- book metadata
European Mathematical Society Publishing House
2018-11-26 00:06:01
189
http://www.ems-ph.org/meta/bmeta-stream.php?update_since=2018-11-26
Function Spaces with Dominating Mixed Smoothness
Hans
Triebel
Friedrich-Schiller Universität Jena, Germany
Functional analysis
Fourier analysis
46-02, 46E35, 42C40, 42B35, 41A55
Functional analysis
Function spaces, dominating mixed smoothness, spaces on domains, wavelets, Faber frames, Haar frames
The first part of this book is devoted to function spaces in Euclidean $n$-space with dominating mixed smoothness. Some new properties are derived and applied in the second part where weighted spaces with dominating mixed smoothness in arbitrary bounded domains in Euclidean $n$-space are introduced and studied. This includes wavelet frames, numerical integration and discrepancy, measuring the deviation of sets of points from uniformity. 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. In particular, it will be of interest for researchers dealing with approximation theory, numerical integration and discrepancy.
1
31
2019
978-3-03719-195-8
978-3-03719-695-3
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/195
http://www.ems-ph.org/doi/10.4171/195
EMS Series of Lectures in Mathematics
2523-5176
2523-5184
The Shock Development Problem
Demetrios
Christodoulou
ETH Zürich, Switzerland
Partial differential equations
Fluid mechanics
35L67; 35L04, 35L65, 76L05, 76N15
Differential equations
Nonlinear hyperbolic partial differential equations, free boundary problems, mechanics of compressible fluids, development of shocks
This monograph addresses the problem of the development of shocks in the context of the Eulerian equations of the mechanics of compressible fluids. The mathematical problem is that of an initial-boundary value problem for a nonlinear hyperbolic system of partial differential equations with a free boundary and singular initial conditions. The free boundary is the shock hypersurface and the boundary conditions are jump conditions relative to a prior solution, conditions following from the integral form of the mass, momentum and energy conservation laws. The prior solution is provided by the author‘s previous work which studies the maximal classical development of smooth initial data. New geometric and analytic methods are introduced to solve the problem. Geometry enters as the acoustical structure, a Lorentzian metric structure defined on the spacetime manifold by the fluid. This acoustical structure interacts with the background spacetime structure. Reformulating the equations as two coupled first order systems, the characteristic system, which is fully nonlinear, and the wave system, which is quasilinear, a complete regularization of the problem is achieved. Geometric methods also arise from the need to treat the free boundary. These methods involve the concepts of bi-variational stress and of variation fields. The main new analytic method arises from the need to handle the singular integrals appearing in the energy identities. Shocks being an ubiquitous phenomenon, occuring also in magnetohydrodynamics, nonlinear elasticity, and the electrodynamics of noninear media, the methods developed in this monograph are likely to be found relevant in these fields as well.
1
31
2019
978-3-03719-192-7
978-3-03719-692-2
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/192
http://www.ems-ph.org/doi/10.4171/192
EMS Monographs in Mathematics
2523-5192
2523-5206
Boundary Behavior of Solutions to Elliptic Equations in General Domains
Vladimir
Maz'ya
Linköping University, Sweden and University of Liverpool, UK
Partial differential equations
Potential theory
35J40, 31B15, 31B25
Differential equations
Wiener test, higher order elliptic equations, elasticity systems, Zaremba problem, weighted positivity, capacity
The present book is a detailed exposition of the author and his collaborators’ work on boundedness, continuity, and differentiability properties of solutions to elliptic equations in general domains, that is, in domains that are not a priori restricted by assumptions such as “piecewise smoothness” or being a “Lipschitz graph”. The description of the boundary behavior of such solutions is one of the most difficult problems in the theory of partial differential equations. After the famous Wiener test, the main contributions to this area were made by the author. In particular, necessary and sufficient conditions for the validity of imbedding theorems are given, which provide criteria for the unique solvability of boundary value problems of second and higher order elliptic equations. Another striking result is a test for the regularity of a boundary point for polyharmonic equations. The book will be interesting and useful for a wide audience. It is intended for specialists and graduate students working in the theory of partial differential equations.
9
30
2018
978-3-03719-190-3
978-3-03719-690-8
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/190
http://www.ems-ph.org/doi/10.4171/190
EMS Tracts in Mathematics
30
Geometric and Topological Aspects of Coxeter Groups and Buildings
Anne
Thomas
The University of Sydney, Australia
Group theory and generalizations
Primary: 20F55; secondary: 20E42, 51E24, 57M07
Groups + group theory
Coxeter groups, buildings, Davis complexes
Coxeter groups are groups generated by reflections, and they appear throughout mathematics. Tits developed the general theory of Coxeter groups in order to develop the theory of buildings. Buildings have interrelated algebraic, combinatorial and geometric structures, and are powerful tools for understanding the groups which act on them. These notes focus on the geometry and topology of Coxeter groups and buildings, especially nonspherical cases. The emphasis is on geometric intuition, and there are many examples and illustrations. Part I describes Coxeter groups and their geometric realisations, particularly the Davis complex, and Part II gives a concise introduction to buildings. This book will be suitable for mathematics graduate students and researchers in geometric group theory, as well as algebra and combinatorics. The assumed background is basic group theory, including group actions, and basic algebraic topology, together with some knowledge of Riemannian geometry.
5
31
2018
978-3-03719-189-7
978-3-03719-689-2
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/189
http://www.ems-ph.org/doi/10.4171/189
Zurich Lectures in Advanced Mathematics
A Brief Introduction to Spectral Graph Theory
Bogdan
Nica
McGill University, Montreal, Canada
Combinatorics
Number theory
Linear and multilinear algebra; matrix theory
Primary: 05-01, 05C50; secondary: 05C25, 11T24, 15A42
Combinatorics + graph theory
Adjacency eigenvalues of graphs, Laplacian eigenvalues of graphs, Cayley graphs, algebraic graphs over finite fields, character sums
Spectral graph theory starts by associating matrices to graphs – notably, the adjacency matrix and the Laplacian matrix. The general theme is then, firstly, to compute or estimate the eigenvalues of such matrices, and secondly, to relate the eigenvalues to structural properties of graphs. As it turns out, the spectral perspective is a powerful tool. Some of its loveliest applications concern facts that are, in principle, purely graph theoretic or combinatorial. This text is an introduction to spectral graph theory, but it could also be seen as an invitation to algebraic graph theory. The first half is devoted to graphs, finite fields, and how they come together. This part provides an appealing motivation and context of the second, spectral, half. The text is enriched by many exercises and their solutions. The target audience are students from the upper undergraduate level onwards. We assume only a familiarity with linear algebra and basic group theory. Graph theory, finite fields, and character theory for abelian groups receive a concise overview and render the text essentially self-contained.
