments: a bivariant-theoretic analogue and a generating series of it, i.e. a zeta function. The motivic Hirzebruch class is a unified theory of the three well-known characteristic classes of singular varieties, i.e. the above MacPherson’s Chern class transformation, Baum–Fulton–MacPherson’s Riemann–Roch and Cappell–Shaneson’s $L$-class transformation, which extends Goresky–MacPherson’s $L$-class. In this paper we discuss a bivariant-theoretic analogue and a zeta function of the motivic Hirzebruch class.
285
343
1
10.4171/118-1/15
http://www.ems-ph.org/doi/10.4171/118-1/15
Minimality of hyperplane arrangements and basis of local system cohomology
Masahiko
Yoshinaga
Kyoto University, KYOTO, JAPAN
Algebraic geometry
Several complex variables and analytic spaces
Convex and discrete geometry
General
The purpose of this paper is applying minimality of hyperplane arrangements to local system cohomology groups. It is well known that twisted cohomology groups with coefficients in a generic rank one local system vanish except in the top degree, and bounded chambers form a basis of the remaining cohomology group. We determine precisely when this phenomenon happens for two-dimensional arrangements.
345
362
1
10.4171/118-1/16
http://www.ems-ph.org/doi/10.4171/118-1/16
Handbook of Teichmüller Theory, Volume IV
Athanase
Papadopoulos
IRMA, Strasbourg, France
Functions of a complex variable
Several complex variables and analytic spaces
Primary 30-00, 32-00, 57-00, 32G13, 32G15, 30F60; Secondary 11F06, 11F75, 14D20, 14H15, 14H60, 14H55, 14J60, 20F14, 20F28, 20F38, 20F65, 20F67, 20H10, 22E46, 30-03, 30C62, 30F20, 30F25, 30F10, 30F15, 30F30, 30F35, 30F40, 30F45, 32-03, 32S30, 37-99, 53A35, 53B35, 53C35, 53C50, 53C80, 53D55, 53Z05, 57M07, 57M20, 57M27, 57M50, 57M60, 57N16
Teichmüller theory is, since several decades, one of the most active research areas in mathematics, with a very wide range of points of view, including Riemann surface theory, hyperbolic geometry, low-dimensional topology, several complex variables, algebraic geometry, arithmetic, partial differential equations, dynamical systems, representation theory, symplectic geometry, geometric group theory, and mathematical physics. The present book is the fourth volume in a Handbook of Teichmüller Theory project that started as an attempt to present, in a most comprehensive and systematic way, the various aspects of this theory with its relations to all the fields mentioned. The handbook is addressed to researchers as well as graduate students. The present volume is divided into five parts: Part A: The metric and the analytic theory. Part B: Representation theory and generalized structures. Part C: Dynamics. Part D: The quantum theory. Part E: Sources. Parts A, B and D are sequels of parts on the same theme in previous volumes. Part E has a new character in the series; it contains the translation together with a commentary of an important paper by Teichmüller which is almost unknown even to specialists. Making clear the original ideas of and motivations for a theory is crucial for many reasons, and rendering available this translation together with the commentary that follows will give a new impulse and will contribute in putting the theory into a broader perspective. The various volumes in this collection are written by experts who have a broad view on the subject. In general, the chapters have an expository character, which is the original purpose of this handbook, while some of them contain new and important results.
5
30
2014
978-3-03719-117-0
978-3-03719-617-5
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/117
http://www.ems-ph.org/doi/10.4171/117
IRMA Lectures in Mathematics and Theoretical Physics
19
Introduction to Teichmüller theory, old and new, IV
Athanase
Papadopoulos
Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE
Several complex variables and analytic spaces
General
This is an extended introduction to the volume. It contains a detailed account of its content, with motivation and background material.
1
39
1
10.4171/117-1/1
http://www.ems-ph.org/doi/10.4171/117-1/1
Local and global aspects of Weil–Petersson geometry
Sumio
Yamada
Gakushuin University, TOKYO, JAPAN
Global analysis, analysis on manifolds
Partial differential equations
General
This is a survey on the topic of Weil–Petersson geometry of Teichmüller spaces. Although historically the subject has been developed as a branch of complex analysis, the treatment here is from the view-point of differential geometry, much influenced by the works of Eells, Earle, Fischer, Tromba and Wolpert over the several decades. The Weil–Petersson geometry has negative sectional curvature, and the main theme of this article is to exploit the convexity of the Weil–Petersson distance function induced by the negativity of the curvature, from Riemannian geometric approaches as well as those of the CAT(0) geometry and the Coxeter theory. Also we discuss the Weil–Petersson geometry of the universal Teichmüller space. In particular, the linear deformation theory of the hyperbolic metrics is treated synthetically in the sense that the universal Teichmüller space and the finite dimensional Teichmüller space are contrasted with each other.
43
111
1
10.4171/117-1/2
http://www.ems-ph.org/doi/10.4171/117-1/2
Simple closed geodesics and the study of Teichmüller spaces
Hugo
Parlier
Université de Fribourg, FRIBOURG, SWITZERLAND
Global analysis, analysis on manifolds
Partial differential equations
General
The goal of the chapter is to present certain aspects of the relationship between the study of simple closed geodesics and Teichmüller spaces. The set of simple closed geodesics is more than a mere curiosity and has been central in the study of surfaces for quite some time. There are two main themes to the chapter. The first one is the study of the set of simple closed geodesics in contrast with the set of closed geodesics and the second is on systoles, their lengths and related quantities such as the lengths of minimal pants decompositions.
113
134
1
10.4171/117-1/3
http://www.ems-ph.org/doi/10.4171/117-1/3
Curve complexes versus Tits buildings: structures and applications
Lizhen
Ji
University of Michigan, ANN ARBOR, UNITED STATES
Differential geometry
Group theory and generalizations
Topological groups, Lie groups
Functions of a complex variable
Tits buildings $\Delta_\mathbb Q(\mathbf G)$ of linear algebraic groups $\mathbf G$ defined over the field of rational numbers $\mathbb Q$ have played an important role in understanding partial compactifications of symmetric spaces and compactifications of locally symmetric spaces, cohomological properties of arithmetic subgroups and S-arithmetic subgroups of $\mathbf G(\mathbb Q)$. Curve complexes $\mathcal C(S_{g,n})$ of surfaces $S_{g,n}$ were introduced to parametrize boundary components of partial compactifications of Teichmüller spaces and were later applied to understand properties of mapping class groups of surfaces and the geometry and topology of 3-dimensional manifolds. Tits buildings are spherical building. Another important class of buildings consists of Euclidean buildings, for example, the Bruhat
135
196
1
10.4171/117-1/4
http://www.ems-ph.org/doi/10.4171/117-1/4
Extremal length geometry
Hideki
Miyachi
Osaka University, OSAKA, JAPAN
General
This is a survey of the development of the extremal length geometry of Teichmüller space after S. Kerckhoff, F. Gardiner and H. Masur’s works. We shall discuss intersection number in extremal length geometry, and we shall view a canonical compactification in extremal length geometry with respect to intersection number. This is an analogy with the Thurston compactification in hyperbolic geometry. We will also obtain a hyperboloid model of Teichmüller space in extremal length geometry. This is an analogy with a realization of Teichmüller space into the space of the geodesic currents observed by F. Bonahon.
197
234
1
10.4171/117-1/5
http://www.ems-ph.org/doi/10.4171/117-1/5
Compactifications of Teichmüller spaces
Ken’ichi
Ohshika
Osaka University Graduate School of Science, OSAKA, JAPAN
Functions of a complex variable
Manifolds and cell complexes
General
We explain three ways of compactifying Teichmüller spaces: the Thurston compactification, the Teichmüller compactification, and the Bers compactification. The mapping class group acts on the Thurston compactification continuously, but neither on the Teichmüller compactification nor on the Bers compactification. We introduce quotient spaces of the Teichmüller boundary and the Bers boundary to which the action of the mapping class group extends continuously, and show that any self-homeomorphism on these quotient spaces is induced from a unique extended mapping class.
235
254
1
10.4171/117-1/6
http://www.ems-ph.org/doi/10.4171/117-1/6
Arc geometry and algebra: foliations, moduli spaces, string topology and field theory
Ralph
Kaufmann
Purdue University, WEST LAFAYETTE, UNITED STATES
Several complex variables and analytic spaces
Category theory; homological algebra
Algebraic topology
Manifolds and cell complexes
By defining natural gluing operations, we combine the classical approach to Teichmüller theory via foliations with the operadic approach to field theory. The result is a universal structure, which governs surprisingly many algebraic and geometric constructions in and around moduli spaces of curves, string (field) theory and string topology. Applications include solutions to Deligne’s conjecture, its cyclic version and the definition of string topology operations. For the algebraic operations, such as the Gerstenhaber bracket and the BV operator, we use a discretized version of the foliations. Interpreted as chain level operations, these give natural definitions of up to homotopy algebras, which have the benefit of being concrete and small enough to be handleable, but large enough to contain all the necessary homotopies, such as those giving rise to brace operations. There are other natural modifications, which we discuss, that lead to moduli space actions, actions of certain compactifications, an open/closed generalization, and the possibility of stabilization in the case of semi-simple algebras.
255
325
1
10.4171/117-1/7
http://www.ems-ph.org/doi/10.4171/117-1/7
The horoboundary and isometry group of Thurston’s Lipschitz metric
Cormac
Walsh
Ecole Polytechnique, PALAISEAU CEDEX, FRANCE
General
We show that the horofunction boundary of Teichmüller space with Thurston’s Lipschitz metric is the same as the Thurston boundary. We use this to determine the isometry group of the Lipschitz metric, apart from in some exceptional cases. We also show that the Teichmüller spaces of different surfaces, when endowed with this metric, are not isometric, again with some possible exceptions of low genus.
327
353
1
10.4171/117-1/8
http://www.ems-ph.org/doi/10.4171/117-1/8
The horofunction compactification of the Teichmüller metric
Lixin
Liu
Sun Yat-sen (Zhongshan) University, GUANGZHOU, GUANGDONG, CHINA
Weixu
Su
Fudan University, SHANGHAI, CHINA
Several complex variables and analytic spaces
Functions of a complex variable
Manifolds and cell complexes
General
We show that the horofunction compactification of Teichmüller space, endowed with the Teichmüller metric, is homeomorphic to the Gardiner–Masur compactification.
355
374
1
10.4171/117-1/9
http://www.ems-ph.org/doi/10.4171/117-1/9
Lipschitz algebras and compactifications of Teichmüller space
Hideki
Miyachi
Osaka University, OSAKA, JAPAN
Functions of a complex variable
Real functions
Functional analysis
General
In this chapter, we develop the function theory on Teichmüller space with the aim of studying the extremal length geometry on that space. We focus on the Lipschitz algebra on Teichmüller space in terms of the Teichmüller distance. We first study the algebraic structure of the Lipschitz algebra by showing a version of the Stone–Weierstrass theorem. Next, we discuss a geometric aspect. Indeed, we show that there is a canonical continuous surjection from the Lipschitz compactification to the Gardiner–Masur compactification of Teichmüller space, and we realize the Gardiner-Masur compactification as a $Q$-compactification, where $Q$ is a subset in the Lipschitz algebra. As a corollary, we conclude that any Lipschitz mapping from a metric space $M$ to Teichmüller space extends continuously from the Lipschitz compactification of $M$ to the Gardiner–Masur compactification. We also define a subalgebra $\mathcal{GM}$ of the Lipschitz algebra on Teichmüller space such that the $\mathcal{GM}$-compactification coincides with the Gardiner–Masur compactification.
375
413
1
10.4171/117-1/10
http://www.ems-ph.org/doi/10.4171/117-1/10
On the geodesic geometry of infinite-dimensional Teichmüller spaces
Zhong
Li
Peking University, BEIJING, CHINA
Functions of a complex variable
Several complex variables and analytic spaces
A finite-dimensional Teichmüller space is a geodesic straight space in the sense of Busemann. However, any infinite-dimensional Teichmüller space is not a geodesic straight space. At some kind of points in infinite-dimensional Teichmüller spaces, the geometric behavior of these points with respect to the base point is very “bad”. Such phenomena were sufficiently investigated. The study led to a complete characterization of such points and promoted a better understanding of the geometry of Teichmüller spaces. This topic is closely connected with the extremal problem of quasi-conformal mappings.
415
437
1
10.4171/117-1/11
http://www.ems-ph.org/doi/10.4171/117-1/11
Holomorphic families of Riemann surfaces and monodromy
Hiroshige
Shiga
Tokyo Institute of Technology, TOKYO, JAPAN
General
Holomorphic families of Riemann surfaces appear in various fields of mathematics. In this chapter, we shall give a comprehensive view of the role of holomorphic families of Riemann surfaces and their monodromies in related fields.
439
458
1
10.4171/117-1/12
http://www.ems-ph.org/doi/10.4171/117-1/12
The deformation of flat affine structures on the two-torus
Oliver
Baues
Georg-August Universität Göttingen, GÖTTINGEN, GERMANY
Differential geometry
Number theory
Group theory and generalizations
Manifolds and cell complexes
The group action which defines the moduli problem for the deformation space of flat affine structures on the two-torus is the action of the affine group Aff(2) on $\mathbb{R}^2$. Since this action has non-compact stabiliser GL(2,$\mathbb{R}$), the underlying locally homogeneous geometry is highly non-Riemannian. In this chapter, we describe the deformation space of all flat affine structures on the two-torus. In this context interesting phenomena arise in the topology of the deformation space, which, for example, is not a Hausdorff space. This contrasts with the case of constant curvature metrics, or conformal structures on surfaces, which are encountered in classical Teichmüller theory. As our main result on the space of deformations of flat affine structures on the two-torus we prove that the holonomy map from the deformation space to the variety of conjugacy classes of homomorphisms from the fundamental group of the two-torus to the affine group is a local homeomorphism.
461
537
1
10.4171/117-1/13
http://www.ems-ph.org/doi/10.4171/117-1/13
Higher Teichmüller spaces: from SL(2,$\mathbb{R}$) to other Lie groups
Marc
Burger
ETH Zentrum, ZÜRICH, SWITZERLAND
Alessandra
Iozzi
ETH Zentrum, ZÜRICH, SWITZERLAND
Anna
Wienhard
Universität Heidelberg, HEIDELBERG, GERMANY
Algebraic geometry
Topological groups, Lie groups
Several complex variables and analytic spaces
We review the Fricke–Klein space of a surface with puncture from the point of view of the bounded Euler class and Goldman’s theorem and we extend this point of view to the study of representations of fundamental groups of punctures surfaces into Lie groups of Hermitian type.
539
618
1
10.4171/117-1/14
http://www.ems-ph.org/doi/10.4171/117-1/14
The theory of quasiconformal mappings in higher dimensions, I
Gaven
Martin
Massey University, AUCKLAND, NEW ZEALAND
Functions of a complex variable
General
This chapter presents a survey of the many and various elements of the modern higher-dimensional theory of quasiconformal mappings and their wide and varied application. It is unified (and limited) by the theme of the author’s interests. Thus we will discuss the basic theory as it developed in the 1960s in the early work of F. W. Gehring and Yu G. Reshetnyak and subsequently explore the connections with geometric function theory, nonlinear partial differential equations, differential and geometric topology and dynamics as they ensued over the following decades. We give few proofs as we try to outline the major results of the area and current research themes. We do not strive to present these results in maximal generality, as to achieve this considerable technical knowledge would be necessary of the reader. We have tried to give a feel of where the area is, what are the central ideas and problems and where are the major current interactions with researchers in other areas. We have also added a bit of history here and there. We have not been able to cover the many recent advances generalising the theory to mappings of finite distortion and to degenerate elliptic Beltrami systems which connects the theory closely with the calculus of variations and nonlinear elasticity, nonlinear Hodge theory and related areas, although the reader may see shadows of this aspect in parts. In the sequel (with Bruce Palka) we will give a more detailed account of the basic techniques and how they are used with a view to providing tools for researchers who may come in contact with higher-dimensional quasiconformal mappings from their own area.
619
677
1
10.4171/117-1/15
http://www.ems-ph.org/doi/10.4171/117-1/15
Infinite-dimensional Teichmüller spaces and modular groups
Katsuhiko
Matsuzaki
Waseda University, TOKYO, JAPAN
Functions of a complex variable
Dynamical systems and ergodic theory
General
In this chapter, we give a synthetic treatment concerning recent theories on the dynamics of Teichmüller modular groups for Riemann surfaces of infinite topological type. We consider quotient spaces of the Teichmüller space by subgroups of the modular group in order to find a moduli space in an appropriate sense. Among a glossary of fundamental results on this subject matter, we emphasize the construction of the stable moduli space and the enlarged moduli space, both of which are regarded as generalizations of the concept of moduli spaces from finite type Riemann surfaces to infinite type surfaces.
681
716
1
10.4171/117-1/16
http://www.ems-ph.org/doi/10.4171/117-1/16
Teichmüller spaces and holomorphic dynamics
Xavier
Buff
Université Paul Sabatier, TOULOUSE CEDEX, FRANCE
Guizhen
Cui
Chinese Academy of Sciences, BEIJING, CHINA
Lei
Tan
Université d'Angers, ANGERS CEDEX, FRANCE
General
One fundamental theorem in the theory of holomorphic dynamics is Thurston’s topological characterization of postcritically finite rational maps. Its proof is a beautiful application of Teichmüller theory. In this chapter we provide a self-contained proof of a slightly generalized version of Thurston’s theorem (the marked Thurston’s theorem). We also mention some applications and related results, as well as the notion of deformation spaces of rational maps introduced by A. Epstein.
717
756
1
10.4171/117-1/17
http://www.ems-ph.org/doi/10.4171/117-1/17
A survey of quantum Teichmüller space and Kashaev algebra
Ren
Guo
Oregon State University, CORVALLIS, UNITED STATES
Manifolds and cell complexes
Group theory and generalizations
General
In this chapter, we survey the algebraic aspects of quantum Teichmüller space, generalized Kashaev algebra and a natural relationship between the two algebras.
759
784
1
10.4171/117-1/18
http://www.ems-ph.org/doi/10.4171/117-1/18
Variable Riemann surfaces
Oswald
Teichmüller
, Berlin, GERMANY
General
This is the English translation of Teichmüller’s article “Veränderliche Riemannsche Flächen”, Deutsche Math. 7 (1944), 344–359.
787
803
1
10.4171/117-1/19
http://www.ems-ph.org/doi/10.4171/117-1/19
A commentary on Teichmüller’s paper Veränderliche Riemannsche Flächen
Annette
A’Campo-Neuen
Universität Basel, BASEL, SWITZERLAND
Norbert
A’Campo
Universität Basel, BASEL, SWITZERLAND
Lizhen
Ji
University of Michigan, ANN ARBOR, UNITED STATES
Athanase
Papadopoulos
Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE
Global analysis, analysis on manifolds
Partial differential equations
General
This is a commentary on Teichmüller’s article “Veränderliche Riemannsche Flächen”, Deutsche Math. 7 (1944), 344–359.
787
814
1
10.4171/117-1/20
http://www.ems-ph.org/doi/10.4171/117-1/20
Tractability of Multivariate Problems
Volume III: Standard Information for Operators
Erich
Novak
University of Jena, Germany
Henryk
Woźniakowski
Columbia University, New York, USA, and University of Warsaw, Poland
Numerical analysis
65-02; 65Y20, 68Q17, 68Q25, 41A63, 65-02, 46E22, 28C20, 46E30, 65N99, 65R20
Multivariate approximation, linear problems, quasilinear problems, Poisson equation, Fredholm equation, power of function values, worst case setting, average case setting, randomized setting, high-dimensional numerical problems, Smolyak and weighted tensor product algorithms, weighted spaces, tractability, curse of dimension
This three-volume set is a comprehensive study of the tractability of multivariate problems. Volume I covers algorithms using linear information consisting of arbitrary continuous linear functionals. Volumes II and III are devoted to algorithms using standard information consisting of function values. Approximation of linear and selected nonlinear functionals is dealt with in volume II, and linear and selected nonlinear operators are studied in volume III. To a large extent, volume III can be read independently of volumes I and II. The most important example studied in volume III is the approximation of multivariate functions. It turns out that many other linear and some nonlinear problems are closely related to the approximation of multivariate functions. While the lower bounds obtained in volume I for the class of linear information also yield lower bounds for the standard class of function values, new techniques for upper bounds are presented in volume III. One of the main issues here is to verify when the power of standard information is nearly the same as the power of linear information. In particular, for the approximation problem defined over Hilbert spaces, the power of standard and linear information is the same in the randomized and average case (with Gaussian measures) settings, whereas in the worst case setting this is not true. The book is of interest to researchers working in computational mathematics, especially in approximation of high-dimensiona problems. It may be well suited for graduate courses and seminars. The text contains 58 open problems for future research in tractability.
10
29
2012
978-3-03719-116-3
978-3-03719-616-8
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/116
http://www.ems-ph.org/doi/10.4171/116
EMS Tracts in Mathematics
18
Derived Categories in Algebraic Geometry
Tokyo 2011
Yujiro
Kawamata
University of Tokyo, Japan
Algebraic geometry
Commutative rings and algebras
Associative rings and algebras
Primary: 13D09, 14-02, 14-06, 14F05, 16E35, 18E30; Secondary: 13D02, 13D10, 14A22, 14E08, 14E16, 14E30, 14J32, 14J33, 14J45, 14K05, 14L24, 14N35, 16E05, 16E40, 16F60, 18G10, 20G05, 20G15
Algebraic variety, derived category, triangulated category, Fourier–Mukai transform, cluster algebra, birational geometry, semi-orthogonal decomposition, exceptional collection, minimal model, flop, McKay correspondence, categorical action, abelian variety, non-commutative algebraic geometry, mirror symmetry, Donaldson–Thomas theory
The study of derived categories is a subject that attracts increasingly many young mathematicians from various fields of mathematics, including abstract algebra, algebraic geometry, representation theory and mathematical physics. The concept of the derived category of sheaves was invented by Grothendieck and Verdier in the 1960s as a tool to express important results in algebraic geometry such as the duality theorem. In the 1970s, Beilinson, Gelfand and Gelfand discovered that a derived category of an algebraic variety may be equivalent to that of a finite dimensional non-commutative algebra, and Mukai found that there are non-isomorphic algebraic varieties that have equivalent derived categories. In this way the derived category provides a new concept that has many incarnations. In the 1990s, Bondal and Orlov uncovered an unexpected parallelism between derived categories and birational geometry. Kontsevich’s homological mirror symmetry provided further motivation for the study of derived categories. This book is the proceedings of a conference held at the University of Tokyo in January 2011 on the current status of the research on derived categories related to algebraic geometry. Most articles are survey papers on this rapidly developing field. The book is suitable for young mathematicians who want to enter this exciting field. Some basic knowledge of algebraic geometry is assumed.
1
7
2013
978-3-03719-115-6
978-3-03719-615-1
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/115
http://www.ems-ph.org/doi/10.4171/115
EMS Series of Congress Reports
Categorical representability and intermediate Jacobians of Fano threefolds
Marcello
Bernardara
Université Paul Sabatier, TOULOUSE CEDEX 9, FRANCE
Michele
Bolognesi
Université de Montpellier, MONTPELLIER, FRANCE
Algebraic geometry
Category theory; homological algebra
General
We define, basing upon semiorthogonal decompositions of ${\rm D}^{\rm b}(X)$, categorical representability of a projective variety $X$ and describe its relation with classical representabilities of the Chow ring. For complex threefolds satisfying both classical and categorical representability assumptions, we reconstruct the intermediate Jacobian from the semiorthogonal decomposition. We discuss finally how categorical representability can give useful information on the birational properties of $X$ by providing examples and stating open questions.
1
25
1
10.4171/115-1/1
http://www.ems-ph.org/doi/10.4171/115-1/1
Fourier–Mukai functors: a survey
Alberto
Canonaco
Università degli Studi di Pavia, PAVIA, ITALY
Paolo
Stellari
Università degli Studi di Milano, MILANO, ITALY
Algebraic geometry
Category theory; homological algebra
General
In this article some recent results about Fourier–Mukai functors are reported. In particular, given an exact functor between the bounded derived categories of coherent sheaves on two smooth projective varieties, we deal with the question whether this functor is of Fourier–Mukai type. Several related questions are answered and many open problems are stated.
27
60
1
10.4171/115-1/2
http://www.ems-ph.org/doi/10.4171/115-1/2
Flops and about: a guide
Sabin
Cautis
University of Southern California, LOS ANGELES, UNITED STATES
Algebraic geometry
Nonassociative rings and algebras
General
Stratified flops show up in the birational geometry of symplectic varieties such as resolutions of nilpotent orbits and moduli spaces of sheaves. Constructing derived equivalences between varieties related by such flops is, strangely enough, related to areas in representation theory and knot homology. In this paper we discuss how to construct such equivalences, explain the main tool for doing this (categorical Lie algebra actions) and comment on various related topics.
61
101
1
10.4171/115-1/3
http://www.ems-ph.org/doi/10.4171/115-1/3
A note on derived categories of Fermat varieties
Akira
Ishii
Hiroshima University, HIROSHIMA, JAPAN
Kazushi
Ueda
Osaka University, OSAKA, JAPAN
Category theory; homological algebra
Associative rings and algebras
General
We show that the quotient stack of a Fermat variety with respect to a natural abelian group action has a full strong exceptional collection consisting of invertible sheaves. We also discuss a description of the derived category of the Fermat variety in terms of the coherent action of the group of characters on the derived category of the quotient stack.