5
31
2018
978-3-03719-188-0
978-3-03719-688-5
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/188
http://www.ems-ph.org/doi/10.4171/188
EMS Textbooks in Mathematics
An Introduction to Kac–Moody Groups over Fields
Timothée
Marquis
Université Catholique de Louvain, Louvain-la-Neuve, Belgium
Group theory and generalizations
20G44, 20E42, 17B67
Groups + group theory
Kac–Moody groups, Kac–Moody algebras, infinite-dimensional Lie theory, highest-weight modules, semisimple algebraic groups, loop groups, affine group schemes, Coxeter groups, buildings, BN pairs, Tits systems, root group data
The interest for Kac–Moody algebras and groups has grown exponentially in the past decades, both in the mathematical and physics communities, and with it also the need for an introductory textbook on the topic. The aims of this book are twofold: - to offer an accessible, reader-friendly and self-contained introduction to Kac–Moody algebras and groups; - to clean the foundations and to provide a unified treatment of the theory. The book starts with an outline of the classical Lie theory, used to set the scene. Part II provides a self-contained introduction to Kac–Moody algebras. The heart of the book is Part III, which develops an intuitive approach to the construction and fundamental properties of Kac–Moody groups. It is complemented by two appendices, respectively offering introductions to affine group schemes and to the theory of buildings. Many exercises are included, accompanying the readers throughout their journey. The book assumes only a minimal background in linear algebra and basic topology, and is addressed to anyone interested in learning about Kac–Moody algebras and/or groups, from graduate (master) students to specialists.
6
15
2018
978-3-03719-187-3
978-3-03719-687-8
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/187
http://www.ems-ph.org/doi/10.4171/187
EMS Textbooks in Mathematics
Non-Linear Partial Differential Equations, Mathematical Physics, and Stochastic Analysis
The Helge Holden Anniversary Volume
Fritz
Gesztesy
Baylor University, Waco, USA
Harald
Hanche-Olsen
The Norwegian University of Science and Technology, Trondheim, Norway
Espen
Jakobsen
The Norwegian University of Science and Technology, Trondheim, Norway
Yurii
Lyubarskii
The Norwegian University of Science and Technology, Trondheim, Norway
Nils Henrik
Risebro
University of Oslo, Norway
Kristian
Seip
Norwegian University of Science and Technology, Trondheim, Norway
Partial differential equations
Linear and multilinear algebra; matrix theory
Dynamical systems and ergodic theory
Fourier analysis
Primary: 15B52, 35J10, 35L65, 35Q41, 35Q51, 35Q53, 37K10, 42B20, 46N20, 46N30, 46T12, 47B36, 47F05, 60H20, 68N30, 76S05; Secondary: 33C45, 35A01, 35A02, 35L80, 37D45, 39A12, 47A10, 47N20, 47N30, 60B20
Differential equations
Infinite-dimensional analysis, partial differential equations, hyperbolic conservation laws, stochastic analysis, spectral theory, discrete evolution, completely integrable systems, random matrix theory, chaotic dynamics
This volume is dedicated to Helge Holden on the occasion of his 60th anniversary. It collects contributions by numerous scientists with expertise in non-linear partial differential equations (PDEs), mathematical physics, and stochastic analysis, reflecting to a large degree Helge Holden’s longstanding research interests. Accordingly, the problems addressed in the contributions deal with a large range of topics, including, in particular, infinite-dimensional analysis, linear and nonlinear PDEs, stochastic analysis, spectral theory, completely integrable systems, random matrix theory, and chaotic dynamics and sestina poetry. They represent to some extent the lectures presented at the conference Non-linear PDEs, Mathematical Physics and Stochastic Analysis, held at NTNU, Trondheim, July 4–7, 2016 (https://wiki.math.ntnu.no/holden60). The mathematical tools involved draw from a wide variety of techniques in functional analysis, operator theory, and probability theory. This collection of research papers will be of interest to any active scientist working in one of the above mentioned areas.
5
31
2018
978-3-03719-186-6
978-3-03719-686-1
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/186
http://www.ems-ph.org/doi/10.4171/186
EMS Series of Congress Reports
2523-515X
2523-5168
Optimal control of forward-backward stochastic Volterra equations
Nacira
Agram
University of Oslo, Norway
Bernt
Øksendal
University of Oslo, Norway
Samia
Yakhlef
University of Biskra, Algeria
Partial differential equations
We study the problem of optimal control of a coupled system of forward-backward stochastic Volterra equations. We use Hida–Malliavin calculus to prove a sufficient and a necessary maximum principle for the optimal control of such systems. Existence and uniqueness of backward stochastic Volterra integral equations are proved. As an application of our methods, we solve a recursive utility optimisation problem in a financial model with memory.
3
35
1
10.4171/186-1/1
http://www.ems-ph.org/doi/10.4171/186-1/1
A unified approach to infinite dimensional integrals of probabilistic and oscillatory type with applications to Feynman path integrals
Sergio
Albeverio
Universität Bonn, Germany
Sonia
Mazzucchi
Università di Trento, Povo (Trento), Italy
Partial differential equations
An unified approach to infinite dimensional integration in terms of linear continuous functional is presented, including the cases of both probabilistic and oscillatory integrals. Applications to the theory of Feynman path integrals and to the study of high-order heat-type equations are also presented.
37
53
1
10.4171/186-1/2
http://www.ems-ph.org/doi/10.4171/186-1/2
The numbers lead a dance. Mathematics of the Sestina
Alan
Champneys
University of Bristol, UK
Poul
Hjorth
Technical University of Denmark, Kgs. Lyngby, Denmark
Harry
Man
Oxford Brookes University, UK
Partial differential equations
Sestinas are poems of 39 lines comprising six verses of six lines each, and a three line final verse or ‘envoi’. The structure of the sestina is built around word repetition rather than strict rhyme. Each verse uses the same set line ending words, but in a permuted order. The form of the permutation is highly specific, and is equivalent to iteration of the tent map. This paper considers for which number $N$ of verses, other than 6, can a sestina-like poem be formed. That is, which $N$ will the prescribed permutation lead to a poem of $N$ verses where no two verses have the same order of their end words. In so doing, a link is found between permutation groups, chaotic dynamics, and Cunningham numbers.
55
71
1
10.4171/186-1/3
http://www.ems-ph.org/doi/10.4171/186-1/3
Compensated compactness in Banach spaces and weak rigidity of isometric immersions of manifolds
Gui-Qiang
Chen
Oxford University, UK
Siran
Li
Rice University, Houston, USA
Partial differential equations
We present a compensated compactness theorem in Banach spaces established recently, whose formulation is originally motivated by the weak rigidity problem for isometric immersions of manifolds with lower regularity. As a corollary, a geometrically intrinsic div–curl lemma for tensor fields on Riemannian manifolds is obtained. Then we show how this intrinsic div–curl lemma can be employed to establish the global weak rigidity of the Gauss–Codazzi–Ricci equations, the Cartan formalism, and the corresponding isometric immersions of Riemannian submanifolds.
73
95
1
10.4171/186-1/4
http://www.ems-ph.org/doi/10.4171/186-1/4
The initial-boundary-value problem for an Ostrovsky–Hunter type equation
Giuseppe Maria
Coclite
Politecnico di Bari, Italy
Lorenzo
di Ruvo
Politecnico di Bari, Italy
Kenneth Hvistendahl
Karlsen
University of Oslo, Norway
Partial differential equations
We consider an Ostrovsky–Hunter type equation. We prove the well-posedness of the entropy solution for the non-homogeneous initial boundary value problem. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method.