103
110
1
10.4171/115-1/4
http://www.ems-ph.org/doi/10.4171/115-1/4
Homology of infinite loop spaces
Dmitry
Kaledin
Steklov Mathematical Institute, MOSCOW, RUSSIAN FEDERATION
Algebraic topology
General
We prove a simple homological expression for the homology of a connected spectrum represented by an infinite loop space via the Segal machine. The expression is essentially due to Pirashvili but not stated explicitly in his work; we give an independent proof.
111
121
1
10.4171/115-1/5
http://www.ems-ph.org/doi/10.4171/115-1/5
Cluster algebras and derived categories
Bernhard
Keller
Université Paris Diderot - Paris 7, PARIS CEDEX 13, FRANCE
Commutative rings and algebras
Associative rings and algebras
General
This is an introductory survey on cluster algebras and their (additive) categorification using derived categories of Ginzburg algebras. After a gentle introduction to cluster combinatorics, we review important examples of coordinate rings admitting a cluster algebra structure. We then present the general definition of a cluster algebra and describe the interplay between cluster variables, coefficients, $c$-vectors and $g$-vectors. We show how $c$-vectors appear in the study of quantum cluster algebras and their links to the quantum dilogarithm. We then present the framework of additive categorification of cluster algebras based on the notion of quiver with potential and on the derived category of the associated Ginzburg algebra. We show how the combinatorics introduced previously lift to the categorical level and how this leads to proofs, for cluster algebras associated with quivers, of some of Fomin–Zelevinsky’s fundamental conjectures.
123
183
1
10.4171/115-1/6
http://www.ems-ph.org/doi/10.4171/115-1/6
Some derived equivalences between noncommutative schemes and algebras
Izuru
Mori
Shizuoka University, SHIZUOKA, JAPAN
Associative rings and algebras
General
It is interesting to find a (non-affine) smooth algebraic variety which is derived equivalent to a noncommutative algebra. A full strong exceptional sequence, a tilting generator, and McKay correspondence provide such examples. In this survey paper, we will show that similar derived equivalences exist between noncommutative schemes and noncommutative algebras.
185
196
1
10.4171/115-1/7
http://www.ems-ph.org/doi/10.4171/115-1/7
Lagrangian-invariant sheaves and functors for abelian varieties
Alexander
Polishchuk
University of Oregon, EUGENE, UNITED STATES
Algebraic geometry
General
We partially generalize the theory of semihomogeneous bundles on an abelian variety $A$ developed by Mukai, Semi-homogeneous vector bundles on an abelian variety, J. Math. Kyoto Univ. 18 (1978), 239–272. This involves considering abelian subvarieties $Y\subset X_A=A\times\hat{A}$ and studying coherent sheaves on $A$ invariant under the action of $Y$. The natural condition to impose on $Y$ is that of being Lagrangian with respect to a certain skew-symmetric biextension $\mathcal{E}$ of $X_A\times X_A$ by $\mathbb{G}_m$. We prove that in this case any $Y$-invariant sheaf is a direct sum of several copies of a single coherent sheaf. We call such sheaves Lagrangian-invariant (or LI-sheaves). We also study LI-functors $D^b(A)\to D^b(B)$ associated with kernels in $D^b(A\times B)$ that are invariant with respect to some Lagrangian subvariety in $X_A\times X_B$. We calculate their composition and prove that in characteristic zero it can be decomposed into a direct sum of LI-functors. In the case $B=A$ this leads to an interesting central extension of the group of symplectic automorphisms of $X_A$ in the category of abelian varieties up to isogeny.
197
250
1
10.4171/115-1/8
http://www.ems-ph.org/doi/10.4171/115-1/8
Generic vanishing filtrations and perverse objects in derived categories of coherent sheaves
Mihnea
Popa
University of Illinois at Chicago, CHICAGO, UNITED STATES
Algebraic geometry
Commutative rings and algebras
General
The paper establishes a natural relationship between generic vanishing in the context of vanishing theorems and birational geometry, and perverse coherent sheaves in the context of derived categories. It includes criteria for checking these conditions, and applications.
251
278
1
10.4171/115-1/9
http://www.ems-ph.org/doi/10.4171/115-1/9
The fundamental group is not a derived invariant
Christian
Schnell
Stony Brook University, STONY BROOK, UNITED STATES
Algebraic geometry
General
We show that the fundamental group is not invariant under derived equivalence of smooth projective varieties.
279
285
1
10.4171/115-1/10
http://www.ems-ph.org/doi/10.4171/115-1/10
Introduction and open problems of Donaldson–Thomas theory
Yukinobu
Toda
The University of Tokyo, Kashiwa, JAPAN
Algebraic geometry
Category theory; homological algebra
General
In this article, we give an introduction to Donaldson–Thomas theory and a survey of its recent developments. We also discuss some open problems related to Donaldson–Thomas theory.
287
318
1
10.4171/115-1/11
http://www.ems-ph.org/doi/10.4171/115-1/11
Notes on formal deformations of abelian categories
Michel
Van den Bergh
Hasselt University, HASSELT, BELGIUM
Commutative rings and algebras
Algebraic geometry
Category theory; homological algebra
General
In these notes we provide the foundation for the deformation-theoretic parts of arXiv:0807.3753 and arXiv:math/0102005.
319
344
1
10.4171/115-1/12
http://www.ems-ph.org/doi/10.4171/115-1/12
Contributions to Algebraic Geometry
Impanga Lecture Notes
Piotr
Pragacz
IM PAN, Warsaw, Poland
Algebraic geometry
Number theory
Several complex variables and analytic spaces
Differential geometry
11S15, 13D10, 14-02, 14B05, 14B12, 14C17, 14C20, 14C35, 14D06, 14D15, 14D20, 14D23, 14E15, 14E30, 14F43, 14H10, 14H40, 14H42, 14J17, 14J28, 14J30, 14J32, 14J50, 14J70, 14K10, 14K25, 14L30, 14M15, 14M17, 14M25, 14N10, 14N15, 32G10, 32Q45, 34A30, 53D05, 55N91; 01-02, 01A70, 05E05, 11S85, 13A35, 13D10, 14C30, 14F18, 14J26, 14J32, 14N20, 19D55, 32S25, 53D20, 57R45
K3 surface, Enriques surface, Calabi–Yau threefold, linear system, Seshadri constant, multiplier ideal, differential form, log canonical treshold, Mori theory, canonical ring, deformation of morphism, Hodge numbers, logarithmic differential form, Prym variety, moduli space, syzygy, Wro´nski determinant, linear ODE, ramification locus, Schubert calculus, Grassmannian, Schubert variety, Schur function, singularity, Thom polynomial, P-ideal, equivariant localization, toric variety, symplectic manifold, toric stack, symplectic quotient, equivariant cohomology, Bloch group, field extension, norm
The articles in this volume are the outcome of the Impanga Conference on Algebraic Geometry in 2010 at the Banach Center in Będlewo. The following spectrum of topics is covered: K3 surfaces and Enriques surfaces; Prym varieties and their moduli; invariants of singularities in birational geometry; differential forms on singular spaces; Minimal Model Program; linear systems; toric varieties; Seshadri and packing constants; equivariant cohomology; Thom polynomials; arithmetic questions. The main purpose of the volume is to give comprehensive introductions to the above topics through texts starting from an elementary level and ending with the discussion of current research. The first four topics are represented by the notes from the minicourses held during the conference. In the articles the reader will find classical results and methods, as well as modern ones. The book is addressed to researchers and graduate students in algebraic geometry, singularity theory and algebraic topology. Most of the material exposed in the volume has not yet appeared in book form.
8
15
2012
978-3-03719-114-9
978-3-03719-614-4
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/114
http://www.ems-ph.org/doi/10.4171/114
EMS Series of Congress Reports
The influence of Oscar Zariski on algebraic geometry
Piotr
Blass
Ulam University, Boynton Beach, UNITED STATES
Algebraic geometry
History and biography
General
We describe some aspects of the life and mathematical work of Oscar Zariski. In particular, we discuss the influence of his work and the work of his students on modern algebraic geometry.
1
15
1
10.4171/114-1/1
http://www.ems-ph.org/doi/10.4171/114-1/1
The geometry of T-varieties
Klaus
Altmann
Freie Universität Berlin, BERLIN, GERMANY
Nathan
Ilten
, BONN, GERMANY
Lars
Petersen
Freie Universität Berlin, BERLIN, GERMANY
Hendrik
Süß
Brandenburgische Technische Universität Cottbus, COTTBUS, GERMANY
Robert
Vollmert
Freie Universität Berlin, BERLIN, GERMANY
Algebraic geometry
General
This is a survey of the language of polyhedral divisors describing T-varieties. This language is explained in parallel to the well established theory of toric varieties. In addition to basic constructions, subjects touched on include singularities, separatedness and properness, divisors and intersection theory, cohomology, Cox rings, polarizations, and equivariant deformations, among others.
17
69
1
10.4171/114-1/2
http://www.ems-ph.org/doi/10.4171/114-1/2
Introduction to equivariant cohomology in algebraic geometry
Dave
Anderson
University of Washington, SEATTLE, UNITED STATES
Algebraic geometry
Combinatorics
General
Introduced by Borel in the late 1950s, equivariant cohomology encodes information about how the topology of a space interacts with a group action. Quite some time passed before algebraic geometers picked up on these ideas, but in the last twenty years, equivariant techniques have found many applications in enumerative geometry, Gromov–Witten theory, and the study of toric varieties and homogeneous spaces. In fact, many classical algebro-geometric notions, going back to the degeneracy locus formulas of Giambelli, are naturally statements about certain equivariant cohomology classes. These lectures survey some of the main features of equivariant cohomology at an introductory level. The first part is an overview, including basic definitions and examples. In the second lecture, I discuss one of the most useful aspects of the theory: the possibility of localizing at fixed points without losing information. The third lecture focuses on Grassmannians, and describes some recent positivity results about their equivariant cohomology rings.
71
92
1
10.4171/114-1/3
http://www.ems-ph.org/doi/10.4171/114-1/3
Recent developments and open problems in linear series
Thomas
Bauer
Philipps-Universität, MARBURG, GERMANY
Cristiano
Bocci
Università di Siena, SIENA, ITALY
Susan
Cooper
University of Nebraska-Lincoln, LINCOLN, UNITED STATES
Sandra
Di Rocco
Royal Institute of Technology, STOCKHOLM, SWEDEN
Marcin
Dumnicki
Jagiellonian University, KRAKÓW, POLAND
Brian
Harbourne
University of Nebraska-Lincoln, LINCOLN, UNITED STATES
Anders
Lindquist
Royal Institute of Technology, STOCKHOLM, SWEDEN
Hans
Munthe-Kaas
University of Bergen, BERGEN, NORWAY
Alex
Küronya
Budapest University of Technology & Economics, BUDAPEST, HUNGARY
Rick
Miranda
Colorado State University, FORT COLLINS, UNITED STATES
Joaquim
Roé
Universitat Autònoma de Barcelona, BELLATERRA, SPAIN
Henry
Schenck
University of Illinois, URBANA, UNITED STATES
Tomasz
Szemberg
Krakow Pedagogical Academy, KRAKÓW, POLAND
Zach
Teitler
Boise State University, BOISE, UNITED STATES
Algebraic geometry
General
This article presents problems discussed during the Mini-Workshop Linear Series on Algebraic Varieties held in the week of October 3–9, 2010, at the Mathematisches Forschungsinstitut at Oberwolfach, as well as results obtained there and shortly after the Workshop. The discussion revolves mainly around the speciality and the postulation problems as well as the containment problems for various powers of ideals. The main motivations originated in the Harbourne-Hirschowitz, Nagata and the Bounded Negativity Conjectures.
93
140
1
10.4171/114-1/4
http://www.ems-ph.org/doi/10.4171/114-1/4
Moduli of map germs, Thom polynomials and the Green–Griffiths conjecture
Gergely
Bérczi
Oxford University, OXFORD, UNITED KINGDOM
Algebraic topology
Algebraic geometry
Several complex variables and analytic spaces
General
This survey paper – based on my IMPANGA lectures given in the Banach Center, Warsaw in January 2011 – studies the moduli of holomorphic map germs from the complex line into complex compact manifolds with applications in global singularity theory and the theory of hyperbolic algebraic varieties.
141
167
1
10.4171/114-1/5
http://www.ems-ph.org/doi/10.4171/114-1/5
The Minimal Model Program revisited
Paolo
Cascini
Imperial College London, LONDON, UNITED KINGDOM
Vladimir
Lazić
Universität Bayreuth, BAYREUTH, GERMANY
General
We give a light introduction to some recent developments in Mori theory, and to our recent direct proof of the finite generation of the canonical ring.
169
187
1
10.4171/114-1/6
http://www.ems-ph.org/doi/10.4171/114-1/6
Invariants of hypersurfaces and logarithmic differential forms
Sławomir
Cynk
Jagiellonian University, KRAKÓW, POLAND
Sławomir
Rams
Jagiellonian University, KRAKÓW, POLAND
Algebraic geometry
Several complex variables and analytic spaces
General
In this survey paper we discuss various results on invariants of a resolution of a singular projective hypersurface. In particular, we prove certain defect-type formulae. In our proofs we make essential use of the sheaf of logarithmic differential forms.
189
213
1
10.4171/114-1/7
http://www.ems-ph.org/doi/10.4171/114-1/7
Prym varieties and their moduli
Gavril
Farkas
Humboldt-Universität zu Berlin, BERLIN, GERMANY
Algebraic geometry
Ordinary differential equations
General
This survey discusses the geometry of the moduli space of Prym varieties. Several applications of Prym in algebraic geometry are presented. The paper begins with with a historical discussion of the life and achievements of Friedrich Prym. Topics treated in subsequent sections include singularities and Kodaira dimension of the moduli space, syzygies of Prym-canonical embedding and the geometry of the moduli space $\mathcal{R}_g$ in small genus.
215
255
1
10.4171/114-1/8
http://www.ems-ph.org/doi/10.4171/114-1/8
On generalized Wrońskians
Letterio
Gatto
Politecnico di Torino, TORINO, ITALY
Inna
Scherbak
Tel Aviv University, TEL AVIV, ISRAEL
Algebraic geometry
Ordinary differential equations
General
The Wroński determinant (Wrońskian), usually introduced in standard courses in Ordinary Differential Equations (ODE), is a very useful tool in algebraic geometry to detect ramification loci of linear systems. The present survey aims to describe some “materializations” of the Wrońskian and of its close relatives, the generalized Wrońskians, in algebraic geometry. Emphasis will be put on the relationships between Schubert Calculus and ODE.
257
295
1
10.4171/114-1/9
http://www.ems-ph.org/doi/10.4171/114-1/9
Lines crossing a tetrahedron and the Bloch group
Kevin
Hutchinson
University College Dublin, DUBLIN, IRELAND
Masha
Vlasenko
Trinity College, DUBLIN, IRELAND
Algebraic geometry
$K$-theory
General
According to B. Totaro (Milnor K-theory is the simplest part of algebraic K-theory, K-theory 6 (1992), 177–189), there is a hope that the Chow groups of a field $k$ can be computed using a very small class of affine algebraic varieties (linear spaces in the right coordinates), whereas the current definition uses essentially all algebraic cycles in affine space. In this note we consider a simple modification of $\mathrm{CH}^2(\operatorname{Spec} (k),3)$ using only linear subvarieties in affine spaces and show that it maps surjectively to the Bloch group $B(k)$ for any infinite field $k$. We also describe the kernel of this map.
297
303
1
10.4171/114-1/10
http://www.ems-ph.org/doi/10.4171/114-1/10
On complex and symplectic toric stacks
Andreas
Hochenegger
Freie Universität Berlin, BERLIN, GERMANY
Frederik
Witt
Universität Münster, MÜNSTER, GERMANY
Algebraic geometry
Differential geometry
General
Toric varieties play an important role both in symplectic and complex geometry. In symplectic geometry, the construction of a symplectic toric manifold from a smooth polytope is due to Delzant [D]. In algebraic geometry, there is a more general construction using fans rather than polytopes. However, in case the fan is induced by a smooth polytope Audin [Au] showed both constructions to give isomorphic projective varieties. For rational but not necessarily smooth polytopes the Delzant construction was refined by Lerman and Tolman [LT], leading to symplectic toric orbifolds or more generally, symplectic toric DM stacks [LM]. We show that the stacks resulting from the Lerman–Tolman construction are isomorphic to the stacks obtained by Borisov et al. [BCS] in case the stacky fan is induced by a polytope. No originality is claimed (cf. also the article by Sakai [S]). Rather we hope that this text serves as an example driven introduction to symplectic toric geometry for the algebraically minded reader.
305
331
1
10.4171/114-1/11
http://www.ems-ph.org/doi/10.4171/114-1/11
Deformation along subsheaves, II
Clemens
Jörder
Universität Freiburg, FREIBURG, GERMANY
Stefan
Kebekus
Universität Freiburg, FREIBURG, GERMANY
Algebraic geometry
Commutative rings and algebras
Several complex variables and analytic spaces
General
Let $f \colon Y \to X$ be the inclusion map of a compact reduced subspace of a complex manifold, and let $\mathcal{F} \subseteq T_X$ be a subsheaf of the tangent bundle which is closed under the Lie bracket, but not necessarily a sheaf of $\mathcal{O}_X$-algebras. This paper discusses criteria to guarantee that infinitesimal deformations of $f$ which are induced by $\mathcal{F}$ lift to positive-dimensional deformations of $f$, where $f$ is deformed “along the sheaf $\mathcal{F}$”. In case where $X$ is complex-symplectic and $\mathcal{F}$ the sheaf of locally Hamiltonian vector fields, this partially reproduces known results on unobstructedness of deformations of Lagrangian submanifolds. The proof is rather elementary and geometric, constructing higher-order liftings of a given infinitesimal deformation using flow maps of carefully crafted time-dependent vector fields.
333
357
1
10.4171/114-1/12
http://www.ems-ph.org/doi/10.4171/114-1/12
Some degenerations of $G_2$ and Calabi–Yau varieties
Michał
Kapustka
Jagiellonian University, KRAKÓW, POLAND
Algebraic geometry
General
We introduce a variety $\hat{G}_2$ parameterizing isotropic five-spaces of a general degenerate four-form in a seven dimensional vector space. It is in a natural way a degeneration of the variety $G_2$, the adjoint variety of the simple Lie group $\mathbb{G}_2$. It occurs that it is also the image of $\mathbb{P}^5$ by a system of quadrics containing a twisted cubic. Degenerations of this twisted cubic to three lines give rise to degenerations of $G_2$ which are toric Gorenstein Fano fivefolds. We use these two degenerations to construct geometric transitions between Calabi–Yau threefolds. We prove moreover that every polarized K3 surface of Picard number 2, genus 10, and admitting a $g^1_5$ appears as linear sections of the variety $\hat{G}_2$.
359
373
1
10.4171/114-1/13
http://www.ems-ph.org/doi/10.4171/114-1/13
Notes on Kebekus’ lectures on differential forms on singular spaces
Mateusz
Michałek
Jagiellonian University, KRAKÓW, POLAND
Algebraic geometry
General
The present paper contains the notes taken by Mateusz Michałek from a series of lectures by Stefan Kebekus during IMPANGA Summer School 2010. The aim of this series of lectures is to give an exposition on the extension of well known results concerning differential forms on manifolds to the case of normal varieties.
375
388
1
10.4171/114-1/14
http://www.ems-ph.org/doi/10.4171/114-1/14
Lecture notes on K3 and Enriques surfaces Notes by Sławomir Rams
Shigeru
Mukai
Kyoto University, KYOTO, JAPAN
Algebraic geometry
General
The main aim of these lectures is to study the connection between symplectic symmetries of $K3$ surfaces and the Mathieu group $M_{24}$, and its Enriques analogy, that is, a conjectural connection between semi-symplectic symmetries of Enriques surfaces and another Mathieu group $M_{12}$.
389
405
1
10.4171/114-1/15
http://www.ems-ph.org/doi/10.4171/114-1/15
IMPANGA lecture notes on log canonical thresholds Notes by Tomasz Szemberg
Mircea
Mustaţă
University of Michigan, ANN ARBOR, UNITED STATES
Algebraic geometry
Commutative rings and algebras
General
We give an introduction to log canonical thresholds, and discuss some open problems and recent progress.
407
442
1
10.4171/114-1/16
http://www.ems-ph.org/doi/10.4171/114-1/16
On Schur function expansions of Thom polynomials
Özer
Öztürk
Mimar Sinan Fine Arts University, BEŞIKTAŞ/ISTANBUL, TURKEY
Piotr
Pragacz
Polish Academy of Sciences, WARSAW 10, POLAND
Combinatorics
Algebraic geometry
Manifolds and cell complexes
General
We discuss computations of the Thom polynomials of singularity classes of maps in the basis of Schur functions. We survey the known results about the bound on the length and a rectangle containment for partitions appearing in such Schur function expansions. We describe several recursions for the coefficients. For some singularities, we give old and new computations of their Thom polynomials.
443
479
1
10.4171/114-1/17
http://www.ems-ph.org/doi/10.4171/114-1/17
A note on the kernel of the norm map
Marek
Szyjewski
, KATOWICE, POLAND
Number theory
Field theory and polynomials
General
We investigate kernel of the norm map on power classes for cyclic field extensions.
481
487
1
10.4171/114-1/18
http://www.ems-ph.org/doi/10.4171/114-1/18
Seshadri and packing constants
Halszka
Tutaj-Gasińska
Jagiellonian University, KRAKÓW, POLAND
Algebraic geometry
Differential geometry
General
This note is about a certain connection between Seshadri constants and symplectic packing.
489
499
1
10.4171/114-1/19
http://www.ems-ph.org/doi/10.4171/114-1/19
Contributions to the History of Number Theory in the 20th Century
Peter
Roquette
University of Heidelberg, Germany
History and biography
01-02, 03-03, 11-03, 12-03 , 16-03, 20-03; 01A60, 01A70, 01A75, 11E04, 11E88, 11R18 11R37, 11U10
The 20th century was a time of great upheaval and great progress, mathematics not excluded. In order to get the overall picture of trends, developments and results it is illuminating to look at their manifestations locally, in the personal life and work of people living at the time. The university archives of Göttingen harbor a wealth of papers, letters and manuscripts from several generations of mathematicians – documents which tell us the story of the historic developments from a local point of view. The present book offers a number of essays based on documents from Göttingen and elsewhere – essays which are not yet contained in the author’s Collected Works. These little pieces, independent from each other, are meant as contributions to the imposing mosaic of history of number theory. They are written for mathematicians but with no special background requirements. Involved are the names of Abraham Adrian Albert, Cahit Arf, Emil Artin, Richard Brauer, Otto Grün, Helmut Hasse, Klaus Hoechsmann, Robert Langlands, Heinrich-Wolfgang Leopoldt, Emmy Noether, Abraham Robinson, Ernst Steinitz, Hermann Weyl and others.
1
24
2013
978-3-03719-113-2
978-3-03719-613-7
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/113
http://www.ems-ph.org/doi/10.4171/113
Heritage of European Mathematics
Large Scale Geometry
Piotr
Nowak
IM PAN, Warsaw, Poland
Guoliang
Yu
Texas A&M University, College Station, USA
Geometry
Group theory and generalizations
51-01; 50F99, 20F69; 19K56, 57-01, 46L87, 58B34, 46L99, 53C20
Large scale geometry, quasi-isometry, coarse geometry, asymptotic dimension, finite decomposition complexity, amenability, Property A, coarse embedding, expanders, a-T-menability, coarse homology, uniformly finite homology, Baum–Connes conjecture, coarse Baum–Connes conjecture, Novikov conjecture, Borel conjecture
Large scale geometry is the study of geometric objects viewed from a great distance. The idea of large scale geometry can be traced back to Mostow’s work on rigidity and the work of Švarc, Milnor and Wolf on growth of groups. In the last decades, large scale geometry has found important applications in group theory, topology, geometry, higher index theory, computer science, and large data analysis. This book provides a friendly approach to the basic theory of this exciting and fast growing subject and offers a glimpse of its applications to topology, geometry, and higher index theory. The authors have made a conscientious effort to make the book accessible to advanced undergraduate students, graduate students, and non-experts.
10
10
2012
978-3-03719-112-5
978-3-03719-612-0
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/112
http://www.ems-ph.org/doi/10.4171/112
EMS Textbooks in Mathematics
Complex Analysis
Translated from the Catalan by Ignacio Monreal
Joaquim
Bruna
Universitat Autònoma de Barcelona, Spain
Julià
Cufí
Universitat Autònoma de Barcelona, Spain
Functions of a complex variable
Several complex variables and analytic spaces
30-01, 31-01
Power series, holomorphic function, line integral, differential form, analytic function, zeros and poles, residues, simply connected domain, harmonic function, Dirichlet problem, Poisson equation, conformal mapping, homographic transformation, meromorphic function, infinite product, entire function, interpolation, band-limited function
The theory of functions of a complex variable is a central theme in mathematical analysis that has links to several branches of mathematics. Understanding the basics of the theory is necessary for anyone who wants to have a general mathematical training or for anyone who wants to use mathematics in applied sciences or technology. The book presents the basic theory of analytic functions of a complex variable and their points of contact with other parts of mathematical analysis. This results in some new approaches to a number of topics when compared to the current literature on the subject. Some issues covered are: a real version of the Cauchy–Goursat theorem, theorems of vector analysis with weak regularity assumptions, an approach to the concept of holomorphic functions of real variables, Green’s formula with multiplicities, Cauchy’s theorem for locally exact forms, a study in parallel of Poisson’s equation and the inhomogeneous Cauchy–Riemann equations, the relationship between Green’s function and conformal mapping, the connection between the solution of Poisson’s equation and zeros of holomorphic functions, and the Whittaker–Shannon theorem of information theory. The text can be used as a manual for complex variable courses of various levels and as a reference book. The only prerequisites for reading it is a working knowledge of the topology of the plane and the differential calculus for functions of several real variables. A detailed treatment of harmonic functions also makes the book useful as an introduction to potential theory.
5
6
2013
978-3-03719-111-8
978-3-03719-611-3
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/111
http://www.ems-ph.org/doi/10.4171/111
EMS Textbooks in Mathematics
A Course on Elation Quadrangles
Koen
Thas
Ghent University, Belgium
Combinatorics
Group theory and generalizations
Geometry
05-02, 20-02, 51-02; 05B25, 05E18, 20B25, 20D15, 20D20, 51B25, 51E12
Generalized quadrangle, elation group, Moufang condition, p-group
The notion of elation generalized quadrangle is a natural generalization to the theory of generalized quadrangles of the important notion of translation planes in the theory of projective planes. Almost any known class of finite generalized quadrangles can be constructed from a suitable class of elation quadrangles. In this book the author considers several aspects of the theory of elation generalized quadrangles. Special attention is given to local Moufang conditions on the foundational level, exploring for instance a question of Knarr from the 1990s concerning the very notion of elation quadrangles. All the known results on Kantor’s prime power conjecture for finite elation quadrangles are gathered, some of them published here for the first time. The structural theory of elation quadrangles and their groups is heavily emphasized. Other related topics, such as p-modular cohomology, Heisenberg groups and existence problems for certain translation nets, are briefly touched. The text starts from scratch and is essentially self-contained. Many alternative proofs are given for known theorems. Containing dozens of exercises at various levels, from very easy to rather difficult, this course will stimulate undergraduate and graduate students to enter the fascinating and rich world of elation quadrangles. The more accomplished mathematician will especially find the final chapters challenging.
6
13
2012
978-3-03719-110-1
978-3-03719-610-6
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/110
http://www.ems-ph.org/doi/10.4171/110
EMS Series of Lectures in Mathematics
Topics in Occupation Times and Gaussian Free Fields
Alain-Sol
Sznitman
ETH Zürich, Switzerland
Probability theory and stochastic processes
Statistical mechanics, structure of matter
60K35, 60J27, 60G15, 82B41
Occupation times, Gaussian free field, Markovian loop, random interlacements
This book grew out of a graduate course at ETH Zurich during the Spring term 2011. It explores various links between such notions as occupation times of Markov chains, Gaussian free fields, Poisson point processes of Markovian loops, and random interlacements, which have been the object of intensive research over the last few years. These notions are developed in the convenient set-up of finite weighted graphs endowed with killing measures. The book first discusses elements of continuous-time Markov chains, Dirichlet forms, potential theory, together with some consequences for Gaussian free fields. Next, isomorphism theorems and generalized Ray-Knight theorems, which relate occupation times of Markov chains to Gaussian free fields, are pre- sented. Markovian loops are constructed and some of their key properties derived. The field of occupation times of Poisson point processes of Markovian loops is investigated. Of special interest are its connection to the Gaussian free field, and a formula of Symanzik. Finally, links between random interlacements and Markovian loops are discussed, and some further connections with Gaussian free fields are mentioned.
5
11
2012
978-3-03719-109-5
978-3-03719-609-0
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/109
http://www.ems-ph.org/doi/10.4171/109
Zurich Lectures in Advanced Mathematics
Lectures on Algebraic Categorification
Volodymyr
Mazorchuk
Uppsala University, Sweden
Category theory; homological algebra
Nonassociative rings and algebras
Manifolds and cell complexes
18-01, 18D05; 17B10, 17B55, 18A40, 18E30, 57M27
Categorification, 2-category, Lie algebra, module, category O, functor, knot, Jones polynomial, Verma module, translation functor
The term “categorification” was introduced by Louis Crane in 1995 and refers to the process of replacing set-theoretic notions by the corresponding category-theoretic analogues. This text mostly concentrates on algebraical aspects of the theory, presented in the historical perspective, but also contains several topological applications, in particular, an algebraic (or, more precisely, representation-theoretical) approach to categorification. It consists of fifteen sections corresponding to fifteen one-hour lectures given during a Master Class at Aarhus University, Denmark in October 2010. There are some exercises collected at the end of the text and a rather extensive list of references. Video recordings of all (but one) lectures are available from the Master Class website. The book provides an introductory overview of the subject rather than a fully detailed monograph. Emphasis is on definitions, examples and formulations of the results. Most proofs are either briefly outlined or omitted. However, complete proofs can be found by tracking references. It is assumed that the reader is familiar with the basics of category theory, representation theory, topology and Lie algebra.
3
16
2012
978-3-03719-108-8
978-3-03719-608-3
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/108
http://www.ems-ph.org/doi/10.4171/108
The QGM Master Class Series
Faber Systems and Their Use in Sampling, Discrepancy, Numerical Integration
Hans
Triebel
University of Jena, Germany
Functional analysis
Approximations and expansions
Fourier analysis
Computer science
46-02, 46E35, 42C40, 42B35, 68Q17, 41A55
Function spaces, Haar bases, Faber bases, Faber frames, numerical integration, discrepancy
This book deals first with Haar bases, Faber bases and Faber frames for weighted function spaces on the real line and the plane. It extends results in the author’s book Bases in Function Spaces, Sampling, Discrepancy, Numerical Integration (EMS, 2010) from unweighted spaces (preferably in cubes) to weighted spaces. The obtained assertions are used to study sampling and numerical integration in weighted spaces on the real line and weighted spaces with dominating mixed smoothness in the plane. A short chapter deals with the discrepancy for spaces on intervals. The book is addressed to graduate students and mathematicians having a working knowledge of basic elements of function spaces and approximation theory.
3
16
2012
978-3-03719-107-1
978-3-03719-607-6
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/107
http://www.ems-ph.org/doi/10.4171/107
EMS Series of Lectures in Mathematics
Concentration Compactness for Critical Wave Maps
Joachim
Krieger
EPFL Lausanne, Switzerland
Wilhelm
Schlag
University of Chicago, USA
Partial differential equations
Differential geometry
35L05, 35L52, 53Z05
Energy critical wave equations, wave maps, concentration compactness, Bahouri–Gérard decomposition, Kenig–Merle method, hyperbolic plane
Wave maps are the simplest wave equations taking their values in a Riemannian manifold $(M,g)$. Their Lagrangian is the same as for the scalar equation, the only difference being that lengths are measured with respect to the metric $g$. By Noether's theorem, symmetries of the Lagrangian imply conservation laws for wave maps, such as conservation of energy. In coordinates, wave maps are given by a system of semilinear wave equations. Over the past 20 years important methods have emerged which address the problem of local and global wellposedness of this system. Due to weak dispersive effects, wave maps defined on Minkowski spaces of low dimensions, such as $\mathbb R^{2+1}_{t,x}$, present particular technical difficulties. This class of wave maps has the additional important feature of being energy critical, which refers to the fact that the energy scales exactly like the equation. Around 2000 Daniel Tataru and Terence Tao, building on earlier work of Klainerman–Machedon, proved that smooth data of small energy lead to global smooth solutions for wave maps from 2+1 dimensions into target manifolds satisfying some natural conditions. In contrast, for large data, singularities may occur in finite time for $M =\mathbb S^2$ as target. This monograph establishes that for $\mathbb H$ as target the wave map evolution of any smooth data exists globally as a smooth function. While we restrict ourselves to the hyperbolic plane as target the implementation of the concentration-compactness method, the most challenging piece of this exposition, yields more detailed information on the solution. This monograph will be of interest to experts in nonlinear dispersive equations, in particular to those working on geometric evolution equations.
2
9
2012
978-3-03719-106-4
978-3-03719-606-9
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/106
http://www.ems-ph.org/doi/10.4171/106
EMS Monographs in Mathematics
Strasbourg Master Class on Geometry
Athanase
Papadopoulos
IRMA, Strasbourg, France
Geometry
Manifolds and cell complexes
Primary 51-01, 51-02, 57-01, 57-02; Secondary 14H30, 14H52, 20F67, 20F69, 22E40, 22D40, 30F10, 30F20, 30F45, 30F60, 32G15, 37E30, 51F99, 51M10, 51E24, 53C21, 53C22, 53C23, 53C35, 53C70, 54C20, 57M15, 57M20, 57M27, 57M50, 57N10
Hyperbolic geometry, hyperbolic space, neutral geometry, Euclid’s axioms, hyperbolic trigonometry, spherical geometry, Khayyam–Saccheri quadrilaterals, trirectangular quadrilaterals, parallelism, horocycle, Lobachevsky parallelism function, Beltrami–Klein
This book contains carefully revised and expanded versions of eight courses that were presented at the University of Strasbourg, during two geometry master classes, in 2008 and 2009. The aim of the master classes was to give to fifth-year students and PhD students in mathematics the opportunity to learn new topics that lead directly to the current research in geometry and topology. The courses were held by leading experts. The subjects treated include hyperbolic geometry, three-manifold topology, representation theory of fundamental groups of surfaces and of three-manifolds, dynamics on the hyperbolic plane with applications to number theory, Riemann surfaces, Teichmüller theory, Lie groups and asymptotic geometry. The text is addressed to students and mathematicians who wish to learn the subject. It can also be used as a reference book and as a textbook for short courses on geometry.
1
18
2012
978-3-03719-105-7
978-3-03719-605-2
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/105
http://www.ems-ph.org/doi/10.4171/105
IRMA Lectures in Mathematics and Theoretical Physics
18
Notes on non-Euclidean geometry
Norbert
A’Campo
Universität Basel, BASEL, SWITZERLAND
Athanase
Papadopoulos
Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE
Differential geometry
Geometry
General
These are notes on plane hyperbolic geometry. The presentation is rather unusual since it is model-free. The methods can be used in the same way for spherical geometry. In particular, we derive the trigonometric formulae without using any Euclidean model. After a brief introduction to the axioms and the basic notions, we study in detail the geometry of triangles and of quadrilaterals. We discuss the notions of area and of dissection, and we derive the trigonometric formulae. Then we present two models of the hyperbolic plane, namely, a disk model whose automorphisms are projective maps, and a model that arises from algebra, whose points are the prime ideals of the ring of polynomials with one real variable. We study parallelism, horocycles, parabolic transformations and the Lobachevsky parallelism function. Finally, we describe a 3-dimensional space which contains the hyperbolic plane, the Euclidean plane and the elliptic space of spherical geometry. In this 3-dimensional space, we can make continuous transitions between objects in the three 2-dimensional geometries (hyperbolic, Euclidean and spherical) and we can watch the transformation of their geometrical properties.
1
182
1
10.4171/105-1/1
http://www.ems-ph.org/doi/10.4171/105-1/1
Crossroads between hyperbolic geometry and number theory
Françoise
Dal’Bo
Université de Rennes I, RENNES CEDEX, FRANCE
Dynamical systems and ergodic theory
Number theory
Group theory and generalizations
Geometry
The motivation of this text is to explain through two elementary examples some links between hyperbolic geometry and another domains of the mathematics. In the first example, we establish relations between the linear orbits of the discrete subgroups of SL(2,R) and the horocyclic trajectories on the hyperbolic surfaces. In the second one, we use the topology of the geodesic trajectories on the modular surface to obtain results in the theory of Diophantine approximations.
183
232
1
10.4171/105-1/2
http://www.ems-ph.org/doi/10.4171/105-1/2
Introduction to origamis in Teichmüller space
Frank
Herrlich
Karlsruhe Institute of Technology, KARLSRUHE, GERMANY
Algebraic geometry
Functions of a complex variable
Several complex variables and analytic spaces
Manifolds and cell complexes
Origamis are translation surfaces that arise from certain coverings of elliptic curves. We give four different characterizations of them, in combinatorial as well as in algebro-geometric terms. By affine variation of the translation structure, an origami defines an embedding of the upper half plane into Teichmüller space; its image in moduli space is an algebraic curve. As a Riemann surface, this algebraic curve is uniformized by the Veech group of the origami, a group closely related to the affine homeomorphisms of the translation surface. We explain these concepts and their interrelations, and illustrate them by examples. In particular we discuss the very helpful example of the quaternion origami or “Eierlegende Wollmilchsau”.
233
253
1
10.4171/105-1/3
http://www.ems-ph.org/doi/10.4171/105-1/3
Five lectures on 3-manifold topology
Philipp
Korablev
Chelyabinsk State University, CHELYABINSK, RUSSIAN FEDERATION
Sergey
Matveev
Chelyabinsk State University, CHELYABINSK, RUSSIAN FEDERATION
Manifolds and cell complexes
General
The article consists of five parts intended for graduate students of mathematics or researchers seeking to extend their mathematical knowledge in mathematics, in particular, in 3-manifold topology. The first section is devoted to describing the most popular classical ways for presenting 3-manifolds: triangulations, Heegaard splittings, surgery along framed links, and special spines. The latter is used in the next section for an elementary description of “quantum” or “state sum” invariants of 3-manifolds discovered by V. Turaev and O. Viro. This description includes an explicit geometric construction of the first nontrivial invariant of TV-type, which can be calculated by looking at a special spine of a given 3-manifold. The third section is short but informative. It contains a very clear explanation of the famous JSJ-decomposition theorem, including its relation to the Thurston geometrization conjecture solved by G. Perelman. The next section is an exposition of the Haken theory of normal surfaces, which is one of the main tools for investigating 3-manifolds. It is very important since it explains how to construct the JSJ-decompositions algorithmically. The last section is devoted to description of the main steps in the proof of the algorithmic classification theorem for sufficiently large 3-manifolds, which includes algorithmic classification of knots.
255
284
1
10.4171/105-1/4
http://www.ems-ph.org/doi/10.4171/105-1/4
An introduction to globally symmetric spaces
Gabriele
Link
Karlsruhe Institute of Technology, KARLSRUHE, GERMANY
Differential geometry
Topological groups, Lie groups
General
These notes provide an introduction to globally symmetric spaces with an emphasis on those of non-compact type. We intend to give a guideline through a part of the landscape rather than proving every detail; however, where proofs are omitted, the reader gets a precise reference. The first part of the text gives an overview on the geometry and algebraic coding of arbitrary globally symmetric spaces. We then restrict ourselves to symmetric spaces of non-compact type and describe the Iwasawa and Cartan decomposition in the second part. The last part is devoted to the study of the geometry at infinity of globally symmetric spaces of non-compact type. We give descriptions of the geometric boundary, the Furstenberg boundary and the Bruhat decomposition, which help to identify the pairs of boundary points which can be joined by a geodesic or flat. Finally we show how invariant Finsler distances on the globally symmetric space can be constructed with the help of Busemann functions.
285
332
1
10.4171/105-1/5
http://www.ems-ph.org/doi/10.4171/105-1/5
Geometry of the representation spaces in SU(2)
Julien
Marché
École Polytechnique, PALAISEAU, FRANCE
Manifolds and cell complexes
Dynamical systems and ergodic theory
Quantum theory
General
These notes of a course given at IRMA in April 2009 cover some aspects of the representation theory of fundamental groups of manifolds of dimension at most 3 in compact Lie groups, mainly SU(2). We give detailed examples, develop the techniques of twisted cohomology and gauge theory. We review Chern–Simons theory and describe an integrable system for the representation space of a surface. Finally, we explain some basic ideas on geometric quantization. We apply them to the case of representation spaces by computing Bohr–Sommerfeld orbits with metaplectic correction.
333
370
1
10.4171/105-1/6
http://www.ems-ph.org/doi/10.4171/105-1/6
Algorithmic construction and recognition of hyperbolic 3-manifolds, links, and graphs
Carlo
Petronio
Università di Pisa, PISA, ITALY
Manifolds and cell complexes
General
This survey article describes the algorithmic approaches successfully used over the time to construct hyperbolic structures on various 3-dimensional topological “objects” (cusped manifolds, manifolds with boundary, links, graphs, orbifolds). It also exposes the classification techniques based on hyperbolic geometry and the experimental classification results of several classes of 3-dimensional topological “objects” obtained over the time.
371
404
1
10.4171/105-1/7
http://www.ems-ph.org/doi/10.4171/105-1/7
An introduction to asymptotic geometry
Viktor
Schroeder
University of Zürich, ZÜRICH, SWITZERLAND
Geometry
Differential geometry
General
This article gives an introduction to certain aspects of the asymptotic geometry of metric spaces. Thus the focus is on the large scale geometry of a space while the local structure is neglected. In particular hyperbolic spaces (in the sense of Gromov) are discussed, for which the asymptotic geometry is encoded in the boundary at infinity. In analogy with the classical hyperbolic space and its boundary, the relation between the metric geometry of a Gromov hyperbolic space and the Möbius geometry of its boundary is studied in detail.
405
454
1
10.4171/105-1/8
http://www.ems-ph.org/doi/10.4171/105-1/8
The mathematical writings of Évariste Galois
Corrected 2nd printing, September 2013
Peter
Neumann
University of Oxford, UK
History and biography
Group theory and generalizations
01-02; 01-55, 01A75, 00B55, 11-03, 11A55, 12-03, 12E12, 12E20, 12F10, 20-02, 20-03, 20B05, 20B15, 20D05, 33-03, 33E05
History of mathematics, Galois, Galois Theory, group, Galois group, equation, theory of equations, Galois field, finite field, elliptic function, modular equation, primitive equation, primitive group, solubility, simple group, soluble group
Although Évariste Galois was only 20 years old when he died, shot in a mysterious early-morning duel in 1832, his ideas, when they were published 14 years later, changed the course of algebra. He invented what is now called Galois Theory, the modern form of what was classically the Theory of Equations. For that purpose, and in particular to formulate a precise condition for solubility of equations by radicals, he also invented groups and began investigating their theory. His main writings were published in French in 1846 and there have been a number of French editions culminating in the great work published by Bourgne & Azra in 1962 containing transcriptions of every page and fragment of the manuscripts that survive. Very few items have been available in English up to now. The present work contains English translations of almost all the Galois material. They are presented alongside a new transcription of the original French, and are enhanced by three levels of commentary. An introduction explains the context of Galois' work, the various publications in which it appears, and the vagaries of his manuscripts. Then there is a chapter in which the five mathematical articles published in his lifetime are reprinted. After that come the Testamentary Letter and the First Memoir (in which Galois expounded the ideas now called Galois Theory), which are the most famous of the manuscripts. There follow the less well known manuscripts, namely the Second Memoir and the many fragments. A short epilogue devoted to myths and mysteries concludes the text. The book is written as a contribution to the history of mathematics but with mathematicans as well as historians in mind. It makes available to a wide mathematical and historical readership some of the most exciting mathematics of the first half of the 19th century, presented in its original form. The primary aim is to establish a text of what Galois wrote. Exegesis would fill another book or books, and little of that is to be found here. This work will be a resource for research in the history of mathematics, especially algebra, as well as a sourcebook for those many mathematicians who enliven their student lectures with reliable historical background.
10
20
2011
978-3-03719-104-0
978-3-03719-604-5
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/104
http://www.ems-ph.org/doi/10.4171/104
Heritage of European Mathematics
Handbook of Teichmüller Theory, Volume III
Athanase
Papadopoulos
IRMA, Strasbourg, France
Functions of a complex variable
Several complex variables and analytic spaces
Primary 30-00, 32-00, 57-00, 32G13, 32G15, 30F60; secondary 11F06, 11F75, 14D20, 14H15, 14H60, 14H55, 14J60, 20F14, 20F28, 20F38, 20F65, 20F67, 20H10, 30C62, 30F20, 30F25, 30F10, 30F15, 30F30, 30F35, 30F40, 30F45, 53A35, 53B35, 53C35, 53C50, 53C80, 53D55, 53Z05, 57M07, 57M20, 57M27, 57M50, 57M60, 57N16
The subject of this handbook is Teichmüller theory in a wide sense, namely the theory of geometric structures on surfaces and their moduli spaces. This includes the study of vector bundles on these moduli spaces, the study of mapping class groups, the relation with 3-manifolds, the relation with symmetric spaces and arithmetic groups, the representation theory of fundamental groups, and applications to physics. Thus the handbook is a place where several fields of mathematics interact: Riemann surfaces, hyperbolic geometry, partial differential equations, several complex variables, algebraic geometry, algebraic topology, combinatorial topology, low-dimensional topology, theoretical physics, and others. This confluence of ideas towards a unique subject is a manifestation of the unity and harmony of mathematics. The present volume contains surveys on the fundamental theory as well as surveys on applications to and relations with the fields mentioned above. It is written by leading experts in the fields. Some of the surveys contain classical material, while others present the latest developments of the theory as well as open problems. The metric and the analytic theory. The group theory. The algebraic topology of mapping class groups and moduli spaces. Teichmüller theory and mathematical physics. The handbook is addressed to graduate students and researchers in all the fields mentioned.
6
8
2012
978-3-03719-103-3
978-3-03719-603-8
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/103
http://www.ems-ph.org/doi/10.4171/103
IRMA Lectures in Mathematics and Theoretical Physics
17
Introduction to Teichmüller theory, old and new, III
Athanase
Papadopoulos
Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE
Several complex variables and analytic spaces
Functions of a complex variable
Manifolds and cell complexes
General
This introduction provides an overview on the contents of this volume, and at the same time is a survey of several aspects of Teichmülller theory with its relation to the moduli spaces of curves, mapping class groups and their various actions, 3-manifold topology and theoretical physics.
1
34
1
10.4171/103-1/1
http://www.ems-ph.org/doi/10.4171/103-1/1
Quasiconformal and BMO-quasiconformal homeomorphisms
Jean-Pierre
Otal
Université Paul Sabatier, TOULOUSE CEDEX 9, FRANCE
Functions of a complex variable
General
We expose a proof of the existence and uniqueness of homeomorphic solutions to the Beltrami equation $\bar \partial f = \mu \partial f$ a.e. in two cases. First, when the $L^\infty$ norm of $\mu$ is $
37
70
1
10.4171/103-1/2
http://www.ems-ph.org/doi/10.4171/103-1/2
Earthquakes on the hyperbolic plane
Jun
Hu
Brooklyn College of CUNY, BROOKLYN, UNITED STATES
Functions of a complex variable
Several complex variables and analytic spaces
Dynamical systems and ergodic theory
General
Based on the recent works of several people, we present a self-contained exposition of the earthquake theory on the hyperbolic plane, initiated by Thurston.
71
122
1
10.4171/103-1/3
http://www.ems-ph.org/doi/10.4171/103-1/3
Kerckhoff’s lines of minima in Teichmüller space
Caroline
Series
University of Warwick, COVENTRY, UNITED KINGDOM
Functions of a complex variable
Several complex variables and analytic spaces
General
We survey the known results about lines of minima introduced by Kerckhoff in [19], and also Rafi’s results about curves which are short on surfaces along Teichmüller geodesics, in particular their use in proving that lines of minima are Teichmüller quasi-geodesics.
123
153
1
10.4171/103-1/4
http://www.ems-ph.org/doi/10.4171/103-1/4
A tale of two groups: arithmetic groups and mapping class groups
Lizhen
Ji
University of Michigan, ANN ARBOR, UNITED STATES
Number theory
Group theory and generalizations
Topological groups, Lie groups
Functions of a complex variable
In this chapter, we discuss similarities, differences and interaction between two natural and important classes of groups: arithmetic subgroups $\Gamma$ of Lie groups $G$ and mapping class groups mod of surfaces of genus $g$ with $n$ punctures. We also mention similar properties and problems for related groups such as outer automorphism groups $\operatorname{Out}(F_n)$, Coxeter groups and hyperbolic groups. Since groups are often effectively studied by suitable spaces on which they act, we also discuss related properties of actions of arithmetic groups on symmetric spaces and actions of mapping class groups on Teichmüller spaces. Interaction between locally symmetric spaces and moduli spaces of Riemann surfaces through the example of the Jacobian map will also be discussed in the last part of this chapter. Since reduction theory, i.e., finding good fundamental domains for proper actions of discrete groups, is crucial to transformation group theory, i.e., to understand the algebraic structures of groups, properties of group actions and geometry, topology and compactifications of the quotient spaces, we discuss many different approaches to reduction theory of arithmetic groups acting on symmetric spaces. These results for arithmetic groups motivate some results on fundamental domains for the action of mapping class groups on Teichmüller spaces. For example, the Minkowski reduction theory of quadratic forms is generalized to the action of $\operatorname{Mod}_g=\operatorname{Mod}_{g, 0}$ on the Teichmüller space $\mathcal{T}_g$ to construct an intrinsic fundamental domain consisting of finitely many cells, solving a weaker version of a folklore conjecture in the theory of Teichmüller spaces.