97
109
1
10.4171/186-1/5
http://www.ems-ph.org/doi/10.4171/186-1/5
Modeling crowd dynamics through hyperbolic – elliptic equations
Rinaldo
Colombo
Università degli Studi di Brescia, Italy
Maria
Gokieli
University of Warsaw, Poland
Massimiliano
Rosini
Maria Curie-Skłodowska University (UMCS), Lublin, Poland
Partial differential equations
Inspired by the works of Hughes [22, 23], we formalize and prove the well-posedness of a hyperbolic–elliptic system whose solutions describe the dynamics of a moving crowd. The resulting model is here shown to be well-posed and the time of evacuation from a bounded environment is proved to be finite. This model also provides a microscopic description of the individuals’ behaviors.
111
128
1
10.4171/186-1/6
http://www.ems-ph.org/doi/10.4171/186-1/6
On the well-posedness of solutions with finite energy for nonlocal equations of porous medium type
Félix
del Teso
The Norwegian University of Science and Technology (NTNU), Trondheim, Norway
Jørgen
Endal
The Norwegian University of Science and Technology (NTNU), Trondheim, Norway
Espen
Jakobsen
The Norwegian University of Science and Technology (NTNU), Trondheim, Norway
Partial differential equations
We study well-posedness and equivalence of different notions of solutions with finite energy for nonlocal porous medium type equations of the form $$\partial_t u – A \varphi (u) = 0.$$ These equations are possibly degenerate nonlinear diffusion equations with a general nondecreasing continuous nonlinearity $\varphi$, and the largest class of linear symmetric nonlocal diffusion operators $A$ considered so far. The operators are defined from a bilinear energy form $\mathcal E$ and may be degenerate and have some $x$-dependence. The fractional Laplacian, symmetric finite differences, and any generator of symmetric pure jump Lévy processes are included. The main results are (i) an Oleĭnik type uniqueness result for energy solutions; (ii) an existence (and uniqueness) result for distributional solutions with finite energy; and (iii) equivalence between the two notions of solution, and as a consequence, new wellposedness results for both notions of solutions. We also obtain quantitative energy and related $L^p$-estimates for distributional solutions. Our uniqueness results are given for a class of functions defined from test functions by completion in a certain topology. We study rigorously several cases where this space coincides with standard function spaces. In particular, for operators comparable to fractional Laplacians, we show that this space is a parabolic homogeneous fractional Sobolev space.
129
167
1
10.4171/186-1/7
http://www.ems-ph.org/doi/10.4171/186-1/7
On the spectrum of leaky surfaces with a potential bias
Pavel
Exner
Czech Academy of Sciences, Řež near Prague, Czechia, and Czech Technical University, Prague, Czechia
Partial differential equations
We discuss operators of the type $H = –\Delta + V(x) – \alpha \delta (x–\Sigma)$ with an attractive interaction, $\alpha > 0$ in $L^2(\mathbb R^3)$, where $\Sigma$ is an infinite surface, asymptotically planar and smooth outside a compact, dividing the space into two regions, of which one is supposed to be convex, and $VB$ is a potential bias being a positive constant $V_0$ in one of the regions and zero in the other. We find the essential spectrum and ask about the existence of the discrete one with a particular attention to the critical case, $V_0 = \alpha^2$. We show that $\sigma_{\mathrm {disc}} (H)$ is then empty if the bias is supported in the ‘exterior’ region, while in the opposite case isolated eigenvalues may exist.
169
181
1
10.4171/186-1/8
http://www.ems-ph.org/doi/10.4171/186-1/8
On the decay of almost periodic solutions of anisotropic degenerate parabolic-hyperbolic equations
Hermano
Frid
Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brazil
Partial differential equations
We prove the well-posedness and decay of Besicovitch almost periodic solutions for nonlinear degenerate anisotropic hyperbolic-parabolic equations. The decay property is proven for the case where the diffusion term is given by a non-degenerate nonlinear $d’’ \times d’’$ diffusion matrix and the complementary $d’$ components of flux-function form a non-degenerate flux in $\mathbb R^{d’}$, with $d’+ d’’ = d$. For this special case we also prove that the strong trace property at the initial time holds, which allows, in particular, to require the assumption of the initial data only in a weak sense, and gives the continuity in time of the solution with values in $L^1_{\mathrm {loc}} (\mathbb R^d)$). So far, for the decay property, we need also to require that the bounded Besicovitch almost periodic initial function can be approximated in the Besicovitch norm by almost periodic functions whose $\epsilon$-inclusion intervals $l_\epsilon$ satisfy $l_\epsilon | \mathrm {log}\: \epsilon |^1/2 \to 0$ as $\epsilon \to 0$. This includes, in particular, generalized limit periodic functions, that is, limits in the Besicovitch norm of purely periodic functions.
183
205
1
10.4171/186-1/9
http://www.ems-ph.org/doi/10.4171/186-1/9
Factorizations and Hardy–Rellich-type inequalities
Fritz
Gesztesy
Baylor University, Waco, USA
Lance
Littlejohn
Baylor University, Waco, USA
Partial differential equations
The principal aim of this note is to illustrate how factorizations of singular, even-order partial differential operators yield an elementary approach to classical inequalities of Hardy–Rellich-type. More precisely, introducing the two-parameter $n$-dimensional homogeneous scalar differential expressions $T_{\alpha,\beta} := - \Delta + \alpha |x|^{-2} x \cdot \nabla + \beta |x|^{-2}$, $\alpha, \beta \in \mathbb R$, $x \in \mathbb R^n \setminus \{0\}$, $n \in \mathbb N$, $n \geq 2$, and its formal adjoint, denoted by $T_{\alpha,\beta}^+$, we show that nonnegativity of $T_{\alpha,\beta}^+ T_{\alpha,\beta}$ on $C_0^{\infty}(\mathbb R^n \setminus \{0\})$ implies the fundamental inequality (*) \begin{equation}\tag{$*$}\label{0.1} \begin{aligned} \int_{\mathbb R^n} [(\Delta f)(x)]^2 \, d^n x & \geq [(n - 4) \alpha - 2 \beta] \int_{\mathbb R^n} |x|^{-2} |(\nabla f)(x)|^2 \, d^n x \\ & \quad - \alpha (\alpha - 4) \int_{\mathbb R^n} |x|^{-4} |x \cdot (\nabla f)(x)|^2 \, d^n x \\ & \quad + \beta [(n - 4) (\alpha - 2) - \beta] \int_{\mathbb R^n} |x|^{-4} |f(x)|^2 \, d^n x,\\ &&\llap {f \in C^{\infty}_0(\mathbb R^n \setminus \{0\}).} \end{aligned} \end{equation} A particular choice of values for $\alpha$ and $\beta$ in (*) yields known Hardy–Rellich-type inequalities, including the classical Rellich inequality and an inequality due to Schmincke. By locality, these inequalities extend to the situation where $\mathbb R^n$ is replaced by an arbitrary open set $\Omega \subseteq \mathbb R^n$ for functions $f \in C^{\infty}_0(\Omega \setminus \{0\})$. Perhaps more importantly, we will indicate that our method, in addition to being elementary, is quite flexible when it comes to a variety of generalized situations involving the inclusion of remainder terms and higher-order operators.