157
295
1
10.4171/103-1/5
http://www.ems-ph.org/doi/10.4171/103-1/5
Simplicial actions of mapping class groups
John
McCarthy
Michigan State University, EAST LANSING, UNITED STATES
Athanase
Papadopoulos
Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE
Several complex variables and analytic spaces
Combinatorics
Group theory and generalizations
Functions of a complex variable
In this chapter, we review the actions of the (extended) mapping class group of a surface of finite type by simplicial automorphisms on various abstract simplicial complexes (the curve complex, the arc complex, the cut system complex, the pants decomposition complex, the ideal triangulation complex, etc.). Each of these complexes has some special combinatorial features, and there are interesting questions that are particular to each one. We study in detail the actions on recently defined complexes, namely, the complex of domains and some of its subcomplexes. In most cases mentioned, the automorphism group of the simplicial automorphism coincides with the (injective) image of the mapping class group in that automorphism group. In the case of the complex of domains, this does not hold, as soon as the surface has at least two boundary components. We compute the automorphism group of the complex of domains, and we introduce a subcomplex of this complex, the truncated complex of domains, whose automorphism group is the extended mapping class group. The survey contains expository sections on surfaces and related objects and on simplicial complexes.
297
423
1
10.4171/103-1/6
http://www.ems-ph.org/doi/10.4171/103-1/6
On the coarse geometry of the complex of domains
Valentina
Disarlo
Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE
Several complex variables and analytic spaces
Group theory and generalizations
Functions of a complex variable
Manifolds and cell complexes
The complex of domains $D(S)$ is a geometric tool with a very rich simplicial structure, it contains the curve complex $C(S)$ as a simplicial subcomplex. In this chapter we shall regard it as a metric space, endowed with the metric which makes each simplex Euclidean with edges of length 1, and we shall discuss its coarse geometry. We prove that for every subcomplex $\Delta(S)$ of $D(S)$ which contains the curve complex $C(S)$, the natural simplicial inclusion $C(S) \to \Delta(S)$ is an isometric embedding and a quasi-isometry. We prove that, except a few cases, the arc complex $A(S)$ is quasi-isometric to the subcomplex $P_\partial(S)$ of $D(S)$ spanned by the vertices which are peripheral pair of pants, and we give necessary and sufficient conditions on $S$ for the simplicial inclusion $P_\partial(S) \to D(S)$ to be a quasi-isometric embedding. We then apply these results to the arc and curve complex $AC(S)$. We give a new proof of the fact that $AC(S)$ is quasi-isometric to $C(S)$, and we discuss the metric properties of the simplicial inclusion $A(S) \to AC(S)$.
425
439
1
10.4171/103-1/7
http://www.ems-ph.org/doi/10.4171/103-1/7
Minimal generating sets for the mapping class group
Mustafa
Korkmaz
Middle East Technical University, ANKARA, TURKEY
Manifolds and cell complexes
Group theory and generalizations
Functions of a complex variable
General
The mapping class group of an orientable surface is the group of orientation-preserving self-diffeomorphisms of the surface modulo isotopy. This family of groups plays a central role in low dimensional topology. Therefore, its algebraic structure is of interest. This chapter focuses on the generating sets of the mapping class groups, specifically various minimal generating sets. After the necessary definitions are given, Dehn twists and relations among them are introduced. Next, the minimal generating sets of the mapping class group are considered: It turns out that mapping class group is generated by two elements, by two torsion elements, and by four involutions. Finally, known results about the minimal generating sets for the extended mapping class groups, the hyperelliptic mapping class groups, and the mapping class groups of nonorientable surface are considered.
441
463
1
10.4171/103-1/8
http://www.ems-ph.org/doi/10.4171/103-1/8
From mapping class groups to monoids of homology cobordisms: a survey
Kazuo
Habiro
Kyoto University, KYOTO, JAPAN
Gwénaël
Massuyeau
Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE
Manifolds and cell complexes
Group theory and generalizations
General
Let $\Sigma$ be a compact oriented surface. A homology cobordism of $\Sigma$ is a cobordism $C$ between two copies of $\Sigma$, such that both the “top” inclusion and the “bottom” inclusion $\Sigma \subset C$ induce isomorphisms in homology. Homology cobordisms of $\Sigma$ form a monoid, into which the mapping class group of $\Sigma$ embeds by the mapping cylinder construction. In this chapter, we survey recent works on the structure of the monoid of homology cobordisms, and we outline their relations with the study of the mapping class group. We are mainly interested in the cases where $\partial \Sigma$ is empty or connected.
465
529
1
10.4171/103-1/9
http://www.ems-ph.org/doi/10.4171/103-1/9
A survey of Magnus representations for mapping class groups and homology cobordisms of surfaces
Manifolds and cell complexes
Group theory and generalizations
General
This is a survey of Magnus representations with particular emphasis on their applications to mapping class groups and monoids (groups) of homology cobordisms of surfaces. In the first half, we begin by recalling the basics of the Fox calculus and overview Magnus representations for automorphism groups of free groups and mapping class groups of surfaces with related topics. In the latter half, we discuss in detail how the theory in the first half extends to homology cobordisms of surfaces and present a number of applications from recent researches.
531
594
1
10.4171/103-1/10
http://www.ems-ph.org/doi/10.4171/103-1/10
Asymptotically rigid mapping class groups and Thompson groups
Louis
Funar
Université Grenoble I, SAINT MARTIN D'HERES CEDEX, FRANCE
Christophe
Kapoudjian
Université Paul Sabatier – Toulouse 3, TOULOUSE CEDEX, FRANCE
Vlad
Sergiescu
Université Grenoble I, SAINT MARTIN D'HERES CEDEX, FRANCE
Manifolds and cell complexes
Group theory and generalizations
General
We consider Thompson's groups from the perspective of mapping class groups of surfaces of infinite type. This point of view leads us to the braided Thompson groups, which are extensions of Thompson’s groups by infinite (spherical) braid groups. We will outline the main features of these groups and some applications to the quantization of Teichmüller spaces. The chapter provides an introduction to the subject with an emphasis on some of the authors results.
595
664
1
10.4171/103-1/11
http://www.ems-ph.org/doi/10.4171/103-1/11
An introduction to moduli spaces of curves and their intersection theory
Dimitri
Zvonkine
Université Pierre et Marie Curie – Paris 6, PARIS CEDEX 05, FRANCE
General
The material of this chapter is based on a series of three lectures for graduate students that the author gave at theJournées mathématiques de Glanon in July 2006. We introduce moduli spaces of smooth and stable curves, the tautological cohomology classes on these spaces, and explain how to compute all possible intersection numbers between these classes.
667
716
1
10.4171/103-1/12
http://www.ems-ph.org/doi/10.4171/103-1/12
Homology of the open moduli space of curves
Ib
Madsen
University of Copenhagen, COPENHAGEN, DENMARK
Algebraic geometry
Algebraic topology
General
This is a survey on the proof of a generalized version of the Mumford conjecture obtained in joint work with M. Weiss stating that a certain map between some classifying spaces which a priori have different natures induces an isomorphism at the level of integral homology. We also discuss our proof of the original Mumford conjecture stating that the stable rational cohomology of the moduli space of Riemann surfaces is a certain polynomial algebra generated by the Mumford–Morita–Miller cohomology classes of even degrees.
717
746
1
10.4171/103-1/13
http://www.ems-ph.org/doi/10.4171/103-1/13
On the $L^p$-cohomology and the geometry of metrics on moduli spaces of curves
Lizhen
Ji
University of Michigan, ANN ARBOR, UNITED STATES
Steven
Zucker
The Johns Hopkins University, BALTIMORE, UNITED STATES
Several complex variables and analytic spaces
Algebraic geometry
Algebraic topology
Global analysis, analysis on manifolds
Let ${\mathcal M}_{g,n}$ be the moduli space of algebraic curves of genus $g$ with $n$ punctures, which is a noncompact orbifold. Let $\overline{\mathcal M}^{DM}_{g,n}$ denote its Deligne–Mumford compactification. Then ${\mathcal M}_{g,n}$ admits a class of canonical Riemannian and Finsler metrics. We probe the analogy between ${\mathcal M}_{g,n}$ (resp. Teichmüller spaces) with these metrics and certain noncompact locally symmetric spaces (resp. symmetric spaces of noncompact type) with their natural metrics. In this chapter, we observe that for all $1 < p < \infty$, the $L^p$-cohomology of ${\mathcal M}_{g,n}$ with respect to these Riemannian metrics that are complete can be identified with the (ordinary) cohomology of $\overline{\mathcal M}^{DM}_{g,n}$, and hence the $L^p$-cohomology is the same for different values of $p$. This suggests a “rank-one nature” of the moduli space ${\mathcal M}_{g,n}$ from the point of view of $L^p$-cohomology. On the other hand, the $L^p$-cohomology of ${\mathcal M}_{g,n}$ with respect to the incomplete Weil Petersson metric is either the cohomology of $\overline{\mathcal M}^{DM}_{g,n}$ or that of ${\mathcal M}_{g,n}$ itself, depending on whether $p\leq \frac{4}{3}$ or not. At the end of the chapter, we pose several natural problems on the geometry and analysis of these complete Riemannian metrics.
747
775
1
10.4171/103-1/14
http://www.ems-ph.org/doi/10.4171/103-1/14
The Weil–Petersson metric and the renormalized volume of hyperbolic 3-manifolds
Kirill
Krasnov
University of Nottingham, NOTTINGHAM, UNITED KINGDOM
Jean-Marc
Schlenker
Université Paul Sabatier, TOULOUSE CEDEX, FRANCE
Several complex variables and analytic spaces
Functions of a complex variable
Differential geometry
General
We survey the renormalized volume of hyperbolic 3-manifolds, as a tool for Teichmüller theory, using simple differential geometry arguments to recover results sometimes first achieved by other means. One such application is McMullen’s quasifuchsian (or more generally Kleinian) reciprocity, for which different arguments are proposed. Another is the fact that the renormalized volume of quasifuchsian (or more generally geometrically finite) hyperbolic 3-manifolds provides a Kähler potential for the Weil–Petersson metric on Teichmüller space. Yet another is the fact that the grafting map is symplectic, which is proved using a variant of the renormalized volume defined for hyperbolic ends.
779
819
1
10.4171/103-1/15
http://www.ems-ph.org/doi/10.4171/103-1/15
Discrete Liouville equation and Teichmüller theory
Rinat
Kashaev
Université de Genève, GENÈVE, SWITZERLAND
Manifolds and cell complexes
Dynamical systems and ergodic theory
Quantum theory
General
The discrete Liouville equation is reviewed both classically and quantum mechanically from the viewpoints of its integrable structure and its relationships to (quantum) Teichmüller theory.
821
851
1
10.4171/103-1/16
http://www.ems-ph.org/doi/10.4171/103-1/16
Frobenius Algebras I
Basic Representation Theory
Andrzej
Skowroński
Nicolaus Copernicus University, Toruń, Poland
Kunio
Yamagata
Tokyo University of Agriculture and Technology, Japan
Associative rings and algebras
16-01; 13E10, 15A63, 15A69, 16Dxx, 16E30, 16G10, 16G20, 16G70, 16K20, 16W30, 51F15
Algebra, module, representation, quiver, ideal, radical, simple module, semisimple module, uniserial module, projective module, injective module, simple algebra, semisimple algebra, separable algebra, hereditary algebra, Nakayama algebra, Frobenius algebra, symmetric algebra, selfinjective algebra, Brauer tree algebra, enveloping algebra, Coxeter group, Coxeter graph, Hecke algebra, coalgebra, comodule, Hopf algebra, Hopf module, syzygy module, periodic module, periodic algebra, irreducible homomorphism, almost split sequence, Auslander–Reiten translation, Auslander–Reiten quiver, extension spaces, projective dimension, injective dimension, category, functor, Nakayama functor, Nakayama automorphism, Morita equivalence, Morita–Azumaya duality
This is the first of two volumes which will provide a comprehensive introduction to the modern representation theory of Frobenius algebras. The first part of the book serves as a general introduction to basic results and techniques of the modern representation theory of finite dimensional associative algebras over fields, including the Morita theory of equivalences and dualities and the Auslander–Reiten theory of irreducible morphisms and almost split sequences. The second part is devoted to fundamental classical and recent results concerning the Frobenius algebras and their module categories. Moreover, the prominent classes of Frobenius algebras, the Hecke algebras of Coxeter groups and the finite dimensional Hopf algebras over fields are exhibited. This volume is self-contained and the only prerequisite is a basic knowledge of linear algebra. It includes complete proofs of all results presented and provides a rich supply of examples and exercises. The text is primarily addressed to graduate students starting research in the representation theory of algebras as well mathematicians working in other fields.
12
13
2011
978-3-03719-102-6
978-3-03719-602-1
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/102
http://www.ems-ph.org/doi/10.4171/102
EMS Textbooks in Mathematics
Representations of Algebras and Related Topics
Andrzej
Skowroński
Nicolaus Copernicus University, Toruń, Poland
Kunio
Yamagata
Tokyo University of Agriculture and Technology, Japan
Associative rings and algebras
13Dxx, 13Fxx, 14Bxx, 14Hxx, 14Lxx, 14Mxx, 14Nxx, 15Axx, 16Dxx, 16Exx, 16Gxx, 16Sxx, 16Wxx, 17Bxx, 18Exx, 19Kxx, 20Cxx, 20Jxx
This book is concerned with recent trends in the representation theory of algebras and its exciting interaction with geometry, topology, commutative algebra, Lie algebras, combinatorics, quantum algebras, and theoretical physics. The collection of articles, written by leading researchers in the field, is conceived as a sort of handbook providing easy access to the present state of knowledge and stimulating further development. The topics under discussion include quivers, quivers with potential, bound quiver algebras, Jacobian algebras, cluster algebras and categories, Calabi–Yau algebras and categories, triangulated and derived categories, quantum loop algebras, Nakajima quiver varieties, Yang–Baxter equations, T-systems and Y-systems, dilogarithm and quantum dilogarithm identities, stable module categories, localizing and colocalizing subcategories, cohomologies of groups, support varieties, fusion systems, Hochschild cohomologies, weighted projective lines, coherent sheaves, Kleinian and Fuchsian singularities, stable categories of vector bundles, nilpotent operators, Artin–Schelter regular algebras, Fano algebras, deformations of algebras, module varieties, degenerations of modules, singularities of orbit closures, coalgebras and comodules, representation types of algebras and coalgebras, Tits and Euler forms of algebras, Galois coverings of algebras, tilting and cluster tilting theory, algebras of small homological dimensions, Auslander–Reiten theory. The book consists of thirteen self-contained expository survey and research articles and is addressed to researchers and graduate students in algebra as well as a broader mathematical community. They contain a large number of examples and open problems and give new perspectives for research in the field.
9
24
2011
978-3-03719-101-9
978-3-03719-601-4
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/101
http://www.ems-ph.org/doi/10.4171/101
EMS Series of Congress Reports
On generalized cluster categories
Claire
Amiot
Université de Strasbourg, STRASBOURG CEDEX, FRANCE
Associative rings and algebras
Commutative rings and algebras
Category theory; homological algebra
General
Cluster categories have been introduced by Buan, Marsh, Reineke, Reiten and Todorov in order to categorify Fomin–Zelevinsky cluster algebras. This survey motivates and outlines the construction of a generalization of cluster categories, and explains different applications of these new categories in representation theory.
1
53
1
10.4171/101-1/1
http://www.ems-ph.org/doi/10.4171/101-1/1
Module categories for finite group algebras
David
Benson
University of Aberdeen, ABERDEEN, UNITED KINGDOM
Srikanth
Iyengar
University of Nebraska-Lincoln, LINCOLN, UNITED STATES
Henning
Krause
Universität Bielefeld, BIELEFELD, GERMANY
Group theory and generalizations
Commutative rings and algebras
Associative rings and algebras
Category theory; homological algebra
This survey article is intended as an introduction to the recent categorical classification theorems of the three authors, restricting to the special case of the category of modules for a finite group.
55
83
1
10.4171/101-1/2
http://www.ems-ph.org/doi/10.4171/101-1/2
On cluster theory and quantum dilogarithm identities
Bernhard
Keller
Université Paris Diderot - Paris 7, PARIS CEDEX 13, FRANCE
Associative rings and algebras
Algebraic geometry
Nonassociative rings and algebras
Category theory; homological algebra
These are expanded notes from three survey lectures given at the 14th International Conference on Representations of Algebras (ICRA XIV) held in Tokyo in August 2010. We first study identities between products of quantum dilogarithm series associated with Dynkin quivers following Reineke. We then examine similar identities for quivers with potential and link them to Fomin–Zelevinsky’s theory of cluster algebras. Here we mainly follow ideas due to Bridgeland, Fock–Goncharov, Kontsevich–Soibelman and Nagao.
85
116
1
10.4171/101-1/3
http://www.ems-ph.org/doi/10.4171/101-1/3
Quantum loop algebras, quiver varieties, and cluster algebras
Bernard
Leclerc
Université de Caen, CAEN CEDEX, FRANCE
General
These notes reflect the contents of three lectures given at the workshop of the 14th International Conference on Representations of Algebras (ICRA XIV), held in August 2010 in Tokyo. We first provide an introduction to quantum loop algebras and their finite-dimensional representations. We explain in particular Nakajima’s geometric description of the irreducible $q$-characters in terms of graded quiver varieties. We then present a recent attempt to understand the tensor structure of the category of finite-dimensional representations by means of cluster algebras. This takes the form of a general conjecture depending on a level $\ell\in \mathbb{N}$. The conjecture for $\ell = 1$ is now proved thanks to some joint work with Hernandez, and a subsequent paper of Nakajima. The general case is still open.
117
152
1
10.4171/101-1/4
http://www.ems-ph.org/doi/10.4171/101-1/4
Weighted projective lines and applications
Helmut
Lenzing
Universität Paderborn, PADERBORN, GERMANY
Algebraic geometry
Associative rings and algebras
General
This survey covers the topics of my mini-course on “Weighted projective lines” given at the Workshop of ICRA XIV (XIV International Conference at Workshop, Tokyo, August 6–15, 2010). Weighted projective lines, and their defining equations, have a long history going back to Klein and Poincaré. Accordingly their study has a high contact surface with many mathematical subjects, classical and modern. Among the many related subjects we mention representation theory of algebras and groups, invariant theory, function theory, orbifolds, 3-manifolds, singularities and the study of nilpotent operators. Since the formal definition of the category of coherent sheaves by W. Geigle and the author in 1987, substantial progress has been made by a number of authors. As a recent application, the analysis of the singularity category of triangle singularities, Kleinian and Fuchsian singularities is covered in some detail. In the center of this analysis is the structure of the corresponding stable categories of vector bundles. The study is in the spirit of Buchweitz (1987) and Orlov (2005) and concerns work in progress with Kussin and Meltzer, recent work by Kajiura, Saito, and Takahashi (2007, 2009), and joint work with J. A. de la Peña (2006). These methods are further applied to the study by C. M. Ringel and M. Schmidmeier (2008) on the invariant subspace problem for nilpotent operators.
153
187
1
10.4171/101-1/5
http://www.ems-ph.org/doi/10.4171/101-1/5
Cohomology of block algebras of finite groups
Markus
Linckelmann
City University, LONDON, UNITED KINGDOM
$K$-theory
General
Block algebras are algebras which arise as indecomposable direct factors of finite group algebras, where we choose as a base ring an algebraically closed field k of prime characteristic p. The purpose of the present notes is to describe some connections between invariants of the module categories of block algebras and of their associated fusion systems, with a particular emphasis on cohomological invariants.
189
250
1
10.4171/101-1/6
http://www.ems-ph.org/doi/10.4171/101-1/6
Algebras with separating Auslander–Reiten components
Piotr
Malicki
Nicolaus Copernicus University, TORUŃ, POLAND
Andrzej
Skowroński
Nicolaus Copernicus University, TORUŃ, POLAND
Associative rings and algebras
General
We survey old and new results on the structure and homological properties of Artin algebras whose Auslander–Reiten quiver admits a separating family of connected components.
251
353
1
10.4171/101-1/7
http://www.ems-ph.org/doi/10.4171/101-1/7
Classification problems in noncommutative algebraic geometry and representation theory
Izuru
Mori
Shizuoka University, SHIZUOKA, JAPAN
Associative rings and algebras
General
In noncommutative algebraic geometry, it is interesting to classify homologically nice classes of connected graded algebras. On the other hand, in representation theory of finite dimensional algebras, it is interesting to classify homologically nice classes of finite dimensional algebras. In this survey paper, we will show that there are strong interactions between these classification problems.
355
406
1
10.4171/101-1/8
http://www.ems-ph.org/doi/10.4171/101-1/8
Periodicities in cluster algebras and dilogarithm identities
Tomoki
Nakanishi
Nagoya University, NAGOYA, CHIKUSA-KU, JAPAN
Commutative rings and algebras
Nonassociative rings and algebras
General
We consider two kinds of periodicities of mutations in cluster algebras. For any sequence of mutations under which exchange matrices are periodic, we define the associated T- and Y-systems. When the sequence is ‘regular’, they are particularly natural generalizations of the known ‘classic’ T- and Y-systems. Furthermore, for any sequence of mutations under which seeds are periodic, we formulate the associated dilogarithm identity. We prove the identities when exchange matrices are skew symmetric.
407
443
1
10.4171/101-1/9
http://www.ems-ph.org/doi/10.4171/101-1/9
The Tits forms of tame algebras and their roots
José Antonio
Peña
Universidad Nacional Autónoma de México, MÉXICO, D.F., MEXICO
Andrzej
Skowroński
Nicolaus Copernicus University, TORUŃ, POLAND
Linear and multilinear algebra; matrix theory
Algebraic geometry
Associative rings and algebras
General
We survey some old and recent results concerning properties of the Tits quadratic forms of triangular finite dimensional algebras of finite and tame representation type over an algebraically closed field, and realization of their roots as dimension vectors of indecomposable modules. In particular, we discuss when we may recover the representation type of a triangular algebra from the combinatorial properties of its Tits quadratic form.
445
499
1
10.4171/101-1/10
http://www.ems-ph.org/doi/10.4171/101-1/10
The minimal representation-infinite algebras which are special biserial
Claus Michael
Ringel
Universität Bielefeld, BIELEFELD, GERMANY
Associative rings and algebras
General
Let $k$ be a field. A finite dimensional k-algebra is said to be minimal representation-infinite provided it is representation-infinite and all its proper factor algebras are representation-finite. Our aim is to classify the special biserial algebras which are minimal representation-infinite. The second part describes the corresponding module categories.
501
560
1
10.4171/101-1/11
http://www.ems-ph.org/doi/10.4171/101-1/11
Coalgebras of tame comodule type, comodule categories, and a tame-wild dichotomy problem
Daniel
Simson
Nicolaus Copernicus University, TORUŃ, POLAND
Associative rings and algebras
General
We study the structure of $K$-coalgebras over a field $K$ and comodule categories. In particular, we discuss the concepts of tame comodule type, of discrete comodule type, of polynomial growth, and of wild comodule type, for $K$-coalgebras $C$, introduced by the author in [83], [84], [93], and intensively studied during the last decade. Among other things, we show that, over an algebraicaly closed field $K$, the tame-wild dichotomy holds, for a wide class of coalgebras $C$ of infinite dimension, including the class of semiperfect coalgebras and the class of incidence coalgebras $K^{\Box} I$ of interval finite posets $I$. Tools and techniques applied in the study of $K$-coalgebras $C$, their comodules, and representation types, are presented. Characterisations of large classes of coalgebras of tame comodule type are presented, including path coalgebras of quivers, string coalgebras, and the incidence coalgebras of interval finite posets.
561
660
1
10.4171/101-1/12
http://www.ems-ph.org/doi/10.4171/101-1/12
Singularities of orbit closures in module varieties
Grzegorz
Zwara
Nicolaus Copernicus University, TORUŃ, POLAND
Associative rings and algebras
Algebraic geometry
General
Finite dimensional modules (or representations of quivers) form affine varieties equipped with algebraic group actions, such that the orbits correspond bijectively to the isomorphism classes of modules. The Zariski-closures of orbits are the objects of our interest. We study their geometric properties and relate them to properties of the corresponding modules.
661
725
1
10.4171/101-1/13
http://www.ems-ph.org/doi/10.4171/101-1/13
Geometric Numerical Integration and Schrödinger Equations
Erwan
Faou
ENS Cachan Bretagne, France
Numerical analysis
Partial differential equations
Dynamical systems and ergodic theory
65P10, 37M15, 35Q41
Geometric numerical integration, symplectic integrators, backward error analysis, Schrödinger equations, long time behavior
The goal of geometric numerical integration is the simulation of evolution equations possessing geometric properties over long times. Of particular importance are Hamiltonian partial differential equations typically arising in application fields such as quantum mechanics or wave propagation phenomena. They exhibit many important dynamical features such as energy preservation and conservation of adiabatic invariants over long time. In this setting, a natural question is how and to which extent the reproduction of such long time qualitative behavior can be ensured by numerical schemes. Starting from numerical examples, these notes provide a detailed analysis of the Schrödinger equation in a simple setting (periodic boundary conditions, polynomial nonlinearities) approximated by symplectic splitting methods. Analysis of stability and instability phenomena induced by space and time discretization are given, and rigorous mathematical explanations for them. The book grew out of a graduate level course and is of interest to researchers and students seeking an introduction to the subject matter.
1
14
2012
978-3-03719-100-2
978-3-03719-600-7
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/100
http://www.ems-ph.org/doi/10.4171/100
Zurich Lectures in Advanced Mathematics
Nonlinear Potential Theory on Metric Spaces
Anders
Björn
Linköping University, Sweden
Jana
Björn
Linköping University, Sweden
Potential theory
31-02, 31E05; 28A12, 30L99, 31C05, 31C15, 31C40, 31C45, 35B45, 35B65, 35D30, 35J20, 35J25, 35J60, 35J67, 35J70, 35J92, 46E35, 47J20, 49J10, 49J27, 49J40, 49J52, 49N60, 49Q20, 58C99, 58J05, 58J32
Boundary regularity, capacity, Dirichlet problem, doubling measure, interior regularity, metric space, minimizer, Newtonian space, nonlinear, obstacle problem, Perron solution, p-harmonic function, Poincaré inequality, potential theory, Sobolev space, upper gradient
The p-Laplace equation is the main prototype for nonlinear elliptic problems and forms a basis for various applications, such as injection moulding of plastics, nonlinear elasticity theory and image processing. Its solutions, called p-harmonic functions, have been studied in various contexts since the 1960s, first on Euclidean spaces and later on Riemannian manifolds, graphs and Heisenberg groups. Nonlinear potential theory of p-harmonic functions on metric spaces has been developing since the 1990s and generalizes and unites these earlier theories. This monograph gives a unified treatment of the subject and covers most of the available results in the field, so far scattered over a large number of research papers. The aim is to serve both as an introduction to the area for an interested reader and as a reference text for an active researcher. The presentation is rather self-contained, but the reader is assumed to know measure theory and functional analysis. The first half of the book deals with Sobolev type spaces, so-called Newtonian spaces, based on upper gradients on general metric spaces. In the second half, these spaces are used to study p-harmonic functions on metric spaces and a nonlinear potential theory is developed under some additional, but natural, assumptions on the underlying metric space. Each chapter contains historical notes with relevant references and an extensive index is provided at the end of the book.
11
5
2011
978-3-03719-099-9
978-3-03719-599-4
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/099
http://www.ems-ph.org/doi/10.4171/099
EMS Tracts in Mathematics
17
Separately Analytic Functions
Marek
Jarnicki
Jagiellonian University, Kraków, Poland
Peter
Pflug
University of Oldenburg, Germany
Several complex variables and analytic spaces
32-02, 32D15, 32A10, 32A17, 32D05, 32D10, 32D26, 32U15
Separately holomorphic/meromorphic functions, Riemann domains, N-fold crosses, generalized crosses, relative extremal functions, holomorphic extension with singularities
The story of separately holomorphic functions began about 100 years ago. During the second half of the 19th century, it became known that a separately continuous function is not necessarily continuous as a function of all variables. At the beginning of the 20th century, the study of separately holomorphic functions started due to the fundamental work of Osgood and Hartogs. This book provides the first self-contained and complete presentation of the study of separately holomorphic functions, starting from its birth up to current research. Most of the results presented have never been published before in book form. The text is divided into two parts. A more elementary one deals with separately holomorphic functions “without singularities”, another addresses the situation of existing singularities. A discussion of the classical results related to separately holomorphic functions leads to the most fundamental result, the classical cross theorem as well as various extensions and generalizations to more complicated “crosses”. Additionally, several applications for other classes of “separately regular” functions are given. A solid background in basic complex analysis is a prerequisite. In order to make the book self-contained, all the results needed for its understanding are collected in special introductory chapters and referred to at the beginning of each section. The book is addressed to students and researchers in several complex variables as well as to mathematicians and theoretical physicists who are interested in this area of mathematics.
8
11
2011
978-3-03719-098-2
978-3-03719-598-7
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/098
http://www.ems-ph.org/doi/10.4171/098
EMS Tracts in Mathematics
16
Mathematical Foundations of Supersymmetry
Claudio
Carmeli
University of Genova, Italy
Lauren
Caston
RAND Corporation, Santa Monica, CA, USA
Rita
Fioresi
University of Bologna, Italy
Global analysis, analysis on manifolds
Algebraic geometry
Nonassociative rings and algebras
Several complex variables and analytic spaces
58-02; 58A50, 58C50, 14M30, 17A70, 32C11, 81Q60
Supermanifolds, supersymmetry, superschemes, Lie superalgebras
Supersymmetry is a highly active area of considerable interest among physicists and mathematicians. It is not only fascinating in its own right, but there is also indication that it plays a fundamental role in the physics of elementary particles and gravitation. The purpose of the book is to lay down the foundations of the subject, providing the reader with a comprehensive introduction to the language and techniques, with a special attention to giving detailed proofs and many clarifying examples. It is aimed ideally at a second year graduate student. After the first three introductory chapters, the text divides into two parts: the theory of smooth supermanifolds and Lie supergroups, including the Frobenius theorem, and the theory of algebraic superschemes and supergroups. There are three appendices, the first introducing Lie superalgebras and representations of classical Lie superalgebras, the second collecting some relevant facts on categories, sheafification of functors and commutative algebra, and the third explaining the notion of Fréchet space in the super context.
8
6
2011
978-3-03719-097-5
978-3-03719-597-0
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/097
http://www.ems-ph.org/doi/10.4171/097
EMS Series of Lectures in Mathematics
Lectures on Duflo Isomorphisms in Lie Algebra and Complex Geometry
Damien
Calaque
ETH Zurich, Switzerland
Carlo
Rossi
Max Planck Institute for Mathematics, Bonn, Germany
Commutative rings and algebras
Algebraic geometry
Nonassociative rings and algebras
13D03, 17B56, 14F43
Lie algebra, Hochschild cohomology, complex manifolds, deformation theory, Kontsevich’s graphical calculus, Atiyah class, Duflo isomorphism, Todd class
Duflo isomorphism first appeared in Lie theory and representation theory. It is an isomorphism between invariant polynomials of a Lie algebra and the center of its universal enveloping algebra, generalizing the pioneering work of Harish-Chandra on semi-simple Lie algebras. Later on, Duflo’s result was refound by Kontsevich in the framework of deformation quantization, who also observed that there is a similar isomorphism between Dolbeault cohomology of holomorphic polyvector fields on a complex manifold and its Hochschild cohomology. The present book, which arose from a series of lectures by the first author at ETH, derives these two isomorphisms from a Duflo-type result for Q-manifolds. All notions mentioned above are introduced and explained in the book, the only prerequisites being basic linear algebra and differential geometry. In addition to standard notions such as Lie (super)algebras, complex manifolds, Hochschild and Chevalley–Eilenberg cohomologies, spectral sequences, Atiyah and Todd classes, the graphical calculus introduced by Kontsevich in his seminal work on deformation quantization is addressed in details. The book is well-suited for graduate students in mathematics and mathematical physics as well as for researchers working in Lie theory, algebraic geometry and deformation theory.
6
11
2011
978-3-03719-096-8
978-3-03719-596-3
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/096
http://www.ems-ph.org/doi/10.4171/096
EMS Series of Lectures in Mathematics
Invariant Manifolds and Dispersive Hamiltonian Evolution Equations
Kenji
Nakanishi
Kyoto University, Japan
Wilhelm
Schlag
University of Chicago, USA
Partial differential equations
35L70, 35Q55, 37D10, 37K40, 37K45
Nonlinear dispersive equations, wave, Klein–Gordon, Schrödinger equations, scattering theory, stability theory, solitons, ground states, global existence, finite time blow up, soliton resolution conjecture, hyperbolic dynamics, stable, unstable, center-stable, invariant manifolds
The notion of an invariant manifold arises naturally in the asymptotic stability analysis of stationary or standing wave solutions of unstable dispersive Hamiltonian evolution equations such as the focusing semilinear Klein–Gordon and Schrödinger equations. This is due to the fact that the linearized operators about such special solutions typically exhibit negative eigenvalues (a single one for the ground state), which lead to exponential instability of the linearized flow and allows for ideas from hyperbolic dynamics to enter. One of the main results proved here for energy subcritical equations is that the center-stable manifold associated with the ground state appears as a hyper-surface which separates a region of finite-time blowup in forward time from one which exhibits global existence and scattering to zero in forward time. Our entire analysis takes place in the energy topology, and the conserved energy can exceed the ground state energy only by a small amount. This monograph is based on recent research by the authors and the proofs rely on an interplay between the variational structure of the ground states on the one hand, and the nonlinear hyperbolic dynamics near these states on the other hand. A key element in the proof is a virial-type argument excluding almost homoclinic orbits originating near the ground states, and returning to them, possibly after a long excursion. These lectures are suitable for graduate students and researchers in partial differential equations and mathematical physics. For the cubic Klein–Gordon equation in three dimensions all details are provided, including the derivation of Strichartz estimates for the free equation and the concentration-compactness argument leading to scattering due to Kenig and Merle.
9
2
2011
978-3-03719-095-1
978-3-03719-595-6
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/095
http://www.ems-ph.org/doi/10.4171/095
Zurich Lectures in Advanced Mathematics
Nonlinear Discrete Optimization
An Algorithmic Theory
Shmuel
Onn
Technion - Israel Institute of Technology, Haifa, Israel
Operations research, mathematical programming
Combinatorics
Linear and multilinear algebra; matrix theory
Computer science
05Axx, 05Cxx, 05Dxx, 05Exx, 11Dxx, 11Hxx, 11Pxx, 13Pxx, 14Qxx, 15Axx, 15Bxx, 51Mxx, 52Axx, 52Bxx, 52Cxx, 62Hxx, 62Kxx, 62Qxx, 65Cxx, 68Qxx, 68Rxx, 68Wxx, 90Bxx, 90Cxx
Integer programming, combinatorial optimization, optimization, linear programming, stochastic programming, randomized algorithm, approximation algorithm, polynomial time, transportation problem, multi index transportation problem, transshipment problem, multicommodity flow, congestion game, spanning tree, matroid, submodular function, matching, partitioning, clustering, polytope, zonotope, edge direction, totally unimodular matrix, test set, Graver base, contingency table, statistical table, multiway table, disclosure control, data security, privacy, algebraic statistics, experimental design, Frobenius number, Grobner base, Hilbert scheme, zero-dimensional ideal
This monograph develops an algorithmic theory of nonlinear discrete optimization. It introduces a simple and useful setup which enables the polynomial time solution of broad fundamental classes of nonlinear combinatorial optimization and integer programming problems in variable dimension. An important part of this theory is enhanced by recent developments in the algebra of Graver bases. The power of the theory is demonstrated by deriving the first polynomial time algorithms in a variety of application areas within operations research and statistics, including vector partitioning, matroid optimization, experimental design, multicommodity flows, multi-index transportation and privacy in statistical databases. The monograph is intended for graduate students and researchers. It is accessible to anyone with standard undergraduate knowledge and mathematical maturity.
9
7
2010
978-3-03719-093-7
978-3-03719-593-2
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/093
http://www.ems-ph.org/doi/10.4171/093
Zurich Lectures in Advanced Mathematics
From Cardano’s great art to Lagrange’s reflections: filling a gap in the history of algebra
Jacqueline
Stedall
University of Oxford, UK
History and biography
01-02; 01A40; 01A45; 01A50
Algebra, equations, renaissance, early modern
This book is an exploration of a claim made by Lagrange in the autumn of 1771 as he embarked upon his lengthy ‘Réflexions sur la résolution algébrique des équations’: that there had been few advances in the algebraic solution of equations since the time of Cardano in the mid sixteenth century. That opinion has been shared by many later historians. The present study attempts to redress that view and to examine the intertwined developments in the theory of equations from Cardano to Lagrange. A similar historical exploration led Lagrange himself to insights that were to transform the entire nature and scope of algebra. Progress was not confined to any one country: at different times mathematicians in Italy, France, the Netherlands, England, Scotland, Russia, and Germany contributed to the discussion and to a gradual deepening of understanding. In particular, the national Academies of Berlin, St Petersburg, and Paris in the eighteenth century were crucial in supporting informed mathematical communities and encouraging the wider dissemination of key ideas. This study therefore truly highlights the existence of a European mathematical heritage. The book is written in three parts. Part I offers an overview of the period from Cardano to Newton (from 1545 to 1707) and is arranged chronologically. Part II covers the period from Newton to Lagrange (from 1707 to 1770) and treats the material according to key themes. Part III is a brief account of the aftermath of the discoveries made in the 1770s. The book attempts throughout to capture the reality of mathematical discovery by inviting the reader to follow in the footsteps of the authors themselves, with as few changes as possible to the original notation and style of presentation.
3
29
2011
978-3-03719-092-0
978-3-03719-592-5
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/092
http://www.ems-ph.org/doi/10.4171/092
Heritage of European Mathematics
Efficient Numerical Methods for Non-local Operators
ℋ2-Matrix Compression, Algorithms and Analysis Corrected 2nd printing, September 2013
Steffen
Börm
Kiel University, Germany
Numerical analysis
65-02; 65F05, 65F30, 65N22, 65N38, 65R20
Hierarchical matrix, data-sparse approximation, boundary element method, preconditioner
Hierarchical matrices present an efficient way of treating dense matrices that arise in the context of integral equations, elliptic partial differential equations, and control theory. While a dense n × n matrix in standard representation requires n2 units of storage, a hierarchical matrix can approximate the matrix in a compact representation requiring only O(nk log n) units of storage, where k is a parameter controlling the accuracy. Hierarchical matrices have been successfully applied to approximate matrices arising in the context of boundary integral methods, to construct preconditioners for partial differential equations, to evaluate matrix functions and to solve matrix equations used in control theory. ℋ2-matrices offer a refinement of hierarchical matrices: using a multilevel representation of submatrices, the efficiency can be significantly improved, particularly for large problems. This books gives an introduction to the basic concepts and presents a general framework that can be used to analyze the complexity and accuracy of ℋ2-matrix techniques. Starting from basic ideas of numerical linear algebra and numerical analysis, the theory is developed in a straightforward and systematic way, accessible to advanced students and researchers in numerical mathematics and scientific computing. Special techniques are only required in isolated sections, e.g., for certain classes of model problems.
12
1
2010
978-3-03719-091-3
978-3-03719-591-8
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/091
http://www.ems-ph.org/doi/10.4171/091
EMS Tracts in Mathematics
14
“Moonshine” of Finite Groups
Koichiro
Harada
The Ohio State University, Columbus, OH, USA
Group theory and generalizations
Number theory
20B05, 11F03
Monster simple group, congruence groups, modular functions, eta function
This is an almost verbatim reproduction of the author’s lecture notes written in 1983–84 at the Ohio State University, Columbus, Ohio, USA. A substantial update is given in the bibliography. Over the last 20 plus years, there has been an energetic activity in the field of finite simple group theory related to the monster simple group. Most notably, influential works have been produced in the theory of vertex operator algebras whose research was stimulated by the moonshine of the finite groups. Still, we can ask the same questions now just as we did some 30–40 years ago: What is the monster simple group? Is it really related to the theory of the universe as it was vaguely so envisioned? What lays behind the moonshine phenomena of the monster group? It may appear that we have only scratched the surface. These notes are primarily reproduced for the benefit of young readers who wish to start learning about modular functions used in moonshine.
9
29
2010
978-3-03719-090-6
978-3-03719-590-1
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/090
http://www.ems-ph.org/doi/10.4171/090
EMS Series of Lectures in Mathematics
math.ch/100
Schweizerische Mathematische Gesellschaft – Société Mathématique Suisse – Swiss Mathematical Society 1910
Bruno
Colbois
University of Neuchâtel, Switzerland
Christine
Riedtmann
University of Bern, Switzerland
Viktor
Schroeder
University of Zurich, Switzerland
History and biography
00Bxx
This book includes twenty-three essays to celebrate the 100th anniversary of the Swiss Mathematical Society. The life and work of outstanding mathematicians, extraordinary conferences held in Switzerland such as the three International Congresses of Mathematicians, the influence of women in Swiss mathematics are among the topics. The articles, including many photographs, old and recent, give a vivid picture of hundred years of mathematical life in Switzerland. Dieses Buch ist eine Festschrift zum 100-jährigen Bestehen der Schweizerischen Mathematischen Gesellschaft. Es enthält dreiundzwanzig Beiträge zur Mathematik in der Schweiz. Geschichtliches und Biographisches über herausragende Mathematiker an Schweizer Universitäten, grosse Tagungen wie etwa die drei Internationalen Mathematiker-Kongresse, die Rolle der Frauen in der Schweizer Mathematik sind nur einige Themen. Insgesamt vermitteln die verschiedenen Essays zusammen mit den zahlreichen Abbildungen ein höchst lebendiges und anschauliches Panorama eines Jahrhunderts Schweizer Mathematik. Cet ouvrage a été édité pour marquer le 100e anniversaire de la Société Mathématique Suisse. Il rassemble vingt-trois articles consacrés aux mathématiques en Suisse. Parmi beaucoup d'autres choses, les écrits évoquent la vie et l'œuvre de grands mathématiciens des universités suisses, les grands événements, dont les trois Congrès internationaux de mathématiques, ou encore la présence des femmes dans les mathématiques suisses. Agrémenté de nombreuses photos, anciennes et récentes, ce livre donne une image très vivante de cent années de vie mathématique en Suisse.
8
24
2010
978-3-03719-089-0
978-3-03719-589-5
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/089
http://www.ems-ph.org/doi/10.4171/089
Mathématiques et Mathématiciens en Suisse (1850–1950)
Michel
Plancherel (1885–1967)
ETH Zürich, , SWITZERLAND
History and biography
General
1
21
1
10.4171/089-1/1
http://www.ems-ph.org/doi/10.4171/089-1/1
100 Jahre Schweizerische Mathematische Gesellschaft
Erwin
Neuenschwander
Universität Zürich, ZÜRICH, SWITZERLAND
History and biography
General
23
105
1
10.4171/089-1/2
http://www.ems-ph.org/doi/10.4171/089-1/2
Ein Mathematikstudium in den Fünfzigerjahren
Christian
Blatter
ETH Zürich, ZÜRICH, SWITZERLAND
History and biography
General
The author, born 1935, spent his youth and student years in Basel (Switzerland). His tale begins when he enters the “Humanistisches Gymnasium” there; it ends with the legendary Colloquium on Differential Geometry and Topology, held 1960 in Zurich on the occasion of the 50-th anniversary of the Swiss Mathematical Society, where our hero for the first time meets the international mathematical scene. In between we make the acquaintance with the relevant mathematical figures (and some other scientists) in Basel at the time, we hear about the author’s struggling with various minors (philosophy, physics, chemistry), about his part time job as proofreader for a local newspaper, about his year as famulus of the great Ostrowski, and about more leisure-oriented student activities. All this converges to serious mathematical work on a theme between complex analysis and differential geometry (a hint of which is explained in the text at hand). The lively account of the author shows that then and there studying meant working in very diverse environments, as it does today, but all in all had an essential quality of liberty – Bologna was still forty years ahead.
107
128
1
10.4171/089-1/3
http://www.ems-ph.org/doi/10.4171/089-1/3
Andreas Speiser (1885–1970)
Johann Jakob
Burckhardt (1903–2006)
Unversität Basel, , SWITZERLAND
Adolf
Schnyder
, THERWIL, SWITZERLAND
History and biography
General
Andreas Speiser is well known for his work in mathematics and philosophy. In mathematics, his main areas of interest were number theory and group theory. His book Die Theorie der Gruppen von endlicher Ordnung was published in 1923 (5th ed. 1980) as volume 5 of Springer's Grundlagen series. From 1928 to 1965 Speiser was “General Redactor” of the Euler edition, 37 volumes of which appeared in this period, among them 11 volumes that were edited by Speiser himself. As far as philosophy is concerned, Speiser published a commentary on Plato’s Parmenides (Ein Parmenideskommentar, 1937) and an account on his involvement with the philosophy of Fichte and Hegel (Elemente der Philosophie und der Mathematik, 1952). Speiser was an intellectual of the old school in the best sense of the word, his interests and his education spread over wide areas, from mathematics and philosophy to music, architecture, poetry and arts. With his article, it is the authors’ intention to explain Speiser’s work to a broader audience.
129
161
1
10.4171/089-1/4
http://www.ems-ph.org/doi/10.4171/089-1/4
Heinz Huber und das Längenspektrum
Peter
Buser
Ecole Polytechnique Fédérale de Lausanne, LAUSANNE, SWITZERLAND
History and biography
General
This is a historical article about the discovery of the length spectrum of a compact Riemann surface. It begins with Euler’s product formula for the prime numbers from 1737, proceeds to Riemann’s zeta function and then focusses on the theorems about length and eigenvalue spectra that were discovered by Heinz Huber in the 1950s. Emphasis is given to the comparison of Wiener’s proof of the prime number theorem with Huber’s proof of the asymptotic formula for the lengths of closed geodesics. The history ends with a brief outlook to current research.
163
193
1
10.4171/089-1/5
http://www.ems-ph.org/doi/10.4171/089-1/5
A glimpse of the de Rham era
Srishti
Chatterji
, LAUSANNE, SWITZERLAND
Manuel
Ojanguren
Ecole Polytechnique Fédérale de Lausanne, LAUSANNE, SWITZERLAND
History and biography
General
The object of this essay is to give an impressionistic overview of the mathematical period during which Georges de Rham played an important role, both through his own mathematical creation as well as his influence on other mathematicians. Our understanding of the man and his work has been much influenced by a study of his far-flung correspondence and his unpublished papers to which we had access. Some general introductory material has been appended to clarify the position of Swiss mathematics in general in order to initiate foreign readers to the life and times of de Rham.
195
240
1
10.4171/089-1/6
http://www.ems-ph.org/doi/10.4171/089-1/6
Les mathématiques appliquées à l'École polytechnique de Lausanne
Jean
Descloux
Ecole Polytechnique Fédérale de Lausanne, LAUSANNE, SWITZERLAND
Dominique
de Werra
Ecole Polytechnique Fédérale de Lausanne, LAUSANNE, SWITZERLAND
History and biography
General
At the request of the Director of the Federal Polytechnic School of Lausanne, Professor Blanc (1910–2006) initiated an ambitious programme in applied mathematics: new courses, the creation of an institute for research, and an orientation towards economics. With the acquisition of a computer in 1958, he promoted numerical analysis, computer science and operational research. When the school received a federal status in 1969, he took this opportunity to endow it with a department of mathematics dedicated to applications.
241
245
1
10.4171/089-1/7
http://www.ems-ph.org/doi/10.4171/089-1/7
Michel Kervaire (1927–2007)
Shalom
Eliahou
Université du Littoral Côte d'Opale, CALAIS, CEDEX, FRANCE
Pierre
de la Harpe
Université de Genève, GENÈVE 4, SWITZERLAND
Jean-Claude
Hausmann
Université de Genève, GENÈVE 4, SWITZERLAND
Claude
Weber
Université de Genève, GENÈVE 4, SWITZERLAND
General
This obituary of Michel Kervaire provides some biographical data, and describes his activity in organising spring seminars in Les Plans-sur-Bex. Some of his mathematical results are briefly discussed: his discovery of a topological manifold without any smooth structure, his results with Milnor on spheres with several smooth structures, his subject-founding paper on high-dimensional knots, the Kervaire conjecture on group presentations, and his results from the period 1987–2007 in algebra, combinatorics and number theory. A short last section discusses dramatic recent progress on the Kervaire invariant problem by Hill, Hopkins and Ravenel in 2009.
247
255
1
10.4171/089-1/8
http://www.ems-ph.org/doi/10.4171/089-1/8
Alexander M. Ostrowski (1893–1986): His life, work, and students
Walter
Gautschi
Purdue University, WEST LAFAYETTE, UNITED STATES
History and biography
General
The life of A. M. Ostrowski, full professor of mathematics at the University of Basel from 1927 to 1958, is sketched, and his extensive and many-sided work is summarized and illustrated by a few selected contributions. A complete list of his Ph.D. students is assembled and the careers of some of them are traced.
257
278
1
10.4171/089-1/9
http://www.ems-ph.org/doi/10.4171/089-1/9
Numerical analysis in Zurich – 50 years ago
Martin
Gutknecht
Eidgenössische Technische Hochschule, ZÜRICH, SWITZERLAND
History and biography
Numerical analysis
General
The Institute for Applied Mathematics at ETH Zurich, founded by Eduard Stiefel in January 1948, made seminal early contributions to numerical analysis and computer science. In numerical analysis the most important ones were the conjugate gradient method and the qd and LR algorithms – which are the predecessors of the ubiquitous QR algorithm. Those in computer science were the construction of the Swiss computer ERMETH, the introduction and development of compilers, and the collaboration at the ground-breaking computer language ALGOL 60. An important side-effect was the education of excellent young numerical analysts, quite a few of which emigrated to the United States –some temporarily, others for good.