207
226
1
10.4171/186-1/10
http://www.ems-ph.org/doi/10.4171/186-1/10
Symmetries and multipeakon solutions for the modified two-component Camassa–Holm system
Katrin
Grunert
The Norwegian University of Science and Technology (NTNU), Trondheim, Norway
Xavier
Raynaud
SINTEF, Oslo, Norway, and The Norwegian University of Science and Technology (NTNU), Trondheim, Norway
Partial differential equations
Compared with the two-component Camassa–Holm system, the modified two-component Camassa–Holm system introduces a regularized density which makes possible the existence of solutions of lower regularity, and in particular of multipeakon solutions. In this paper, we derive a new pointwise invariant for the modified two-component Camassa–Holm system. The derivation of the invariant uses directly the symmetry of the system, following the classical argument of Noether’s theorem. The existence of the multipeakon solutions can be directly inferred from this pointwise invariant. This derivation shows the strong connection between symmetries and the existence of special solutions. The observation also holds for the scalar Camassa–Holm equation and, for comparison, we have also included the corresponding derivation. Finally, we compute explicitly the solutions obtained for the peakon-antipeakon case. We observe the existence of a periodic solution which has not been reported in the literature previously. This case shows the attractive effect that the introduction of an elastic potential can have on the solutions.
227
260
1
10.4171/186-1/11
http://www.ems-ph.org/doi/10.4171/186-1/11
Vanishing viscosity solutions of Riemann problems for models of polymer flooding
Graziano
Guerra
Università degli Studi di Milano-Bicocca, Italy
Wen
Shen
The Pennsylvania State University, University Park, USA
Partial differential equations
We consider the solutions of Riemann problems for polymer flooding models. In a suitable Lagrangian coordinate the systems take a triangular form, where the equation for thermodynamics is decoupled from the hydrodynamics, leading to the study of scalar conservation laws with discontinuous flux functions. We prove three equivalent admissibility conditions for shocks for scalar conservation laws with discontinuous flux. Furthermore, we show that a variation of minimum path of [10] proposed in [18] is the vanishing viscosity limit of a partially viscous model with viscosity only in the hydro-dynamics.
261
285
1
10.4171/186-1/12
http://www.ems-ph.org/doi/10.4171/186-1/12
Efficient computation of all speed flows using an entropy stable shock-capturing space-time discontinuous Galerkin method
Andreas
Hiltebrand
ANSYS Switzerland, Zürich, Switzerland
Siddhartha
Mishra
ETH Zürich, Switzerland and University of Oslo, Norway
Partial differential equations
We present a shock-capturing space-time discontinuous Galerkin method to approximate all speed flows modeled by systems of conservation laws with multiple time scales. The method provides a very general and computationally efficient framework for approximating such systems on account of its ability to incorporate large time steps. Numerical examples ranging from computing the incompressible limit (robustness with respect to Mach number) of the Euler equations to accelerating convergence to steady state are presented for illustrating the method.
287
318
1
10.4171/186-1/13
http://www.ems-ph.org/doi/10.4171/186-1/13
Dispersion estimates for spherical Schrödinger equations with critical angular momentum
Markus
Holzleitner
Universität Wien, Austria
Aleksey
Kostenko
Universität Wien, Austria
Gerald
Teschl
Universität Wien, Austria and Erwin Schrödinger Institut, Wien, Austria
Partial differential equations
We derive a dispersion estimate for one-dimensional perturbed radial Schrödinger operators, where the angular momentum takes the critical value $l = –1/2$. We also derive several new estimates for solutions of the underlying differential equation and investigate the behavior of the Jost function near the edge of the continuous spectrum.
319
347
1
10.4171/186-1/14
http://www.ems-ph.org/doi/10.4171/186-1/14
Sixty years of moments for random matrices
Werner
Kirsch
FernUniversität Hagen, Germany
Thomas
Kriecherbauer
Universität Bayreuth, Germany
Partial differential equations
This is an elementary review, aimed at non-specialists, of results that have been obtained for the limiting distribution of eigenvalues and for the operator norms of real symmetric random matrices via the method of moments. This method goes back to a remarkable argument of Eugene Wigner some sixty years ago which works best for independent matrix entries, as far as symmetry permits, that are all centered and have the same variance. We then discuss variations of this classical result for ensembles for which the variance may depend on the distance of the matrix entry to the diagonal, including in particular the case of band random matrices, and/or for which the required independence of the matrix entries is replaced by some weaker condition. This includes results on ensembles with entries from Curie–Weiss random variables or from sequences of exchangeable random variables that have been obtained quite recently.
349
379
1
10.4171/186-1/15
http://www.ems-ph.org/doi/10.4171/186-1/15
Bound states of Schrödinger type operators with Heisenberg sub-Laplacian
Ari
Laptev
Imperial College London, UK
Andrei
Velicu
Imperial College London, UK
Partial differential equations
Using the technique from [8] we find a new constant in a Cwikel–Lieb–Rozenblum type inequality that estimate the number of negative eigenvalues of a Schrödinger operator involving the Heisenberg sub-Laplacian with a potential that is proportional to the characteristic function of a measurable set.
381
387
1
10.4171/186-1/16
http://www.ems-ph.org/doi/10.4171/186-1/16
On Holden’s seven guidelines for scientific computing and development of open-source community software
Knut-Andreas
Lie
SINTEF, Oslo, Norway and NTNU, Trondheim, Norway
Partial differential equations
Two decades ago, Helge Holden proposed seven guidelines to improve the way new achievements and results in scientific computing were presented, evaluated, and compared in contemporary scientific literature. In this essay, written as a tribute to Helge on his 60th birthday, I revisit the guidelines and point out why they are still valid today seen from my perspective, working as a contract researcher at the interface between mathematics and applications in industry. Developing new computational methods usually involves a lot of experimental programming. Over the past decade, my research group has developed an open-source community code that today has hundreds of users worldwide. I discuss some considerations that have gone into this development and present a few lessons learned. Moreover, based on this experience, as well as from development of professional software for our clients, I present advice on how you can be more productive in your experimental programming and increase the impact of your scientific results.
389
422
1
10.4171/186-1/17
http://www.ems-ph.org/doi/10.4171/186-1/17
Sharp uniqueness results for discrete evolutions
Yurii
Lyubarskii
The Norwegian University of Science and Technology (NTNU), Trondheim, Norway
Eugenia
Malinnikova
Norwegian University of Science and Technology (NTNU), Trondheim, Norway
Partial differential equations
We prove sharp uniqueness results for a wide class of one-dimensional discrete evolutions. The proof is based on a construction from the theory of complex Jacobi matrices combined with growth estimates of entire functions.
423
436
1
10.4171/186-1/18
http://www.ems-ph.org/doi/10.4171/186-1/18
Spatial analyticity of solutions to nonlinear dispersive PDE
Sigmund
Selberg
University of Bergen, Norway
Partial differential equations
Given a nonlinear dispersive PDE, for example the KdV equation, we consider the Cauchy problem with real analytic initial data. For data with a uniform radius of analyticity, we are interested in obtaining lower bounds on the radius of analyticity at later times. A rather general approach to this problem is presented, based on Bourgain’s Fourier restriction norm method. Applications to the KdV equation (periodic and non-periodic) and the Dirac–Klein–Gordon equations are discussed.