279
290
1
10.4171/089-1/10
http://www.ems-ph.org/doi/10.4171/089-1/10
Armand Borel (1923–2003)
André
Haefliger
Université de Genève, GENÈVE 4, SWITZERLAND
History and biography
General
291
302
1
10.4171/089-1/11
http://www.ems-ph.org/doi/10.4171/089-1/11
Bericht über meine Zeit in der Schweiz in den Jahren 1948–1950
Friedrich
Hirzebruch
, BONN, GERMANY
History and biography
General
I was released as a prisoner of war on July 1, 1945, still 17 years old, and was lucky enough to be able to begin my study of mathematics, mathematical logic, and physics at the University of Münster already in the winter semester 1945/46. The University and the town were terribly destroyed, but the University began to operate. My teachers were Heinrich Behnke and Heinrich Scholz, later Karl Stein and Friedrich Karl Schmidt joined the faculty. In 1948 I participated in a Swiss program for German students (three weeks of hard work on a farm, followed by one week to be spent freely in Switzerland). Upon recommendation of Behnke I was invited by Heinz and Anja Hopf to stay with them in their house in Zollikon near Zurich for this fourth week. I learnt a lot from Heinz Hopf about algebraic topology and complex manifolds and returned to Münster full of new ideas which I told Heinrich Behnke and Karl Stein. I spoke in their seminars. This was already basic for my dissertation under Behnke, but much inspired by Hopf. I received a stipend for the ETH Zurich for three semesters (summer 1949 to summer 1950) and became a student of Heinz Hopf, Beno Eckmann and Paul Bernays at the ETH. For a German student who had lived through the war, Switzerland was a paradise academically and otherwise. I enjoyed the international atmosphere with visitors from many countries and the excellent courses and seminars offered. But most important were my frequent discussions with Heinz Hopf at the ETH and at his home. My dissertation developed, won a prize of the ETH and was accepted in Münster by Behnke. In July 1950 I was promoted to Dr. rer. nat. by the University of Münster. In my report I try to explain how important my study in Zurich was for my academic career and how fortunate I was that such a chance was given to me shortly after the war. My dissertation was published in two parts in Mathematische Annalen 1951 and 1953. These are the first two papers in my Collected Papers, Springer-Verlag 1987. In my report I indicate the contents of Part II of the dissertation.
303
315
1
10.4171/089-1/12
http://www.ems-ph.org/doi/10.4171/089-1/12
Michel Plancherel, une vie pour les mathématiques et pour le prochain
Norbert
Hungerbühler
ETH Zentrum, HG E63.1, ZÜRICH, SWITZERLAND
Martine
Schmutz
Université de Fribourg, FRIBOURG, SWITZERLAND
General
Michel Plancherel was one of the leading figures of Swiss mathematics of the 20th century. He is best known for his fundamental results in harmonic analysis and applications in PDE theory and the calculus of variations. A milestone in mathematical physics was his proof that mechanical systems cannot be ergodic. He also left his traces in algebra, in particular in the context of quadratic forms and the theory of commutative Hilbert algebras. Not less remarkable is Plancherel’s unfatiguing dedication to the community: He served as a president of the Swiss Mathematical Society, as vice-president of the International Congress of Mathematicians 1932 in Zurich, he was rector of ETH, president and co-founder of the foundation for the advancement of the mathematical sciences in Switzerland, and he served in many other institutions. He raised funds to help the 550 Hungarian students who escaped in 1956 to Switzerland, he was president of the Swiss Winterhilfe, an organization to help people during hard winters, and he presided the Mission Catholique Française in Zurich. With his wife Cécile, née Tercier, he had 9 children, five boys and four girls.
317
342
1
10.4171/089-1/13
http://www.ems-ph.org/doi/10.4171/089-1/13
Zur Geschichte des Mathematischen Instituts der Universität Freiburg (Schweiz)
General
The story of the development of the Institute of Mathematics of the University of Fribourg begins 1896 when Mathias Lerch and Matthieu Franz Daniëls were nominated professors at the newly founded Faculty of Science. There is a long list of mathematicians who spent their career, or at least part of it, in Fribourg. Many began their professional activity in Fribourg and changed later to greener meadows at another university. Some spent only their last few active years at the Fribourg Mathematics Department. However they all left their traces and helped to put Fribourg on the mathematical map. Among the former was Michel Plancherel who was one of the great Swiss mathematician of the 20th century. He held a position in Fribourg from 1911 to 1921. Among the latter I should mention Peter Thullen who in his youth was already a distinguished mathematician in the field of complex analysis. He emigrated 1933 from Germany to Italy and then to South America where he changed his research to statistics and became an expert in Social Security. He held the position of professor of mathematical statistics in Fribourg from 1972 until his retirement in 1976.
343
350
1
10.4171/089-1/14
http://www.ems-ph.org/doi/10.4171/089-1/14
Martin Eichler – Leben und Werk
Jürg
Kramer
Humboldt-Universität zu Berlin, BERLIN, GERMANY
General
In this paper, the life and work of Martin Eichler, who taught at the University of Basel from 1958 until his retirement in 1980, is presented. It gives an overview of his fundamental contributions to the arithmetic of algebras and quadratic forms, the theory of modular forms and Jacobi forms, as well as the Riemann–Roch Theorem. In particular, one of his most original contributions, namely the confirmation the Taniyama–Shimura conjecture by first non-trivial examples, is discussed, which experienced a renaissance in connection with A. Wiles’ proof of the Fermat Conjecture. The article contains a complete list of Martin Eichler’s publications.
351
371
1
10.4171/089-1/15
http://www.ems-ph.org/doi/10.4171/089-1/15
Mathematik an der Universität Bern im 19. und 20. Jahrhundert
Peter
Mani
, Hünibach/BE, SWITZERLAND
History and biography
General
373
388
1
10.4171/089-1/16
http://www.ems-ph.org/doi/10.4171/089-1/16
An interview with Beno Eckmann
Martin
Raussen
Aalborg Universitet, AALBORG 0ST, DENMARK
Alain
Valette
Université de Neuchâtel, NEUCHÂTEL, SWITZERLAND
History and biography
General
389
401
1
10.4171/089-1/17
http://www.ems-ph.org/doi/10.4171/089-1/17
Wege von Frauen: Mathematikerinnen in der Schweiz
Christine
Riedtmann
Universität Bern, BERN, SWITZERLAND
History and biography
General
403
421
1
10.4171/089-1/18
http://www.ems-ph.org/doi/10.4171/089-1/18
L'Institut de mathématiques de Neuchâtel 1950–90
Alain
Robert
Université de Neuchâtel, NEUCHÂTEL, SWITZERLAND
History and biography
General
In the second half of the 20th century, mathematics at the University of Neuchâtel occupied four different sites. Its teaching staff increased from two to five. The author of this article relates a few personal recollections concerning the actors and the atmosphere during this exceptional period.
423
439
1
10.4171/089-1/19
http://www.ems-ph.org/doi/10.4171/089-1/19
Hermann Weyl, Heinz Hopf und das Jahr 1930 an der ETH
Urs
Stammbach
ETH Zürich, ZÜRICH, SWITZERLAND
History and biography
General
The year 1930 was a year of great significance for the ETH in Zurich, for the development of mathematics in Switzerland, and also for mathematics in general. Firstly, in this year the ETH celebrated its 75th anniversary; secondly, Hermann Weyl, left Zurich after 17 fruitful years at ETH and accepted the call to succeed David Hilbert in Göttingen; thirdly, to replace Weyl the ETH appointed Heinz Hopf, who subsequently built up an important school in algebraic topology; and finally, just before accepting the call to ETH in 1930, Heinz Hopf completed his influential paper, in which he described and classified the essential maps from the three dimensional sphere onto the two dimensionals sphere. The present text is concerned with these four significant developments of the year 1930; central to the presentation are the eminent personalities of Hermann Weyl and Heinz Hopf.
441
469
1
10.4171/089-1/20
http://www.ems-ph.org/doi/10.4171/089-1/20
Rolf Nevanlinna in Zurich
Kurt
Strebel
Universität Zürich, ZÜRICH, SWITZERLAND
History and biography
General
The article starts with a proof of Denjoy’s conjecture by Lars Ahlfors who was then a twenty years old student of Rolf Nevanlinna. This is followed by an account of the first year of Nevanlinna in Zürich. He stimulated the mathematical and even the musical life of Zurich. In 1963 he retired and went back to Finland, where he became chancellor of the University of Turcu. In order not to lose him permanently Künzi and Strebel founded the Nevanlinna Colloqium, which took place about every other year and to which Nevanlinna was always invited.
471
485
1
10.4171/089-1/21
http://www.ems-ph.org/doi/10.4171/089-1/21
Quelques souvenirs sur le troisième cycle romand de mathématiques et le séminaire des Plans-sur-Bex
Claude
Weber
Université de Genève, GENÈVE 4, SWITZERLAND
History and biography
General
In this paper the author recalls, in a very subjective way, some events which were at the origin of the “Troisième Cycle Romand de Mathématiques” and of the “Séminaire des Plans-sur-Bex”. It deals with the main actors who worked for their success. The paper ends with the list of the subjects and of the participants of Les Plans-sur-Bex.
487
503
1
10.4171/089-1/22
http://www.ems-ph.org/doi/10.4171/089-1/22
Jürgen Moser (1928–1999)
Eduard
Zehnder
ETH Zürich, LEO D 3, ZÜRICH, SWITZERLAND
General
This is a brief account on Moser’s important contributions to mathematics and physics, ranging from dynamical systems, celestial mechanics, spectral theory, calculus of variations, partial differential equations to differential geometry and complex analysis.
505
517
1
10.4171/089-1/23
http://www.ems-ph.org/doi/10.4171/089-1/23
The Blind Spot
Lectures on Logic
Jean-Yves
Girard
Institut de Mathématiques de Luminy, Marseille, France
Mathematical logic and foundations
Category theory; homological algebra
Computer science
03-01, 03A05, 03B40, 03F(03-05-07-10-20-40-52-99), 18A15, 68Q(05-10-12-15-55)
Logic, proof-theory, incompleteness, sequent calculus, natural deduction, lambda-calculus, Curry–Howard isomorphism, system F, coherent spaces, linear logic, proof-nets, ludics, implicit complexity, geometry of interaction
These lectures on logic, more specifically proof theory, are basically intended for postgraduate students and researchers in logic. The question at stake is the nature of mathematical knowledge and the difference between a question and an answer, i.e., the implicit and the explicit. The problem is delicate mathematically and philosophically as well: the relation between a question and its answer is a sort of equality where one side is “more equal than the other”: one thus discovers essentialist blind spots. Starting with Gödel’s paradox (1931) – so to speak, the incompleteness of answers with respect to questions – the book proceeds with paradigms inherited from Gentzen’s cut-elimination (1935). Various settings are studied: sequent calculus, natural deduction, lambda calculi, category-theoretic composition, up to geometry of interaction (GoI), all devoted to explicitation, which eventually amounts to inverting an operator in a von Neumann algebra. Mathematical language is usually described as referring to a preexisting reality. Logical operations can be given an alternative procedural meaning: typically, the operators involved in GoI are invertible, not because they are constructed according to the book, but because logical rules are those ensuring invertibility. Similarly, the durability of truth should not be taken for granted: one should distinguish between imperfect (perennial) and perfect modes. The procedural explanation of the infinite thus identifies it with the unfinished, i.e., the perennial. But is perenniality perennial? This questioning yields a possible logical explanation for algorithmic complexity. This highly original course on logic by one of the world’s leading proof theorists challenges mathematicians, computer scientists, physicists and philosophers to rethink their views and concepts on the nature of mathematical knowledge in an exceptionally profound way.
9
25
2011
978-3-03719-088-3
978-3-03719-588-8
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/088
http://www.ems-ph.org/doi/10.4171/088
Nikolai I. Lobachevsky, Pangeometry
Athanase
Papadopoulos
IRMA, Strasbourg, France
History and biography
Geometry
Differential geometry
01-02, 51-02; 01A55, 51-01, 51-03, 53-02, 53-01, 53-03, 53A35
Lobachevsky geometry, hyperbolic geometry, non-Euclidean geometry, pangeometry
Lobachevsky wrote his Pangeometry in 1855, the year before his death. This memoir is a résumé of his work on non-Euclidean geometry and its applications, and it can be considered as his clearest account on the subject. It is also the conclusion of his lifework, and the last attempt he made to acquire recognition. The treatise contains basic ideas of hyperbolic geometry, including the trigonometric formulae, the techniques of computation of arc length, of area and of volume, with concrete examples. It also deals with the applications of hyperbolic geometry to the computation of new definite integrals. The techniques are different from those found in most modern books on hyperbolic geometry since they do not use models. Besides its historical importance, Lobachevsky’s Pangeometry is a beautiful work, written in a simple and condensed style. The material that it contains is still very alive, and reading this book will be most useful for researchers and for students in geometry and in the history of science. It can be used as a textbook, as a source book and as a repository of inspiration. The present edition provides the first complete English translation of the Pangeometry that appears in print. It contains facsimiles of both the Russian and the French original versions. The translation is accompanied by notes, followed by a biography of Lobachevky and an extensive commentary.
12
1
2010
978-3-03719-087-6
978-3-03719-587-1
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/087
http://www.ems-ph.org/doi/10.4171/087
Heritage of European Mathematics
Homotopy Quantum Field Theory
With Appendices by Michael Müger and Alexis Virelizier
Vladimir
Turaev
Indiana University, Bloomington, USA
Manifolds and cell complexes
Associative rings and algebras
Category theory; homological algebra
Quantum theory
57-02, 81-02; 16-02, 18-02, 57M27
Topological Quantum Field Theory, surfaces, 3-manifolds, Frobenius algebras, monoidal categories, sections of fiber bundles
Homotopy Quantum Field Theory (HQFT) is a branch of Topological Quantum Field Theory founded by E. Witten and M. Atiyah. It applies ideas from theoretical physics to study principal bundles over manifolds and, more generally, homotopy classes of maps from manifolds to a fixed target space. This book is the first systematic exposition of Homotopy Quantum Field Theory. It starts with a formal definition of an HQFT and provides examples of HQFTs in all dimensions. The main body of the text is focused on 2-dimensional and 3-dimensional HQFTs. A study of these HQFTs leads to new algebraic objects: crossed Frobenius group-algebras, crossed ribbon group-categories, and Hopf group-coalgebras. These notions and their connections with HQFTs are discussed in detail. The text ends with several appendices including an outline of recent developments and a list of open problems. Three appendices by M. Müger and A. Virelizier summarize their work in this area. The book is addressed to mathematicians, theoretical physicists, and graduate students interested in topological aspects of quantum field theory. The exposition is self-contained and well suited for a one-semester graduate course. Prerequisites include only basics of algebra and topology.
6
1
2010
978-3-03719-086-9
978-3-03719-586-4
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/086
http://www.ems-ph.org/doi/10.4171/086
EMS Tracts in Mathematics
10
Bases in Function Spaces, Sampling, Discrepancy, Numerical integration
Hans
Triebel
University of Jena, Germany
Functional analysis
Approximations and expansions
Fourier analysis
Computer science
46-02, 46E35, 42C40, 42B35, 68Q17, 41A55
Function spaces, Haar bases, Faber bases, sampling, numerical integration, discrepancy
The first chapters of this book deal with Haar bases, Faber bases and some spline bases for function spaces in Euclidean n-space and n-cubes. This is used in the subsequent chapters to study sampling and numerical integration preferably in spaces with dominating mixed smoothness. The subject of the last chapter is the symbiotic relationship between numerical integration and discrepancy, measuring the deviation of sets of points from uniformity. This book is addressed to graduate students and mathematicians having a working knowledge of basic elements of function spaces and approximation theory, and who are interested in the subtle interplay between function spaces, complexity theory and number theory (discrepancy).
6
1
2010
978-3-03719-085-2
978-3-03719-585-7
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/085
http://www.ems-ph.org/doi/10.4171/085
EMS Tracts in Mathematics
11
Tractability of Multivariate Problems
Volume II: Standard Information for Functionals
Erich
Novak
University of Jena, Germany
Henryk
Woźniakowski
Columbia University, New York, USA, and University of Warsaw, Poland
Numerical analysis
Computer science
65Y20, 68Q17, 68Q25, 41A63, 65-02, 46E22, 28C20, 46E30, 11K38, 65D32
Discrepancy, lattice rules, Smolyak/sparse grid algorithms, decomposable reproducing kernels, curse of dimensionality, tractability, high-dimensional numerical problems, worst case, average case, probabilistic and randomized settings, weighted spaces
This three-volume set is a comprehensive study of the tractability of multivariate problems. The present second volume deals with algorithms using standard information consisting of function values for the approximation of linear and selected nonlinear functionals. An important example is numerical multivariate integration. The proof techniques used in volumes I and II are quite different. It is especially hard to establish meaningful lower error bounds for the approximation of functionals by using finitely many function values. Here, the concept of decomposable reproducing kernels is helpful, allowing it to find matching lower and upper error bounds for some linear functionals. It is then possible to conclude tractability results from such error bounds. Tractability results even for linear functionals are very rich in variety. There are infinite-dimensional Hilbert spaces for which the approximation with an arbitrarily small error of all linear functionals requires only one function value. There are Hilbert spaces for which all nontrivial linear functionals suffer from the curse of dimensionality. This holds for unweighted spaces, where the role of all variables and groups of variables is the same. For weighted spaces one can monitor the role of all variables and groups of variables. Necessary and sufficient conditions on the decay of the weights are given to obtain various notions of tractability. The text contains extensive chapters on discrepancy and integration, decomposable kernels and lower bounds, the Smolyak/sparse grid algorithms, lattice rules and the CBC (component-by-component) algorithms. This is done in various settings. Path integration and quantum computation are also discussed. The book is of interest for researchers working in computational mathematics, especially in approximation of high-dimensional problems. It is also well suited for graduate courses and seminars. 61 open problems are listed to stimulate future research in tractability.
7
3
2010
978-3-03719-084-5
978-3-03719-584-0
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/084
http://www.ems-ph.org/doi/10.4171/084
EMS Tracts in Mathematics
12
Nonabelian Algebraic Topology
Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids
Ronald
Brown
Bangor University, UK
Philip
Higgins
Durham University, UK
Rafael
Sivera
Universitat de València, Spain
Algebraic topology
Category theory; homological algebra
55-02, 18B40, 18G50, 55P15, 55U50; 18A30, 18A40, 18D15, 18D30, 18D30, 18G55, 19D23, 20F65, 20L05,55Q45, 55S37, 55S45, 55U10, 55U15, 55U35, 57M05, 57M10, 57M15, 57M20
Algebraic topology, homotopy theory, nonabelian methods, van Kampen theorem, groupoids, cubical homotopy groupoids, crossed complexes, filtered spaces, crossed modules, double groupoids, cubical sets with connections, monoidal closed categories, higher category theory, homotopy classification of maps, classifying spaces
The main theme of this book is that the use of filtered spaces rather than just topological spaces allows the development of basic algebraic topology in terms of higher homotopy groupoids; these algebraic structures better reflect the geometry of subdivision and composition than those commonly in use. Exploration of these uses of higher dimensional versions of groupoids has been largely the work of the first two authors since the mid 1960s. The structure of the book is intended to make it useful to a wide class of students and researchers for learning and evaluating these methods, primarily in algebraic topology but also in higher category theory and its applications in analogous areas of mathematics, physics and computer science. Part I explains the intuitions and theory in dimensions 1 and 2, with many figures and diagrams, and a detailed account of the theory of crossed modules. Part II develops the applications of crossed complexes. The engine driving these applications is the work of Part III on cubical ω-groupoids, their relations to crossed complexes, and their homotopically defined examples for filtered spaces. Part III also includes a chapter suggesting further directions and problems, and three appendices give accounts of some relevant aspects of category theory. Endnotes for each chapter give further history and references.
8
11
2011
978-3-03719-083-8
978-3-03719-583-3
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/083
http://www.ems-ph.org/doi/10.4171/083
EMS Tracts in Mathematics
15
Geometrisation of 3-Manifolds
Laurent
Bessières
Université Joseph Fourier, Grenoble, France
Gérard
Besson
Université Joseph Fourier, Grenoble, France
Michel
Boileau
Université Paul Sabatier, Toulouse, France
Sylvain
Maillot
Université Montpellier II, France
Joan
Porti
Universitat Autònoma de Barcelona, Spain
Manifolds and cell complexes
Differential geometry
57-02; 57M50, 53C44
Ricci flow, geometrisation, hyperbolisation, three-dimensional topology, geometric structure, Haken manifolds, JSJ decomposition
The Geometrisation Conjecture was proposed by William Thurston in the mid 1970s in order to classify compact 3-manifolds by means of a canonical decomposition along essential, embedded surfaces into pieces that possess geometric structures. It contains the famous Poincaré Conjecture as a special case. In 2002, Grigory Perelman announced a proof of the Geometrisation Conjecture based on Richard Hamilton’s Ricci flow approach, and presented it in a series of three celebrated arXiv preprints. Since then there has been an ongoing effort to understand Perelman’s work by giving more detailed and accessible presentations of his ideas or alternative arguments for various parts of the proof. This book is a contribution to this endeavour. Its two main innovations are first a simplified version of Perelman’s Ricci flow with surgery, which is called Ricci flow with bubbling-off, and secondly a completely different and original approach to the last step of the proof. In addition, special effort has been made to simplify and streamline the overall structure of the argument, and make the various parts independent of one another. A complete proof of the Geometrisation Conjecture is given, modulo pre-Perelman results on Ricci flow, Perelman’s results on the ℒ-functional and κ-solutions, as well as the Colding–Minicozzi extinction paper. The book can be read by anyone already familiar with these results, or willing to accept them as black boxes. The structure of the proof is presented in a lengthy introduction, which does not require knowledge of geometric analysis. The bulk of the proof is the existence theorem for Ricci flow with bubbling-off, which is treated in parts I and II. Part III deals with the long time behaviour of Ricci flow with bubbling-off. Part IV finishes the proof of the Geometrisation Conjecture.
9
29
2010
978-3-03719-082-1
978-3-03719-582-6
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/082
http://www.ems-ph.org/doi/10.4171/082
EMS Tracts in Mathematics
13
Lectures on Dynamical Systems
Hamiltonian Vector Fields and Symplectic Capacities
Eduard
Zehnder
ETH Zurich, Switzerland
Dynamical systems and ergodic theory
Ordinary differential equations
Differential geometry
Mechanics of particles and systems
37-01; 37A25, 37B25, 37C10, 37C20, 37C27, 37C29,37C50, 37C57, 37D05, 37J05, 37J35, 37J45, 37N05, 34C25, 34C28, 34C37, 34C45, 34D20, 70H15,70H05, 70H25, 53D35, 53C24, 53D05, 58E05, 58E30
Dynamical systems, ergodicity, transitivity, structural stability, stable and unstable manifolds, hyperbolic sets, shadowing lemma, homoclinic and heteroclinic orbits, Bernoulli systems, chaos, limit sets, Lyapunov function, gradient systems, Morse inequalities, symplectic maps, Hamiltonian vector fields, Hamiltonian formalism, integrable systems, symplectic capacities, Hofer–Zehnder capacity, action functional, mini-max principles, periodic motions on prescribed energy surfaces
This book originated from an introductory lecture course on dynamical systems given by the author for advanced students in mathematics and physics at the ETH Zurich. The first part centres around unstable and chaotic phenomena caused by the occurrence of homoclinic points. The existence of homoclinic points complicates the orbit structure considerably and gives rise to invariant hyperbolic sets nearby. The orbit structure in such sets is analyzed by means of the shadowing lemma, whose proof is based on the contraction principle. This lemma is also used to prove S. Smale’s theorem about the embedding of Bernoulli systems near homoclinic orbits. The chaotic behavior is illustrated in the simple mechanical model of a periodically perturbed mathematical pendulum. The second part of the book is devoted to Hamiltonian systems. The Hamiltonian formalism is developed in the elegant language of the exterior calculus. The theorem of V. Arnold and R. Jost shows that the solutions of Hamiltonian systems which possess sufficiently many integrals of motion can be written down explicitly and for all times. The existence proofs of global periodic orbits of Hamiltonian systems on symplectic manifolds are based on a variational principle for the old action functional of classical mechanics. The necessary tools from variational calculus are developed. There is an intimate relation between the periodic orbits of Hamiltonian systems and a class of symplectic invariants called symplectic capacities. From these symplectic invariants one derives surprising symplectic rigidity phenomena. This allows a first glimpse of the fast developing new field of symplectic topology.
5
19
2010
978-3-03719-081-4
978-3-03719-581-9
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/081
http://www.ems-ph.org/doi/10.4171/081
EMS Textbooks in Mathematics
Lectures on Gaussian Integral Operators and Classical Groups
Yurii
Neretin
University of Vienna, Austria, and Moscow State University, Russia
Topological groups, Lie groups
Several complex variables and analytic spaces
Operator theory
Differential geometry
22-02; 22E46, 53C35, 22E50, 47G10, 32M15, 51N30, 14K25, 47B50
Semisimple Lie groups and their representations, classical groups, symmetric spaces, Fourier transform, p-adic groups, integral operators, Hilbert spaces of holomorphic functions, holomorphic discrete series, reproducing kernels, theta-functions, Cartier
This book is an elementary self-contained introduction to some constructions of representation theory and related topics of differential geometry and analysis. Topics covered include the theory of various Fourier-like integral operators as Segal–Bargmann transforms, Gaussian integral operators in $L^2$ and in the Fock space, integral operators with theta-kernels, the geometry of real and $p$-adic classical groups and symmetric spaces. The heart of the book is the Weil representation of the symplectic group (real and complex realizations, relations with theta-functions and modular forms, $p$-adic and adelic constructions) and representations in Hilbert spaces of holomorphic functions of several complex variables. The book is addressed to graduate students and researchers in representation theory, differential geometry, and operator theory. The reader is supposed to be familiar with standard university courses in linear algebra, functional analysis, and complex analysis.