437
454
1
10.4171/186-1/19
http://www.ems-ph.org/doi/10.4171/186-1/19
Local Representation Theory and Simple Groups
Radha
Kessar
City University of London, UK
Gunter
Malle
Universität Kaiserslautern, Germany
Donna
Testerman
EPF Lausanne, Switzerland
Group theory and generalizations
20BXX, 20CXX, 20GXX
Groups + group theory
Finite reductive groups, Deligne-Lusztig varities, Brauer $p$-blocks, local-global conjectures, base size, fixed-point ratios, random walks
The book contains extended versions of seven short lecture courses given during a semester programme on "Local Representation Theory and Simple Groups" held at the Centre Interfacultaire Bernoulli of the EPF Lausanne. These focussed on modular representation theory of finite groups, modern Clifford theoretic methods, the representation theory of finite reductive groups, as well as on various applications of character theory and representation theory, for example to base sizes and to random walks. These lectures are intended to form a good starting point for graduate students and researchers who wish to familiarize themselves with the foundations of the topics covered here. Furthermore they give an introduction to current research directions, including the state of some open problems in the field.
4
30
2018
978-3-03719-185-9
978-3-03719-685-4
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/185
http://www.ems-ph.org/doi/10.4171/185
EMS Series of Lectures in Mathematics
2523-5176
2523-5184
Basic local representation theory
Burkhard
Külshammer
Friedrich-Schiller-Universität Jena, Germany
Group theory and generalizations
1
22
1
10.4171/185-1/1
http://www.ems-ph.org/doi/10.4171/185-1/1
Reduction theorems for some global–local conjectures
Britta
Späth
Bergische Universität Wuppertal, Germany
Group theory and generalizations
23
61
1
10.4171/185-1/2
http://www.ems-ph.org/doi/10.4171/185-1/2
A first guide to the character theory of finite groups of Lie type
Meinolf
Geck
Universität Stuttgart, Germany
Group theory and generalizations
63
106
1
10.4171/185-1/3
http://www.ems-ph.org/doi/10.4171/185-1/3
Lectures on modular Deligne–Lusztig theory
Olivier
Dudas
Université Paris Diderot Paris 7, France
Group theory and generalizations
107
177
1
10.4171/185-1/4
http://www.ems-ph.org/doi/10.4171/185-1/4
Local methods for blocks of finite simple groups
Marc
Cabanes
Université Paris Diderot Paris 7, France
Group theory and generalizations
179
265
1
10.4171/185-1/5
http://www.ems-ph.org/doi/10.4171/185-1/5
Simple groups, fixed point ratios and applications
Timothy
Burness
University of Bristol, UK
Group theory and generalizations
267
322
1
10.4171/185-1/6
http://www.ems-ph.org/doi/10.4171/185-1/6
Applications of character theory of finite simple groups
Martin
Liebeck
Imperial College, London, UK
Group theory and generalizations
323
352
1
10.4171/185-1/7
http://www.ems-ph.org/doi/10.4171/185-1/7
Lectures in Model Theory
Franziska
Jahnke
Universität Münster, Germany
Daniel
Palacín
The Hebrew University of Jerusalem, Israel
Katrin
Tent
Universität Münster, Germany
Mathematical logic and foundations
Combinatorics
Field theory and polynomials
Algebraic geometry
Primary: 03C45, 03C60, 03C98. Secondary: 05E15, 12J20, 12L12, 14E18, 20E18
Mathematical logic
Algebraic geometry
Model theory, stability theory, NIP theories, definably amenable groups, profinite groups, valuation theory, algebraically closed valued fields, motivic integration
Model theory is a thriving branch of mathematical logic with strong connections to other fields of mathematics. Its versatility has recently led to spectacular applications in areas ranging from diophantine geometry, algebraic number theory and group theory to combinatorics. This volume presents lecture notes from a spring school in model theory which took place in Münster, Germany. The notes are aimed at PhD students but should also be accessible to undergraduates with some basic knowledge in model theory. They contain the core of stability theory (Bays, Palacín), two chapters connecting generalized stability theory with group theory (Clausen and Tent, Simon), as well as introductions to the model theory of valued fields (Hils, Jahnke) and motivic integration (Halupczok).
4
30
2018
978-3-03719-184-2
978-3-03719-684-7
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/184
http://www.ems-ph.org/doi/10.4171/184
Münster Lectures in Mathematics
2523-5230
2523-5249
An introduction to stability theory
Daniel
Palacín
The Hebrew University of Jerusalem, Israel
Mathematical logic and foundations
These lecture notes are based on the first section of Pillay's book [3] and they cover fundamental notions of stability theory such as defi nable types, forking calculus and canonical bases, as well as stable groups and homogeneous spaces. The approach followed here is originally due to Hrushovski and Pillay [2], who presented stability from a local point of view. Throughout the notes, some general knowledge of model theory is assumed. I recommend the book of Tent and Ziegler [4] as an introduction to model theory. Furthermore, the texts of Casanovas [1] and Wagner [5] may also be useful to the reader to obtain a di fferent approach to stability theory.
1
27
1
10.4171/184-1/1
http://www.ems-ph.org/doi/10.4171/184-1/1
Geometric stability theory
Martin
Bays
Universität Münster, Germany
Mathematical logic and foundations
These notes cover some of the foundational results of geometric stability theory. We focus on the geometry of minimal sets. The main aim is an account of Hrushovski's result that unimodular (in particular, locally finite or pseudo finite) minimal sets are locally modular; along the way, we discuss the Zilber trichotomy and the group and field confi gurations. We assume the basics of stability theory (forking calculus, U-rank, canonical bases, stable groups and homogeneous spaces), as can be found e.g. in Daniel Palac ín's chapter in this volume [5].
29
58
1
10.4171/184-1/2
http://www.ems-ph.org/doi/10.4171/184-1/2
NIP and definably amenable groups
Pierre
Simon
University of California, Berkeley, USA
Mathematical logic and foundations
This text is an introduction to de finably amenable NIP groups. It is based on a number of papers, mainly [6], [7] and [4]. This subject has two origins, the fi rst one is the theory of stable groups and in particular generic types, which were fi rst defi ned by Poizat (see [12]) and have since played a central role throughout stability theory. Later, part of the theory was generalized to groups in simple theories, where generic types are de ned as types, none of whose translates forks over the empty set.
59
82
1
10.4171/184-1/3
http://www.ems-ph.org/doi/10.4171/184-1/3
Some model theory of profinite groups
Tim
Clausen
Universität Münster, Germany
Katrin
Tent
Universität Münster, Germany
Mathematical logic and foundations
The main purpose of these notes is to give more background and details for the results obtained in [13] which rely heavily on deep results by Lazard, Lubotzky, Mann, du Sautoy and others. At the center is Lazard's purely group theoretic characterization of $p$-adic analytic groups given in [8] (see Section 4.2 below). By Lazard's result a compact topological group is a $p$-adic analytic group if and only if it has an open uniformly powerful pro-$p$ subgroup (see Section 3).