2
25
2011
978-3-03719-080-7
978-3-03719-580-2
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/080
http://www.ems-ph.org/doi/10.4171/080
EMS Series of Lectures in Mathematics
Handbook of Pseudo-Riemannian Geometry and Supersymmetry
Vicente
Cortés
University of Hamburg, Germany
Differential geometry
Quantum theory
Relativity and gravitational theory
53-00; 53C26, 53C50, 81T60, 83E30, 83E50, 53C25, 53C29, 53C35, 53C10
Pseudo-Riemannian manifold, supersymmetry, special geometry, quaternionic Kähler manifold, c-map, generalized geometry, sigma-model, skew-symmetric torsion, nearly Kähler manifold, Euclidian supersymmetry, black hole, para-geometries
The purpose of this handbook is to give an overview of some recent developments in differential geometry related to supersymmetric field theories. The main themes covered are: special geometry and supersymmetry generalized geometry geometries with torsion para-geometries holonomy theory symmetric spaces and spaces of constant curvature conformal geometry wave equations on Lorentzian manifolds D-branes and K-theory The intended audience consists of advanced students and researchers working in differential geometry, string theory and related areas. The emphasis is on geometrical structures occurring on target spaces of supersymmetric field theories. Some of these structures can be fully described in the classical framework of pseudo-Riemannian geometry. Others lead to new concepts relating various fields of research, such as special Kähler geometry or generalized geometry.
6
10
2010
978-3-03719-079-1
978-3-03719-579-6
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/079
http://www.ems-ph.org/doi/10.4171/079
IRMA Lectures in Mathematics and Theoretical Physics
16
Quaternion-Kähler spaces, hyper-Kähler cones, and the c-map
Martin
Roček
Stony Brook University, STONY BROOK, UNITED STATES
Cumrun
Vafa
Harvard University, Cambridge, UNITED STATES
Stefan
Vandoren
Universiteit Utrecht, UTRECHT, NETHERLANDS
Differential geometry
Quantum theory
General
Under the action of the c-map, special Kähler manifolds are mapped into a class of quaternion-Kähler spaces. We explicitly construct the corresponding Swann bundle or hyper-Kähler cone, and determine the hyper-Kähler potential in terms of the prepotencies of the special Kähler geometry.
3
13
1
10.4171/079-1/1
http://www.ems-ph.org/doi/10.4171/079-1/1
Differential forms on quaternionic Kähler manifolds
Gregor
Weingart
Universidad Nacional Autónoma de México, CUERNAVACA, MORELOS, MEXICO
Differential geometry
General
The bundle of differential forms on a hyper-Kähler manifold is a representation of a natural Lie algebra so4,1ℝ of parallel operators. After constructing this algebra from the point of view of representation theory we calculate the minimal parallel decomposition of the bundle of differential forms on hyper-Kähler and quaternionic Kähler manifolds.
15
37
1
10.4171/079-1/2
http://www.ems-ph.org/doi/10.4171/079-1/2
Sasakian geometry, holonomy, and supersymmetry
Charles
Boyer
University of New Mexico, ALBUQUERQUE, UNITED STATES
Krzysztof
Galicki†
University of New Mexico, ALBUQUERQUE, UNITED STATES
Differential geometry
Quantum theory
General
The main purpose of this review is to describe geometric properties of Sasaki–Einstein manifolds which make them interesting in modern theoretical physics. In spite of the fact that it is supersymmetry that connects Sasaki–Einstein spaces to physics, it is not the purpose of this article to describe what this concept really means to either physicists or mathematicians. There have been many recent attempts to frame these important notions of theoretical physics in precise mathematical terms. This enormous task is far beyond the scope of this article. Here we content ourselves with providing the main theorems and results concerning Killing spinors.
39
83
1
10.4171/079-1/3
http://www.ems-ph.org/doi/10.4171/079-1/3
Special geometry for arbitrary signatures
María
Lledó
Universidad de Valencia, BURJASSOT (VALENCIA), SPAIN
Óscar
Maciá
Universidad de Valencia, BURJASSOT (VALENCIA), SPAIN
Antoine
Van Proeyen
Katholieke Universiteit Leuven, LEUVEN, BELGIUM
Veeravalli
Varadarajan
UCLA, LOS ANGELES, UNITED STATES
Differential geometry
General
In this paper we generalize special geometry to arbitrary signatures in target space. We formulate the definitions in a precise mathematical setting and give a translation to the coordinate formalism used in physics. For the projective case, we first discuss in detail projective Kähler manifolds, appearing in N = 1 supergravity. We develop a new point of view based on the intrinsic construction of the line bundle. The topological properties are then derived and the Levi-Civita connection in the projective manifold is obtained as a particular projection of a Levi-Civita connection in a ‘mother’ manifold with one extra complex dimension. The origin of this approach is in the superconformal formalism of physics, which is also explained in detail. Finally, we specialize these results to projective special Kähler manifolds and provide explicit examples with different choices of signature.
85
147
1
10.4171/079-1/4
http://www.ems-ph.org/doi/10.4171/079-1/4
Special geometry, black holes and Euclidean supersymmetry
Thomas
Mohaupt
University of Liverpool, Liverpool, UNITED KINGDOM
Differential geometry
Relativity and gravitational theory
Quantum theory
General
We review recent developments in special geometry and explain its role in the theory of supersymmetric black holes. To make this article self-contained, a short introduction to black holes is given, with emphasis on the laws of black hole mechanics and black hole entropy. We also summarize the existing results on the para-complex version of special geometry, which occurs in Euclidean supersymmetry. The role of real coordinates in special geometry is illustrated, and we briefly indicate how Euclidean supersymmetry can be used to study stationary black hole solutions via dimensional reduction over time.
149
181
1
10.4171/079-1/5
http://www.ems-ph.org/doi/10.4171/079-1/5
Generalized geometry – an introduction
Nigel
Hitchin
University of Oxford, OXFORD, UNITED KINGDOM
Several complex variables and analytic spaces
Algebraic geometry
Differential geometry
General
“Generalized geometry” is an approach to differential geometric structures which seems remarkably well-adapted to some of the concepts in String Theory and Supergravity, for example: 3-form flux, gauged sigma-models, D-branes, connections with skew torsion. It also incorporates in a natural way the role of the B-field as a symmetry. Here we shall offer an introduction to this geometry and a few of its applications, showing how in particular some relatively old results in the physics literature acquire a natural meaning within this new setting.
185
208
1
10.4171/079-1/6
http://www.ems-ph.org/doi/10.4171/079-1/6
Generalizing geometry – algebroids and sigma models
Alexei
Kotov
Université du Luxembourg, LUXEMBOURG, LUXEMBOURG
Thomas
Strobl
Université Claude Bernard Lyon 1, VILLEURBANNE CEDEX, FRANCE
General
In this contribution we review some of the interplay between sigma models in theoretical physics and novel geometrical structures such as Lie (n-)algebroids. The first part of the article contains the mathematical background, the definition of various algebroids as well as of Dirac structures, a joint generalization of Poisson, presymplectic, but also complex structures. Proofs are given in detail. The second part deals with sigma models. Topological ones, in particular the AKSZ and the Dirac sigma models, as generalizations of the Poisson sigma models to higher dimensions and to Dirac structures, respectively, but also physical ones, that reduce to standard Yang Mills theories for the “flat” choice of a Lie algebra: Lie algebroid Yang–Mills theories and possible action functionals for nonabelian gerbes and general higher gauge theories. Characteristic classes associated to Dirac structures and to higher principal bundels are also mentioned.
209
262
1
10.4171/079-1/7
http://www.ems-ph.org/doi/10.4171/079-1/7
A potential for generalized Kähler geometry
Ulf
Lindström
Uppsala Universitet, UPPSALA, SWEDEN
Martin
Roček
Stony Brook University, STONY BROOK, UNITED STATES
Rikard
von Unge
Masaryk University, BRNO, CZECH REPUBLIC
Maxim
Zabzine
Uppsala Universitet, UPPSALA, SWEDEN
Differential geometry
General
We show that, locally, in the neighbourhood of a regular point, all geometric objects of Generalized Kähler Geometry can be derived from a function K, the “generalized Kähler potential”. The metric g and two-form B are determined as nonlinear functions of second derivatives of K. These nonlinearities are shown to arise via a quotient construction from an auxiliary local product (ALP) space.
263
273
1
10.4171/079-1/8
http://www.ems-ph.org/doi/10.4171/079-1/8
Non-integrable geometries, torsion, and holonomy
Ilka
Agricola
Philipps-Universität, MARBURG, GERMANY
Differential geometry
Quantum theory
General
This review article intends to introduce the reader to non-integrable geometric structures on Riemannian manifolds and invariant metric connections with torsion, and to discuss recent aspects of mathematical physics – in particular superstring theory – where these naturally appear.
277
346
1
10.4171/079-1/9
http://www.ems-ph.org/doi/10.4171/079-1/9
Connections with totally skew-symmetric torsion and nearly-Kähler geometry
Paul-Andi
Nagy
University of Auckland, AUCKLAND, NEW ZEALAND
Differential geometry
General
We study almost Hermitian structures admitting a Hermitian connection with totally skew-symmetric torsion or equivalently, those almost Hermitian structures with totally skew-symmetric Nijenhuis tensor. We investigate up to what extent the Nijenhuis tensor fails to be parallel with respect to the characteristic connection. This is naturally described by means of an extension of the notion of Killing form to almost Hermitian geometry. In this context, we also make an essentially self-contained survey of nearly-Kähler geometry, from the perspective of non-integrable holonomy systems.
347
398
1
10.4171/079-1/10
http://www.ems-ph.org/doi/10.4171/079-1/10
Homogeneous nearly Kähler manifolds
Jean-Baptiste
Butruille
Université de Cergy-Pontoise, CERGY-PONTOISE CEDEX, FRANCE
Differential geometry
General
We classify six-dimensional homogeneous nearly Kähler manifolds and give a positive answer to Gray and Wolf’s conjecture: every homogeneous nearly Kähler manifold is a Riemannian 3-symmetric space equipped with its canonical almost Hermitian structure. The only four examples in dimension 6 are S3 × S3, the complex projective space ℂP3, the flag manifold
399
423
1
10.4171/079-1/11
http://www.ems-ph.org/doi/10.4171/079-1/11
Nearly pseudo-Kähler and nearly para-Kähler six-manifolds
Lars
Schäfer
Leibniz Universität Hannover, HANNOVER, GERMANY
Fabian
Schulte-Hengesbach
Universität Hamburg, HAMBURG, GERMANY
Differential geometry
General
The subject of this chapter is six-dimensional nearly (para-)Kähler geometry with pseudo-Riemannian metrics. Firstly, we derive the analogue of the well-known exterior differential system characterising a nearly Kähler manifold and prove applications to the automorphism group of a nearly (para-)Kähler structure. Secondly, we prove existence and uniqueness results for left-invariant nearly (para-)Kähler structures on Lie groups G × G where G is three-dimensional and simple.
425
453
1
10.4171/079-1/12
http://www.ems-ph.org/doi/10.4171/079-1/12
Quaternionic geometries from superconformal symmetry
Andrew
Swann
University of Southern Denmark, ODENSE M, DENMARK
Differential geometry
Quantum theory
General
The geometry of hyper-Kähler manifolds with torsion is surveyed, particularly in the cases arising from Michelson and Strominger’s description of superconformal symmetry of one-dimensional quantum mechanics.
455
474
1
10.4171/079-1/13
http://www.ems-ph.org/doi/10.4171/079-1/13
Twistor and reflector spaces of almost para-quaternionic manifolds
Stefan
Ivanov
University of Sofia, SOFIA, BULGARIA
Ivan
Minchev
University of Sofia, SOFIA, BULGARIA
Simeon
Zamkovoy
University of Sofia, SOFIA, BULGARIA
Differential geometry
General
We investigate the integrability of almost complex structures on the twistor space of an almost para-quaternionic manifold as well as the integrability of natural almost paracomplex structures on the reflector space of an almost para-quaternionic manifold constructed with the help of an arbitrary para-quaternionic connection. We show that if there exists an integrable structure then it is independent on the para-quaternionic connection. In dimension four, we express the ant-self-duality condition in terms of the Riemannian Ricci forms with respect to the associated para-quaternionic structure.
477
496
1
10.4171/079-1/14
http://www.ems-ph.org/doi/10.4171/079-1/14
Para-pluriharmonic maps and twistor spaces
Matthias
Krahe
, Frankfurt, GERMANY
Differential geometry
General
Twistor theory goes back to the 1960s, where twistor methods were introduced by R. Penrose to provide an approach to quantum gravity. The general idea is to translate a problem on a differentiable manifold M endowed with a certain geometric structure to a problem on a complex manifold Z, the twistor space. The purpose of this work is to transfer some of these methods to paracomplex geometry.
497
557
1
10.4171/079-1/15
http://www.ems-ph.org/doi/10.4171/079-1/15
Maximally homogeneous para-CR manifolds of semisimple type
Dmitri
Alekseevsky
University of Edinburgh, EDINBURGH, UNITED KINGDOM
Costantino
Medori
Universit, PARMA, ITALY
Adriano
Tomassini
Università degli Studi di Parma, PARMA, ITALY
Differential geometry
Global analysis, analysis on manifolds
General
An almost para-CR structure on a manifold M is given by a distribution HM ⊂ TM together with a field K ∈ (End(HM)) of involutive endomorphisms of HM. If K satisfies an integrability condition, then (HM,K) is called a para-CR structure. The notion of maximally homogeneous para-CR structure of a semisimple type is given. A classification of such maximally homogeneous para-CR structures is given in terms of appropriate gradations of real semisimple Lie algebras.
559
577
1
10.4171/079-1/16
http://www.ems-ph.org/doi/10.4171/079-1/16
Recent developments in pseudo-Riemannian holonomy theory
Anton
Galaev
Masaryk University, BRNO, CZECH REPUBLIC
Thomas
Leistner
The University of Adelaide, , AUSTRALIA
Differential geometry
General
We review recent results in the theory of holonomy groups of pseudo-Riemannian manifolds, i.e., manifolds with indefinite metrics. First we present the classification of Lorentzian holonomy groups, that is a list of possible groups and metrics which realise these as holonomy groups. This is followed by applications and some remarks about holonomy related structures on Lorentzian manifolds. Then we review partial results in signature (2, n + 2), in particular the classification of unitary holonomy groups, again presenting the groups and the realising metrics. Then we turn to results in neutral signature (n, n), focussing on the situation of a para-Kähler structure. Finally, the classification in signature (2, 2) obtained by Bérard-Bergery and Ikemakhen is presented. As a new result we prove the existence of metrics in cases for which the realisation as holonomy group was left open in their article.
581
627
1
10.4171/079-1/17
http://www.ems-ph.org/doi/10.4171/079-1/17
Geometric applications of irreducible representations of Lie groups
Antonio
Di Scala
Politecnico di Torino, TORINO, ITALY
Thomas
Leistner
The University of Adelaide, , AUSTRALIA
Thomas
Neukirchner
Humboldt-Universität zu Berlin, BERLIN, GERMANY
Differential geometry
Group theory and generalizations
Topological groups, Lie groups
General
In this note we give proofs of the following three algebraic facts which have applications in the theory of holonomy groups and homogeneous spaces: Any irreducibly acting connected subgroup G ⊂ GL(n,ℝ) is closed. Moreover, if G admits an invariant bilinear form of Lorentzian signature, G is maximal, i.e. it is conjugated to SO(1,n − 1)0. We calculate the vector space of G-invariant symmetric bilinear forms, show that it is at most 3-dimensional, and determine the maximal stabilizers for each dimension. Finally, we give some applications and present some open problems.
629
651
1
10.4171/079-1/18
http://www.ems-ph.org/doi/10.4171/079-1/18
Surface holonomy
Konrad
Waldorf
University of California, BERKELEY, UNITED STATES
Differential geometry
Quantum theory
General
The notion of holonomy of a line bundle (endowed with a connection) around a loop is extended to the holonomy of a gerbe along a closed oriented surface, which corresponds to the interaction of a string with a three-form gauge field.
653
682
1
10.4171/079-1/19
http://www.ems-ph.org/doi/10.4171/079-1/19
Classification results for pseudo-Riemannian symmetric spaces
Ines
Kath
Ernst-Moritz-Arndt-Universität Greifswald, GREIFSWALD, GERMANY
Differential geometry
General
We give an introduction to the long-standing classification problem for pseudo- Riemannian symmetric spaces and summarise some old and new approaches and results.
685
702
1
10.4171/079-1/20
http://www.ems-ph.org/doi/10.4171/079-1/20
Pseudo-Kähler and para-Kähler symmetric spaces
Dmitri
Alekseevsky
University of Edinburgh, EDINBURGH, UNITED KINGDOM
Differential geometry
General
We study simply connected pseudo-Kähler and para-Kähler symmetric spaces that is simply connected pseudo-Riemannian symmetric spaces (M,g) which admit a parallel skew-symmetric complex or para-complex structure J ( such that J2 = −Id or J2 = Id respectively.) The classification of para-Kähler symmetric spaces reduces to description of 3-graded Lie algebras g = g−1 + g0 + g1 such that g0-modules g−1 and g1 are dual. The classification of pseudo-Kähler symmetric spaces reduces to description of some anti-involutive automorphisms of complex 3-graded Lie algebras. We classify para-Kähler and pseudo-Kähler Ricci-flat symmetric spaces of dimension ≤ 6. A class of Ricci-flat para-Kähler and pseudo-Kähler symmetric spaces with commutative holonomy group is described in terms of rank four tensors.
703
729
1
10.4171/079-1/21
http://www.ems-ph.org/doi/10.4171/079-1/21
Prehomogeneous affine representations and flat pseudo-Riemannian manifolds
Oliver
Baues
Georg-August Universität Göttingen, GÖTTINGEN, GERMANY
Differential geometry
Number theory
Group theory and generalizations
Topological groups, Lie groups
The theory of flat pseudo-Riemannian manifolds and flat affine manifolds is closely connected to the topic of prehomogeneous affine representations of Lie groups. In this article, we exhibit several aspects of this correspondence. At the heart of our presentation is a development of the theory of characteristic classes and characters of prehomogeneous affine representations. We give applications concerning flat affine, as well as pseudo-Riemannian and symplectic affine flat manifolds.
731
817
1
10.4171/079-1/22
http://www.ems-ph.org/doi/10.4171/079-1/22
The conformal analog of Calabi–Yau manifolds
Helga
Baum
Humboldt-Universität zu Berlin, BERLIN, GERMANY
Differential geometry
General
This survey intends to introduce the reader to holonomy theory of Cartan connections. Special attention is given to the normal conformal Cartan connection, uniquely defined for a class of conformally equivalent metrics, and to its holonomy group – the ‘conformal holonomy group’. We explain the relation between conformal holonomy group and existence of Einstein metrics in the conformal class as well as the relation between conformal holonomy group and existence of conformal Killing spinors. In particular, we describe Lorentzian manifolds with conformal holonomy group in SU(1, m), which can be viewed as conformal analog of Calabi–Yau manifolds. Such Lorentzian metrics, known as Fefferman metrics, appear on S1-bundles over strictly pseudoconvex CR spin manifolds.
821
861
1
10.4171/079-1/23
http://www.ems-ph.org/doi/10.4171/079-1/23
Nondegenerate conformal structures, CR structures and quaternionic CR structures on manifolds
Yoshinobu
Kamishima
Tokyo Metropolitan University, TOKYO, JAPAN
Differential geometry
General
In this expository article, we study integrable, nondegenerate geometric structures on odd dimensional smooth manifolds simultaneously. We are mainly interested in conformal structure on n-manifolds, CR structure on (2n + 1)-manifolds and a new structure, pseudo-conformal quaternionic CR structure on (4n + 3)-manifolds.
863
893
1
10.4171/079-1/24
http://www.ems-ph.org/doi/10.4171/079-1/24
Linear wave equations on Lorentzian manifolds
Christian
Bär
Universität Potsdam, POTSDAM, GERMANY
Global analysis, analysis on manifolds
Quantum theory
General
We summarize the analytic theory of linear wave equations on globally hyperbolic Lorentzian manifolds.
897
914
1
10.4171/079-1/25
http://www.ems-ph.org/doi/10.4171/079-1/25
Survey of D-branes and K-theory
Daniel
Freed
University of Texas at Austin, AUSTIN, UNITED STATES
Differential geometry
Quantum theory
General
In this chapter we explain the relation between D-branes in string theory and K-theory. We will not review the purely mathematical developments, but will instead recount the justification for positing K-theory as the home for D-brane charges.
915
925
1
10.4171/079-1/26
http://www.ems-ph.org/doi/10.4171/079-1/26
Splitting Methods for Partial Differential Equations with Rough Solutions
Analysis and MATLAB programs
Helge
Holden
Norwegian University of Science and Technology, Trondheim, Norway
Kenneth
Karlsen
University of Oslo, Norway
Knut-Andreas
Lie
University of Oslo, Norway
Nils Henrik
Risebro
University of Oslo, Norway
Numerical analysis
Partial differential equations
Operator theory
35L65, 35K65, 65M99, 65M15, 65M12; 35L67, 35D30, 47N40, 35A35
Operator splitting, nonlinear partial differential equations, evolution equations, hyperbolic conservation laws, degenerate convection-diffusion equations, finite difference methods, finite volume methods, front tracking, Matlab codes
Operator splitting (or the fractional steps method) is a very common tool to analyze nonlinear partial differential equations both numerically and analytically. By applying operator splitting to a complicated model one can often split it into simpler problems that can be analyzed separately. In this book one studies operator splitting for a family of nonlinear evolution equations, including hyperbolic conservation laws and degenerate convection-diffusion equations. Common for these equations is the prevalence of rough, or non-smooth, solutions, e.g., shocks. Rigorous analysis is presented, showing that both semi-discrete and fully discrete splitting methods converge. For conservation laws, sharp error estimates are provided and for convection-diffusion equations one discusses a priori and a posteriori correction of entropy errors introduced by the splitting. Numerical methods include finite difference and finite volume methods as well as front tracking. The theory is illustrated by numerous examples. There is a dedicated web page that provides MATLAB codes for many of the examples. The book is suitable for graduate students and researchers in pure and applied mathematics, physics, and engineering.
4
30
2010
978-3-03719-078-4
978-3-03719-578-9
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/078
http://www.ems-ph.org/doi/10.4171/078
EMS Series of Lectures in Mathematics
European Congress of Mathematics Amsterdam, 14–18 July, 2008
A.C.M.
Ran
Vrije Universiteit Amsterdam, The Netherlands
Herman
te Riele
Centrum Wiskunde en Informatica, Amsterdam, The Netherlands
Jan
Wiegerinck
University of Amsterdam, The Netherlands
00Bxx
The European Congress of Mathematics, held every four years, has established itself as a major international mathematical event. Following those in Paris (1992), Budapest (1996), Barcelona (2000) and Stockholm (2004), the Fifth European Congress of Mathematics (5ECM) took place in Amsterdam, The Netherlands, July 14–18, 2008, with about 1000 participants from 68 different countries. Ten plenary and thirty-three invited lectures were delivered. Three science lectures outlined applications of mathematics in other sciences: climate change, quantum information theory and population dynamics. As in the four preceding EMS congresses, ten EMS prizes were granted to very promising young mathematicians. In addition, the Felix Klein Prize was awarded, for the second time, for an application of mathematics to a concrete and difficult industrial problem. There were twenty-two minisymposia, spread over the whole mathematical area. Two round table meetings were organized: one on industrial mathematics and one on mathematics and developing countries. As part of the 44th Nederlands Mathematisch Congres, which was embedded in 5ECM, the so-called Brouwer lecture was presented. It is the Netherland's most prestigious award in mathematics, organized every three years by the Royal Dutch Mathematical Society. Information about Brouwer was given in an invited historical lecture during the congress. These proceedings contain a selection of the contributions to the congress, providing a permanent record of the best what mathematics offers today.
4
24
2010
978-3-03719-077-7
978-3-03719-577-2
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/077
http://www.ems-ph.org/doi/10.4171/077
Uniqueness of bounded solutions to aggregation equations by optimal transport methods
José
Carrillo
Imperial College London, LONDON, UNITED KINGDOM
Jesús
Rosado
Universidad Autonoma de Barcelona, BELLATERRA, SPAIN
Partial differential equations
General
We show how to extend the method used in [22] to prove uniqueness of solutions to a family of several nonlocal equations containing aggregation terms and aggregation/diffusion competition. They contain several mathematical biology models proposed in macroscopic descriptions of swarming and chemotaxis for the evolution of mass densities of individuals or cells. Uniqueness is shown for bounded nonnegative mass-preserving weak solutions without diffusion. In diffusive cases, we use a coupling method [16], [33], and thus we need a stochastic representation of the solution to hold. In summary, our results show, modulo certain technical hypotheses, that nonnegative mass-preserving solutions remain unique as long as their L∞-norm is controlled in time.