83
118
1
10.4171/184-1/4
http://www.ems-ph.org/doi/10.4171/184-1/4
An introduction to valued fields
Franziska
Jahnke
Universität Münster, Germany
Mathematical logic and foundations
The aim of this chapter is to give a short introduction to the algebra of valued fields and thereby to provide the necessary background for the following two chapters. The material presented here is heavily based on the book "Valued Fields" by Engler and Prestel, as well as (unpublished) lectures given by Jochen Koenigsmann at the University of Oxford in Hilary 2010. Many of the proofs presented are taken from (or at least inspired by) one of these two sources.
119
149
1
10.4171/184-1/5
http://www.ems-ph.org/doi/10.4171/184-1/5
Model theory of valued fields
Martin
Hils
Universität Münster, Germany
Mathematical logic and foundations
This chapter presents a variety of classical results on the model theory of valued fi elds.
151
180
1
10.4171/184-1/6
http://www.ems-ph.org/doi/10.4171/184-1/6
An introduction to motivic integration
Immanuel
Halupczok
Heinrich-Heine-Universität Düsseldorf, Germany
Mathematical logic and foundations
This introduction to motivic integration is aimed at readers who have some base knowledge of model theory of valued fields, as provided e.g. by the notes [9] by Martin Hils in this volume. I will not assume a lot of knowledge about valued fields.
181
202
1
10.4171/184-1/7
http://www.ems-ph.org/doi/10.4171/184-1/7
Linear Forms in Logarithms and Applications
Yann
Bugeaud
Université de Strasbourg, France
Number theory
Primary: 11-02, 11J86, 11D; Secondary: 11B37, 11D25, 11D41, 11D59, 11D61, 11D75, 11D88, 11J25, 11J81, 11J82
Number theory
Baker's theory, linear form in logarithms, Diophantine equation, Thue equation, $abc$-conjecture, primitive divisor, irrationality measure, $p$-adic analysis
The aim of this book is to serve as an introductory text to the theory of linear forms in the logarithms of algebraic numbers, with a special emphasis on a large variety of its applications. We wish to help students and researchers to learn what is hidden inside the blackbox ‚Baker's theory of linear forms in logarithms' (in complex or in $p$-adic logarithms) and how this theory applies to many Diophantine problems, including the e ffective resolution of Diophantine equations, the $abc$-conjecture, and upper bounds for the irrationality measure of some real numbers. Written for a broad audience, this accessible and self-contained book can be used for graduate courses (some 30 exercises are supplied). Specialists will appreciate the inclusion of over 30 open problems and the rich bibliography of over 450 references.
3
14
2018
978-3-03719-183-5
978-3-03719-683-0
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/183
http://www.ems-ph.org/doi/10.4171/183
IRMA Lectures in Mathematics and Theoretical Physics
2523-5133
2523-5141
28
Schubert Varieties, Equivariant Cohomology and Characteristic Classes
Impanga 15
Jarosław
Buczyński
Polish Academy of Sciences and University of Warsaw, Poland
Mateusz
Michałek
Polish Academy of Sciences, Warsaw, Poland and Max Planck-Institute, Leipzig, Germany
Elisa
Postinghel
Loughborough University, UK
Algebraic geometry
Several complex variables and analytic spaces
Algebraic topology
Primary 14-06; secondary 32L10, 14M15, 55N91, 14C17, 14G17
Analytic geometry
IMPANGA, vector bundles, Schubert varieties and degeneracy loci, homogeneous spaces, equivariant cohomology, Thom polynomials, characteristic classes, symmetric functions and polynomials, quasi-elliptic surfaces
IMPANGA stands for the activities of Algebraic Geometers at the Institute of Mathematics, Polish Academy of Sciences, including one of the most important seminars in algebraic geometry in Poland. The topics of the lectures usually fit within the framework of complex algebraic geometry and neighboring areas of mathematics. This volume is a collection of contributions by the participants of the conference IMPANGA15, organized by participants of the seminar, as well as notes from the major lecture series of the seminar in the period 2010–2015. Both original research papers and self-contained expository surveys can be found here. The articles circulate around a broad range of topics within algebraic geometry such as vector bundles, Schubert varieties, degeneracy loci, homogeneous spaces, equivariant cohomology, Thom polynomials, characteristic classes, symmetric functions and polynomials, and algebraic geometry in positive characteristic.
1
21
2018
978-3-03719-182-8
978-3-03719-682-3
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/182
http://www.ems-ph.org/doi/10.4171/182
EMS Series of Congress Reports
2523-515X
2523-5168
Introduction
Jarosław
Buczyński
Polish Academy of Sciences, Warsaw, Poland
Mateusz
Michałek
Polish Academy of Sciences, Warsaw, Poland
Elisa
Postinghel
Loughborough University, UK
1
7
1
10.4171/182-1/1
http://www.ems-ph.org/doi/10.4171/182-1/1
Friedrich Hirzebruch – a handful of reminiscences
Piotr
Pragacz
Polish Academy of Sciences, Warsaw, Poland
9
23
1
10.4171/182-1/2
http://www.ems-ph.org/doi/10.4171/182-1/2
Pieri rule for the factorial Schur $P$-functions
Soojin
Cho
Ajou University, Suwon, Republic of Korea
Takeshi
Ikeda
Okayama University of Science, Japan
Maximal orthogonal Grassmannian, Pieri rule, factorial Schur $P$-functions
Algebraic geometry
Combinatorics
We prove an identity expressing the product of two factorial Schur $P$-functions as an alternating sum of the factorial Schur $P$-functions with explicitly defined coefficients depending on the deformation parameters. As an application, we derive the Pieri rule for the factorial Schur $P$-functions.
25
48
1
10.4171/182-1/3
http://www.ems-ph.org/doi/10.4171/182-1/3
Restriction varieties and the rigidity problem
Izzet
Coskun
University of Illinois at Chicago, USA
Homogeneous varieties, Schubert varieties, quadratic forms, restriction coefficients
Algebraic geometry
Commutative rings and algebras
This is a survey paper based on the author's lectures given at IMPAN in December 2013. We will discuss recent results on the restriction and rigidity problems. The purpose of the lectures was to develop a more geometric approach to the study of isotropic flag varieties. As an illustration of the techniques, we compute the map induced in cohomology of the inclusion of $OG(k,n)$ and $SG(k,n)$ in $G(k,n)$ via an explicit sequence of rational equivalences. We also discuss applications to classifying representatives of Schubert classes.
49
95
1
10.4171/182-1/4
http://www.ems-ph.org/doi/10.4171/182-1/4
On Plücker equations characterizing Grassmann cones
Letterio
Gatto
Politecnico di Torino, Italy
Parham
Salehyan
IBILCE/UNESP, São José do Rio Preto, Brazil
Algebraic geometry
The KP hierarchy (after Kadomtsev and Petshiasvily) is a system of infinitely many PDEs in Lax form defining a universal family of iso-spectral deformation of an ordinary linear differential operator. It is a classical result due to Sato's japanese school that the rational solutions to the KP hierarchy are parametrized by a cone over an infinite-dimensional Grassmann variety. The present survey will revisit this fact from the point of view of Schubert derivations on a Grassmann algebra. These enable to encode the classical Plücker equations of Grassmannians of $r$-dimensional subspaces in a formula whose limit for $r \to \infty$ coincides with the KP hierarchy, phrased in terms of vertex operators, showing in particular how the latter is intimately related to Schubert calculus.
97
125
1
10.4171/182-1/5
http://www.ems-ph.org/doi/10.4171/182-1/5
Kempf–Laksov Schubert classes for even infinitesimal cohomology theories
Thomas
Hudson
Bergische Universität Wuppertal, Germany
Tomoo
Matsumura
Okayama University of Science, Japan
Grassmann bundles, Schubert varieties, Oriented cohomology theories, Chern classes
Algebraic geometry
Combinatorics
n this paper we prove a generalisation of Kempf–Laksov formula for the degeneracy loci classes in the even infinitesimal cohomology theories of Grassmann bundles and Lagrangian Grassmann bundles.
127
151
1
10.4171/182-1/6
http://www.ems-ph.org/doi/10.4171/182-1/6
On the multicanonical systems of quasi-elliptic surfaces in characteristic 3
Toshiyuki
Katsura
Hosei University, Tokyo, Japan
Quasi-elliptic surfaces, multicanonical systems, positive characteristic
Algebraic geometry
We consider the multicanonical systems $\vert mK_S \vert$ of quasi-elliptic surfaces with Kodaira dimension $1$ in characteristic 3. We show that for any $m \geq 5$ $\vert mK_S \vert$ gives the structure of quasi-elliptic fiber space, and the number $5$ is best possible to give the structure for any such surfaces.
153
157
1
10.4171/182-1/7
http://www.ems-ph.org/doi/10.4171/182-1/7
Characteristic classes of mixed Hodge modules and applications
Laurentiu
Maxim
University of Wisconsin, Madison, USA
Jörg
Schürmann
Universität Münster, Germany
Characteristic classes, Atiyah–Singer classes, mixed Hodge modules, $V$-filtration, nearby and vanishing cycles, singularities, toric varieties, hypersurface, symmetric product, generating series.
Manifolds and cell complexes
Algebraic geometry
Associative rings and algebras
Several complex variables and analytic spaces
We give an overview, with an emphasis on applications, of recent developments on the interaction between characteristic class theories for singular spaces and Saito's theory of mixed Hodge modules in the complex algebraic context.
159
202
1
10.4171/182-1/8
http://www.ems-ph.org/doi/10.4171/182-1/8
On a certain family of $U(\mathfrak b)$-modules
Piotr
Pragacz
Polish Academy of Sciences, Warsaw, Poland
$U(\mathfrak b)$-module, Demazure module, KP module, KP filtration, cyclic module, character, Schur function, Schur functor, Schubert polynomial, subquotient, positivity, ample bundle
Algebraic geometry
Group theory and generalizations
We report on results of Kraskiewicz and the author, and Watanabe on KP modules materializing Schubert polynomials, and filtrations having KP modules as their sub quotients. We discuss applications of the bundles $S_w(E)$ for filtered ample bundles $E$ and KP filtrations to positivity due to Fulton and Watanabe respectively.
203
224
1
10.4171/182-1/9
http://www.ems-ph.org/doi/10.4171/182-1/9
Equivariant Chern–Schwartz–MacPherson classes in partial flag varieties: interpolation and formulae
Richárd
Rimányi
University of North Carolina at Chapel Hill, USA
Alexander
Varchenko
University of North Carolina at Chapel Hill, USA
Equivariant Chern–Schwartz–MacPherson class, Schubert calculus, weight function
Algebraic geometry
Nonassociative rings and algebras
Consider the natural torus action on a partial flag manifold $\mathcal F}_\lambda$. Let $\Omega_I\subset \mathcal F}_\lambda$ be an open Schubert variety, and let $c^{sm}(\Omega_I)\in H_T^*(\mathcal F}_\lambda)$ be its torus equivariant Chern–Schwartz–MacPherson class. We show a set of interpolation properties that uniquely determine $c^{sm}(\Omega_I)$, as well as a formula, of 'localization type', for $c^{sm}(\Omega_I)$. In fact, we proved similar results for a class $\kappa_I\in H_T^*(\mathcal F}_\lambda)$ – in the context of quantum group actions on the equivariant cohomology groups of partial flag varieties. In this note we show that $c^{sm}(\Omega_I)=\kappa_I$.
225
235
1
10.4171/182-1/10
http://www.ems-ph.org/doi/10.4171/182-1/10
Thom polynomials in $\mathcal A$-classification I: counting singular projections of a surface
Takahisa
Sasajima
Kyoto, Japan
Toru
Ohmoto
Hokkaido University, Sapporo, Japan
$\mathcal A$-classification of map-germs, Thom polynomials, classical enumerative geometry, projective surfaces
Algebraic geometry
Manifolds and cell complexes
We study universal polynomials of characteristic classes associated to the $\mathcal A$-classification of map-germs $(\mathbb C^2,0) \to (\mathbb C^n, 0)$ $(n=2,3)$, that enable us to systematically generalize enumerative formulae in classical algebraic geometry of projective surfaces in 3 and 4-spaces.
237
259
1
10.4171/182-1/11
http://www.ems-ph.org/doi/10.4171/182-1/11
Schubert polynomials and degeneracy locus formulas
Harry
Tamvakis
University of Maryland, College Park, USA
Schubert polynomials, theta polynomials, symmetric functions, Weyl groups, divided difference operators, flag varieties, degeneracy loci, equivariant cohomology
Algebraic geometry
Combinatorics
In previous work, we employed the approach to Schubert polynomials by Fomin, Stanley, and Kirillov to obtain simple, uniform proofs that the double Schubert polynomials of Lascoux and Schützenberger and Ikeda, Mihalcea, and Naruse represent degeneracy loci for the classical groups in the sense of Fulton. Using this as our starting point, and purely combinatorial methods, we obtain a new proof of the general formulas of [T5], which represent the degeneracy loci coming from any isotropic partial flag variety. Along the way, we also find several new formulas and elucidate the connections between some earlier ones.
261
314
1
10.4171/182-1/12
http://www.ems-ph.org/doi/10.4171/182-1/12
Hirzebruch $\chi_y$-genera of complex algebraic fiber bundles – the multiplicativity of the signature modulo 4
Shoji
Yokura
Kagoshima University, Japan
315
330
1
10.4171/182-1/13
http://www.ems-ph.org/doi/10.4171/182-1/13
Pushing-forward Schur classes using iterated residues at infinity
Magdalena
Zielenkiewicz
University of Warsaw, Poland
Gysin homomorphism, equivariant cohomology, torus action, homogeneous space
Algebraic geometry
Differential geometry
In this paper we review the results presented during the IMPANGA 15 Conference on the author's approach to equivariant Gysin homomorphism via iterated residues at infinity, with connections to the Jeffrey–Kirwan nonabelian localization theorem in symplectic geometry. We show examples of computations using the formulas of our previous paper, which express the push-forwards in equivariant cohomology as iterated residues at infinity. As an example we consider the equivariant cohomology of the complex Lagrangian Grassmannian $LG(n)$ with the action of the maximal torus in the symplectic group $Sp(n)$. In particular, we obtain, via our methods, analogues in equivariant cohomology of some well-known results due to Pragacz and Ratajski on push-forwards of Schur classes on $LG(n)$.