3
16
1
10.4171/077-1/1
http://www.ems-ph.org/doi/10.4171/077-1/1
On the computation of the coefﬁcients of modular forms
Bas
Edixhoven
Universiteit Leiden, LEIDEN, NETHERLANDS
Number theory
Algebraic geometry
General
An overview for a non-specialised audience is given of joint work with Jean-Marc Couveignes, Robin de Jong, Franz Merkl, and Johan Bosman. This joint work concerns fast computation of coefﬁcients of modular forms, via the computation of associated Galois representations. For example, for p prime, Ramanujan’s τ(p) can be computed in time polynomial in log p. The overview focuses on the main results and ideas. Developments since 2006 are included: more examples by Johan Bosman, generalisation to forms of level one and of arbitrary weight, and an application to theta functions of lattices. Some future developments are mentioned.
17
29
1
10.4171/077-1/2
http://www.ems-ph.org/doi/10.4171/077-1/2
Effective equidistribution and spectral gap
Manfred
Einsiedler
ETH Zürich, HG G 64.2, ZÜRICH, SWITZERLAND
Dynamical systems and ergodic theory
Number theory
General
In these notes we discuss some equidistribution problems with the aim to give reasonable error rates, i.e., we are interested in effective statements. We motivate some arguments by studying a concrete problem on a two-torus, and then describe recent results on the equidistribution of semisimple orbits obtained in joint work with G. Margulis and A. Venkatesh. We end by studying the relationship between equidistribution of closed orbits and mixing properties. This leads to a way of transporting spectral gap from one group – via an effective equidistribution result on a quotient by an irreducible lattice – to another group. The latter topic is ongoing joint work with G. Margulis and A. Venkatesh.
31
51
1
10.4171/077-1/3
http://www.ems-ph.org/doi/10.4171/077-1/3
Survey on aspherical manifolds
Wolfgang
Lück
Universität Bonn, BONN, GERMANY
Manifolds and cell complexes
$K$-theory
Group theory and generalizations
General
This is a survey on known results and open problems about closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. Many examples come from certain kinds of non-positive curvature conditions. The property aspherical, which is a purely homotopy theoretical condition, implies many striking results about the geometry and analysis of the manifold or its universal covering, and the ring theoretic properties and the K- and L-theory of the group ring associated to its fundamental group. The Borel Conjecture predicts that closed aspherical manifolds are topologically rigid. The article contains new results about product decompositions of closed aspherical manifolds and an announcement of a result joint with Arthur Bartels and Shmuel Weinberger about hyperbolic groups with spheres of dimension ≥ 6 as boundary. At the end we describe (winking) our universe of closed manifolds.
53
82
1
10.4171/077-1/4
http://www.ems-ph.org/doi/10.4171/077-1/4
Wheeled props in algebra, geometry and quantization
Sergei
Merkulov
Stockholm University, STOCKHOLM, SWEDEN
Category theory; homological algebra
Functional analysis
General
Wheeled props is one the latest species found in the world of operads and props. We attempt to give an elementary introduction to the main ideas of the theory of wheeled props for beginners, and also a survey of its most recent major applications (ranging from algebra and geometry to deformation theory and Batalin–Vilkovisky quantization) which might be of interest to experts.
83
114
1
10.4171/077-1/5
http://www.ems-ph.org/doi/10.4171/077-1/5
Positive deﬁnite functions in distance geometry
Oleg
Musin
University of Texas at Brownsville, BROWNSVILLE, UNITED STATES
Fourier analysis
Convex and discrete geometry
General
I. J. Schoenberg proved that a function is positive deﬁnite in the unit sphere if and only if this function is a nonnegative linear combination of Gegenbauer polynomials. This fact plays a crucial role in Delsarte’s method for ﬁnding bounds for the density of sphere packings on spheres and Euclidean spaces. One of the most exciting applications of Delsarte’s method is a solution of the kissing number problem in dimensions 8 and 24. However, 8 and 24 are the only dimensions in which this method gives a precise result. For other dimensions (for instance, three and four) the upper bounds exceed the lower. We have found an extension of Delsarte’s method that allows to solve the kissing number problem (as well as the one-sided kissing number problem) in dimensions three and four. In this paper we also will discuss the maximal cardinalities of spherical two-distance sets. With help of the so-called polynomial method and Delsarte’s method these cardinalities can be determined for all dimensions n < 40. Recently, extensions were found of Schoenberg’s theorem for multivariate positive-deﬁnite functions. With help of these extensions and semideﬁnite programming some upper bounds for spherical codes can be improved.
115
134
1
10.4171/077-1/6
http://www.ems-ph.org/doi/10.4171/077-1/6
From sparse graphs to nowhere dense structures: decompositions, independence, dualities and limits
Jaroslav
Nešetřil
Charles University, PRAHA 1, CZECH REPUBLIC
Patrice
Ossona de Mendez
Ecole des hautes études en sciences sociales, PARIS, FRANCE
Combinatorics
General
This paper surveys the recent research related to the structural properties of sparse graphs and general ﬁnite relational structures. We provide a classiﬁcation of classes of ﬁnite relational structures which is deﬁned by means of a sequence of derived classes (deﬁned by local changes). The resulting classiﬁcation is surprisingly robust and it leads to many equivalent formulations which play an important role in applications in model theory, algorithmic graph theory and structural theory. One of these equivalent formulations guarantees a quick test for a ﬁnite type decomposition (called low tree depth decomposition). We give numerous examples from geometry and extremal graph theory and apply the result to dualities for special classes of graphs. Finally we characterize the existence of all restricted dualities in terms of limit objects induced by the convergence deﬁned on the homomorphism order of graphs.
135
165
1
10.4171/077-1/7
http://www.ems-ph.org/doi/10.4171/077-1/7
Bundle gerbes and surface holonomy
Jürgen
Fuchs
Karlstads Universitet, KARLSTAD, SWEDEN
Thomas
Nikolaus
Universität Regensburg, REGENSBURG, GERMANY
Christoph
Schweigert
Universität Hamburg, HAMBURG, GERMANY
Konrad
Waldorf
University of California, BERKELEY, UNITED STATES
Differential geometry
General
Hermitian bundle gerbes with connection are geometric objects for which a notion of surface holonomy can be deﬁned for closed oriented surfaces. We systematically introduce bundle gerbes by closing the pre-stack of trivial bundle gerbes under descent. Inspired by structures arising in a representation theoretic approach to rational conformal ﬁeld theories, we introduce geometric structure that is appropriate to deﬁne surface holonomy in more general situations: Jandl gerbes for unoriented surfaces, D-branes for surfaces with boundaries, and bi-branes for surfaces with defect lines.
167
195
1
10.4171/077-1/8
http://www.ems-ph.org/doi/10.4171/077-1/8
Topological ﬁeld theories in 2 dimensions
Constantin
Teleman
University of California, BERKELEY, UNITED STATES
Manifolds and cell complexes
Algebraic geometry
Quantum theory
General
We discuss a formality result for 2-dimensional topological ﬁeld theories which are based on a semi-simple Frobenius algebra: namely, when sufﬁciently constrained by structural axioms, the complete theory is determined by the Frobenius algebra and the grading information. The structural constraints apply to Gromov–Witten theory of a variety whose quantum cohomology is semi-simple. Some open questions about semi-simple ﬁeld theories are mentioned in the ﬁnal section.
197
210
1
10.4171/077-1/9
http://www.ems-ph.org/doi/10.4171/077-1/9
The revolution of 1907 – Brouwer’s dissertation
Dirk
van Dalen
Utrecht University, UTRECHT, NETHERLANDS
History and biography
Mathematical logic and foundations
General
Brouwer’s dissertation contains both the germs of his topological and his foundational work. We concentrate here on the latter. The extraordinary rich thesis contains comments and critique on his contemporaries, and a novel approach to many foundational issues. Between the lines one ﬁnds the genesis of the continuum and the natural numbers via the ur-intuition, the constructive interpretation of logic, choice sequences, and a precise discussion of the language, logic, and mathematics levels. The present paper provides a survey of the material and comments on Brouwer’s innovations.
213
230
1
10.4171/077-1/10
http://www.ems-ph.org/doi/10.4171/077-1/10
New developments in combinatorial number theory and applications
Jean
Bourgain
Institute for Advanced Study, PRINCETON, UNITED STATES
Number theory
Group theory and generalizations
General
This is a survey of a line of research in arithmetic combinatorics. It is centered around so-called sum-product phenomena in various settings and its applications to problems in number theory, computer science, spectral and ergodic theory. More speciﬁcally, the sum-product results in ﬁnite ﬁelds and residue rings lead to new bounds on exponential sums of various types and in fact provide the ﬁrst non-trivial estimates. A typical result in this spirit are bounds on Gauss sums for small multiplicative subgroups. Product theorems in matrix spaces derived from the scalar theory enable one to prove various conjectures on the expansion of Cayley graphs and the existence of spectral gaps for Hecke operators, most notably in SL2(q) and SU(2). Those in turn lead to an extension of Selberg’s theorem for congruence subgroups and new results on prime sieving in non-elementary subgroups of SL2(ℤ). Finally, Furstenberg’s “stiffness problem” for toral actions of subgroups of SL2(ℤ) as well as a quantitative equidistibution property of the orbits are described.
233
251
1
10.4171/077-1/11
http://www.ems-ph.org/doi/10.4171/077-1/11
Large random planar maps and their scaling limits
Jean-François
Le Gall
Université Paris-Sud, ORSAY CEDEX, FRANCE
Probability theory and stochastic processes
Combinatorics
General
We discuss scaling limits of random planar maps chosen uniformly at random in a certain class. This leads to a universal limiting space called the Brownian map, which is viewed as a random compact metric space. The Brownian map can be obtained as a quotient of the continuous random tree called the CRT, for an equivalence relation which is deﬁned in terms of Brownian labels assigned to the vertices of the CRT. We discuss the known properties of the Brownian map. In particular, we give a complete description of the geodesics starting from the distinguished point called the root. We also discuss applications to various properties of large random planar maps.
253
276
1
10.4171/077-1/12
http://www.ems-ph.org/doi/10.4171/077-1/12
Geometry and non-archimedean integrals
François
Loeser
Université Pierre et Marie Curie, PARIS CEDEX 05, FRANCE
Number theory
Algebraic geometry
Several complex variables and analytic spaces
General
Non-archimedean integrals are ubiquitous in various parts of mathematics. Motivic integration allows to understand them geometrically and to get strong uniformity statements. In these notes, intended for a general audience, we start by giving various examples of situations where one can get new geometric results by using p-adic or motivic integrals. We then present some more recent results in this area, in particular a Transfer Principle allowing to transfer identities involving functions deﬁned by integrals from one class of local ﬁelds to another. Orbital integrals occurring in the Fundamental Lemma of Langlands Theory form a natural family of functions falling within the range of application of this Transfer Principle.
277
292
1
10.4171/077-1/13
http://www.ems-ph.org/doi/10.4171/077-1/13
Feynman integrals and motives
Matilde
Marcolli
California Institute of Technology, PASADENA, UNITED STATES
Quantum theory
$K$-theory
General
This article gives an overview of recent results on the relation between quantum ﬁeld theory and motives, with an emphasis on two different approaches: a “bottom-up” approach based on the algebraic geometry of varieties associated to Feynman graphs, and a “top-down” approach based on the comparison of the properties of associated categorical structures. This survey is mostly based on joint work of the author with Paolo Alufﬁ, along the lines of the ﬁrst approach, and on previous work of the author with Alain Connes on the second approach.
293
332
1
10.4171/077-1/14
http://www.ems-ph.org/doi/10.4171/077-1/14
Topological quantum ﬁeld theory: 20 years later
Nicolai
Reshetikhin
University of California, BERKELEY, UNITED STATES
Manifolds and cell complexes
Quantum theory
General
This article is an overview of the developments in topological quantum ﬁeld theory, and, in particular on the progress in the Chern–Simons theory.
333
377
1
10.4171/077-1/15
http://www.ems-ph.org/doi/10.4171/077-1/15
Computational complexity and numerical stability of linear problems
Olga
Holtz
University of California, BERKELEY, UNITED STATES
Noam
Shomron
Massachusetts Institute of Technology, CAMBRIDGE, UNITED STATES
Numerical analysis
Algebraic geometry
Group theory and generalizations
Computer science
We survey classical and recent developments in numerical linear algebra, focusing on two issues: computational complexity, or arithmetic costs, and numerical stability, or performance under roundoff error. We present a brief account of the algebraic complexity theory as well as the general error analysis for matrix multiplication and related problems. We emphasize the central role played by the matrix multiplication problem and discuss historical and modern approaches to its solution.
381
400
1
10.4171/077-1/16
http://www.ems-ph.org/doi/10.4171/077-1/16
High-dimensional distributions with convexity properties
Bo'az
Klartag
Tel-Aviv University, TEL-AVIV, ISRAEL
Functional analysis
Convex and discrete geometry
Probability theory and stochastic processes
General
We review recent advances in the understanding of probability measures with geometric characteristics on Rn , for large n. These advances include the central limit theorem for convex sets, according to which the uniform measure on a high-dimensional convex body has marginals that are approximately gaussian.
401
417
1
10.4171/077-1/17
http://www.ems-ph.org/doi/10.4171/077-1/17
Some recent results about the sixth problem of Hilbert: hydrodynamic limits of the Boltzmann equation
Laure
Saint-Raymond
Ecole Normale Superieure, PARIS CEDEX 05, FRANCE
Partial differential equations
Fluid mechanics
Statistical mechanics, structure of matter
General
The behaviour of a gas can be described by different models depending on the time and space scales to be considered. A natural question is therefore to know if there exist continuous transitions between these models: this is precisely the matter of the sixth problem asked by Hilbert on the occasion of the International Congress of Mathematicians held in Paris in 1900. Recent works allow to understand rigorously the passage from Boltzmann’s equation (which provides a statistical description of the microscopic state of the gas) to continuous models of ﬂuid mechanics. The goal of the present paper is to present brieﬂy the mathematical framework of the study, and to exhibit some difﬁculties of the asymptotic analysis related to the physics of the problem.
419
439
1
10.4171/077-1/18
http://www.ems-ph.org/doi/10.4171/077-1/18
Graded algebras associated to algebraic algebras need not be algebraic
Agata
Smoktunowicz
University of Edinburgh, EDINBURGH, UNITED KINGDOM
Associative rings and algebras
General
It is shown that the associated graded algebras of afﬁne nil algebras need not be nil. A similar argument shows that ﬁltered algebras generated in degree one of nil algebras need not be nil. This answers a question of Riley [5].
441
449
1
10.4171/077-1/19
http://www.ems-ph.org/doi/10.4171/077-1/19
Lectures on the ℒ2-Sobolev Theory of the ∂-Neumann problem
Emil
Straube
Texas A&M University, College Station, USA
Several complex variables and analytic spaces
Partial differential equations
32-02; 32W05, 35N15
∂-Neumann – problem, ℒ2-theory, pseudoconvex domains, compactness, global regularity
This book provides a thorough and self-contained introduction to the ∂-Neumann problem, leading up to current research, in the context of the ℒ2-Sobolev theory on bounded pseudoconvex domains in ℂn. It grew out of courses for advanced graduate students and young researchers given by the author at the Erwin Schrödinger International Institute for Mathematical Physics and at Texas A&M University. The introductory chapter provides an overview of the contents and puts them in historical perspective. The second chapter presents the basic ℒ2-theory. Following is a chapter on the subelliptic estimates on strictly pseudoconvex domains. The two final chapters on compactness and on regularity in Sobolev spaces bring the reader to the frontiers of research. Prerequisites are a solid background in basic complex and functional analysis, including the elementary ℒ2-Sobolev theory and distributions. Some knowledge in several complex variables is helpful. Concerning partial differential equations, not much is assumed. The elliptic regularity of the Dirichlet problem for the Laplacian is quoted a few times, but the ellipticity results needed for elliptic regularization in the third chapter are proved from scratch.
2
26
2010
978-3-03719-076-0
978-3-03719-576-5
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/076
http://www.ems-ph.org/doi/10.4171/076
ESI Lectures in Mathematics and Physics
Decorated Teichmüller Theory
Robert
Penner
Centre for Quantum Geometry of Moduli Spaces, Aarhus, Denmark, and Caltech, Pasadena, USA
Functions of a complex variable
Several complex variables and analytic spaces
30-02, 30F60, 32G15, 30F10, 30Fxx
Teichmüller space, mapping class group, Riemann’s moduli space, quantum Teichmüller theory
There is an essentially “tinker-toy” model of a trivial bundle over the classical Teichmüller space of a punctured surface, called the decorated Teichmüller space, where the fiber over a point is the space of all tuples of horocycles, one about each puncture. This model leads to an extension of the classical mapping class groups called the Ptolemy groupoids and to certain matrix models solving related enumerative problems, each of which has proved useful both in mathematics and in theoretical physics. These spaces enjoy several related parametrizations leading to a rich and intricate algebro-geometric structure tied to the already elaborate combinatorial structure of the tinker-toy model. Indeed, the natural coordinates give the prototypical examples not only of cluster algebras but also of tropicalization. This interplay of combinatorics and coordinates admits further manifestations, for example, in a Lie theory for homeomorphisms of the circle, in the geometry underlying the Gauss product, in profinite and pronilpotent geometry, in the combinatorics underlying conformal and topological quantum field theories, and in the geometry and combinatorics of macromolecules. This volume gives the story and wider context of these decorated Teichmüller spaces as developed by the author over the last two decades in a series of papers, some of them in collaboration. Sometimes correcting errors or typos, sometimes simplifying proofs and sometimes articulating more general formulations than the original research papers, this volume is self-contained and requires little formal background. Based on a master’s course at Aarhus University, it gives the first treatment of these works in monographic form.
1
2
2012
978-3-03719-075-3
978-3-03719-575-8
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/075
http://www.ems-ph.org/doi/10.4171/075
The QGM Master Class Series
Cohomological Theory of Crystals over Function Fields
Gebhard
Böckle
Interdisciplinary Center for Scientific Computing, Heidelberg, Germany
Richard
Pink
ETH Zürich, Switzerland
Number theory
Algebraic geometry
11-02, 14-02
Characteristic $p$ cohomologies, Drinfeld modules, $A$-motives, $L$-functions, Weil conjecture
This book develops a new cohomological theory for schemes in positive characteristic p and it applies this theory to give a purely algebraic proof of a conjecture of Goss on the rationality of certain L-functions arising in the arithmetic of function fields. These L-functions are power series over a certain ring A, associated to any family of Drinfeld A-modules or, more generally, of A-motives on a variety of finite type over the finite field Fp. By analogy to the Weil conjecture, Goss conjectured that these L-functions are in fact rational functions. In 1996 Taguchi and Wan gave a first proof of Goss’s conjecture by analytic methods à la Dwork. The present text introduces A-crystals, which can be viewed as generalizations of families of A-motives, and studies their cohomology. While A-crystals are defined in terms of coherent sheaves together with a Frobenius map, in many ways they actually behave like constructible étale sheaves. A central result is a Lefschetz trace formula for L-functions of A-crystals, from which the rationality of these L-functions is immediate. Beyond its application to Goss’s L-functions, the theory of A-crystals is closely related to the work of Emerton and Kisin on unit root F-crystals, and it is essential in an Eichler–Shimura type isomorphism for Drinfeld modular forms as constructed by the first author. The book is intended for researchers and advanced graduate students interested in the arithmetic of function fields and/or cohomology theories for varieties in positive characteristic. It assumes a good working knowledge in algebraic geometry as well as familiarity with homological algebra and derived categories, as provided by standard textbooks. Beyond that the presentation is largely self-contained.
10
1
2009
978-3-03719-074-6
978-3-03719-574-1
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/074
http://www.ems-ph.org/doi/10.4171/074
EMS Tracts in Mathematics
9
Renormalization and Galois Theories
Alain
Connes
IHÉS, Bures-sur-Yvette, France
Frédéric
Fauvet
IRMA, Strasbourg, France
Jean-Pierre
Ramis
Institut de Mathématiques de Toulouse, France
Algebraic geometry
Quantum theory
14F42, 34M30, 34M35, 34M37, 34M40, 81T15, 81T18, 81T75; 11S40, 33B30, 34M50
Renormalization, motives, noncommutative geometry, Hopf algebras, differential Galois groups, Borel summation, Gevrey asymptotics, Stokes phenomenon, resurgent functions
This volume is the outcome of a CIRM Workshop on Renormalization and Galois Theories held in Luminy, France, in March 2006. The subject of this workshop was the interaction and relationship between four currently very active areas: renormalization in quantum field theory (QFT), differential Galois theory, noncommutative geometry, motives and Galois theory. The last decade has seen a burst of new techniques to cope with the various mathematical questions involved in QFT, with notably the development of a Hopf-algebraic approach and insights into the classes of numbers and special functions that systematically appear in the calculations of perturbative QFT (pQFT). The analysis of the ambiguities of resummation of the divergent series of pQFT, an old problem, has been renewed, using recent results on Gevrey asymptotics, generalized Borel summation, Stokes phenomenon and resurgent functions. The purpose of the present book is to highlight, in the context of renormalization, the convergence of these various themes, orchestrated by diverse Galois theories. It contains three lecture courses together with five research articles and will be useful to both reseachers and graduate students in mathematics and physics.
9
24
2009
978-3-03719-073-9
978-3-03719-573-4
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/073
http://www.ems-ph.org/doi/10.4171/073
IRMA Lectures in Mathematics and Theoretical Physics
15
Noncommutative geometry and motives (à quoi servent les endomotifs?)
Caterina
Consani
The Johns Hopkins University, BALTIMORE, UNITED STATES
Global analysis, analysis on manifolds
Number theory
Algebraic geometry
General
This paper gives a short and historical survey on the theory of pure motives in algebraic geometry and reviews some of the recent developments of this theory in noncommutative geometry. The second part of the paper outlines the new theory of endomotives and some of its relevant applications in number theory.
1
37
1
10.4171/073-1/1
http://www.ems-ph.org/doi/10.4171/073-1/1
Renormalisation of non-commutative field theories
Vincent
Rivasseau
Université Paris Sud-XI, ORSAY CEDEX, FRANCE
Fabien
Vignes-Tourneret
Université Claude Bernard Lyon 1, VILLEURBANNE CEDEX, FRANCE
General
The first renormalisable quantum field theories on non-commutative space have been found recently. We review this rapidly growing subject.
39
82
1
10.4171/073-1/2
http://www.ems-ph.org/doi/10.4171/073-1/2
Mould expansions for the saddle-node and resurgence monomials
David
Sauzin
, PARIS, FRANCE
General
This article is an introduction to some aspects of Écalle’s mould calculus, a powerful combinatorial tool which yields surprisingly explicit formulas for the normalising series attached to an analytic germ of singular vector field or of map. This is illustrated on the case of the saddle-node, a two-dimensional vector field which is formally conjugate to Euler’s vector field x2 ∂/∂x + (x + y) ∂/∂y, and for which the formal normalisation is shown to be resurgent in 1/x. Resurgence monomials adapted to alien calculus are also described as another application of mould calculus.
83
163
1
10.4171/073-1/3
http://www.ems-ph.org/doi/10.4171/073-1/3
Galois theory, motives and transcendental numbers
Yves
André
Projet de théorie des nombres, tour 25 4ème étage, PARIS, FRANCE
General
From its early beginnings up to nowadays, algebraic number theory has evolved in symbiosis with Galois theory: indeed, one could hold that it consists in the very study of the absolute Galois group of the field of rational numbers. Nothing like that can be said of transcendental number theory. Nevertheless, could one not associate conjugates and a Galois group to transcendental numbers such as π? Beyond, can one not envision an appropriate Galois theory in the field of transcendental number theory? In which role? The aim of this text is to indicate what Grothendieck’s theory of motives has to say, at least conjecturally, on these questions.
165
177
1
10.4171/073-1/4
http://www.ems-ph.org/doi/10.4171/073-1/4
The combinatorics of Bogoliubov’s recursion in renormalization
Kurusch
Ebrahimi-Fard
Université de Haute Alsace, MULHOUSE CEDEX 07, FRANCE
Dominique
Manchon
Université Blaise Pascal, AUBIÈRE CEDEX, FRANCE
General
We describe various combinatorial aspects of the Birkhoff–Connes–Kreimer factorization in perturbative renormalization. The analog of Bogoliubov’s preparation map on the Lie algebra of Feynman graphs is identified with the pre-Lie Magnus expansion. Our results apply to any connected filtered Hopf algebra, based on the pro-nilpotency of the Lie algebra of infinitesimal characters.
179
207
1
10.4171/073-1/5
http://www.ems-ph.org/doi/10.4171/073-1/5
(Non)commutative Hopf algebras of trees and (quasi)symmetric functions
Michael
Hoffman
United States Naval Academy, ANNAPOLIS, UNITED STATES