331
345
1
10.4171/182-1/14
http://www.ems-ph.org/doi/10.4171/182-1/14
Regular, Quasi-regular and Induced Representations of Infinite-dimensional Groups
Alexander
Kosyak
National Academy of Science of Ukraine, Kiev, Ukraine
Topological groups, Lie groups
Measure and integration
Probability theory and stochastic processes
22E66, 22E65; 60B15, 28C20
Topology
Quasi-invariant measure on infinite-dimensional group, ergodic measure, Hilbert–Lie group, unitary, irreducible, regular, quasi-regular, induced representations, Ismagilov conjecture, Schur–Weil duality, Kirillov orbit method, von Neumann algebras, factor, type of factors, C*-group algebras, finite field
Almost all harmonic analysis on locally compact groups is based on the existence (and uniqueness) of a Haar measure. Therefore, it is very natural to attempt a similar construction for non-locally compact groups. The essential idea is to replace the non-existing Haar measure on an infinite-dimensional group by a suitable quasi-invariant measure on an appropriate completion of the initial group or on the completion of a homogeneous space. The aim of the book is a systematic development, by example, of noncommutative harmonic analysis on infinite-dimensional (non-locally compact) matrix groups. We generalize the notion of regular, quasi-regular and induced representations for arbitrary infinite-dimensional groups. The central idea to verify the irreducibility is the Ismagilov conjecture. We also extend the Kirillov orbit method for the group of upper triangular matrices of infinite order. In order to make the content accessible to a wide audience of nonspecialists, the exposition is essentially self-contained and very few prerequisites are needed. The book is aimed at graduate and advanced undergraduate students, as well as mathematicians who wish an introduction to representations of infinite-dimensional groups.
5
31
2018
978-3-03719-181-1
978-3-03719-681-6
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/181
http://www.ems-ph.org/doi/10.4171/181
EMS Tracts in Mathematics
Higher-Dimensional Knots According to Michel Kervaire
Françoise
Michel
Université Paul Sabatier, Toulouse, France
Claude
Weber
Université de Genève, Switzerland
Manifolds and cell complexes
Several complex variables and analytic spaces
57Q45, 57R65, 32S55
Algebraic topology
Knots in high dimensions, homotopy spheres, complex singularities
Michel Kervaire wrote six papers which can be considered fundamental to the development of higher-dimensional knot theory. They are not only of historical interest but naturally introduce to some of the essential techniques in this fascinating theory. This book is written to provide graduate students with the basic concepts necessary to read texts in higher-dimensional knot theory and its relations with singularities. The first chapters are devoted to a presentation of Pontrjagin’s construction, surgery and the work of Kervaire and Milnor on homotopy spheres. We pursue with Kervaire’s fundamental work on the group of a knot, knot modules and knot cobordism. We add developments due to Levine. Tools (like open books, handlebodies, plumbings, …) often used but hard to find in original articles are presented in appendices. We conclude with a description of the Kervaire invariant and the consequences of the Hill–Hopkins–Ravenel results in knot theory.
7
25
2017
978-3-03719-180-4
978-3-03719-680-9
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/180
http://www.ems-ph.org/doi/10.4171/180
EMS Series of Lectures in Mathematics
2523-5176
2523-5184
A Course In Error-Correcting Codes
Second edition
Jørn
Justesen
Technical University of Denmark, Lyngby, Denmark
Tom
Høholdt
Technical University of Denmark, Lyngby, Denmark
Information and communication, circuits
Field theory and polynomials
94-01;12-01
Mathematical theory of computation
Fields + rings
Error-correcting codes, Reed–Solomon codes, convolutional codes, product codes, graph codes, algebraic geometry codes
This book, updated and enlarged for the second edition, is written as a text for a course aimed at 3rd or 4th year students. Only some familiarity with elementary linear algebra and probability is directly assumed, but some maturity is required. The students may specialize in discrete mathematics, computer science, or communication engineering. The book is also a suitable introduction to coding theory for researchers from related fields or for professionals who want to supplement their theoretical basis. The book gives the coding basics for working on projects in any of the above areas, but material specific to one of these fields has not been included. The chapters cover the codes and decoding methods that are currently of most interest in research, development, and application. They give a relatively brief presentation of the essential results, emphasizing the interrelations between different methods and proofs of all important results. A sequence of problems at the end of each chapter serves to review the results and give the student an appreciation of the concepts. In addition, some problems and suggestions for projects indicate direction for further work. The presentation encourages the use of programming tools for studying codes, implementing decoding methods, and simulating performance. Specific examples of programming exercises are provided on the book's home page.
7
11
2017
978-3-03719-179-8
978-3-03719-679-3
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/179
http://www.ems-ph.org/doi/10.4171/179
EMS Textbooks in Mathematics
Shape Variation and Optimization
A Geometrical Analysis
Antoine
Henrot
Université de Lorraine, Vandœuvre-lès-Nancy, France
Michel
Pierre
ENS Cachan Bretagne, Bruz, France
Calculus of variations and optimal control; optimization
Partial differential equations
Differential geometry
Global analysis, analysis on manifolds
49Q10, 49Q05, 49Q12, 49K20, 49K40, 53A10, 35R35, 35J20, 58E25, 31B15, 65K10, 93B27, 74P20, 74P15, 74G65, 76M30
Calculus + mathematical analysis
Differential equations
Differential + Riemannian geometry
Shape optimization, optimum design, calculus of variations, variations of domains, Hausdorff convergence, continuity with respect to domains, G-convergence, shape derivative, geometry of optimal shapes, Laplace-Dirichlet problem, Neumann problem, overdetermined problems, isoperimetric inequality, capacity, potential theory, spectral theory, homogenization
Optimizing the shape of an object to make it the most efficient, resistant, streamlined, lightest, noiseless, stealthy or the cheapest is clearly a very old task. But the recent explosion of modeling and scientific computing have given this topic new life. Many new and interesting questions have been asked. A mathematical topic was born – shape optimization (or optimum design). This book provides a self-contained introduction to modern mathematical approaches to shape optimization, relying only on undergraduate level prerequisite but allowing to tackle open questions in this vibrant field. The analytical and geometrical tools and methods for the study of shapes are developed. In particular, the text presents a systematic treatment of shape variations and optimization associated with the Laplace operator and the classical capacity. Emphasis is also put on differentiation with respect to domains and a FAQ on the usual topologies of domains is provided. The book ends with geometrical properties of optimal shapes, including the case where they do not exist.
2
15
2018
978-3-03719-178-1
978-3-03719-678-6
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/178
http://www.ems-ph.org/doi/10.4171/178
EMS Tracts in Mathematics
28
Interviews with the Abel Prize Laureates 2003–2016
Martin
Raussen
Aalborg University, Denmark
Christian
Skau
Norwegian University of Science and Technology (NTNU), Trondheim, Norway
History and biography
General
01A70, 01A60, 01A61, 01A80, 00A35
History of mathematics
Abel prize, laureates, interviews, history of mathematics, appreciation of mathematics
The Abel Prize was established in 2002 by the Norwegian Ministry of Education and Research. It has been awarded annually to mathematicians in recognition of pioneering scientific achievements. Since the first occasion in 2003, Martin Raussen and Christian Skau have had the opportunity to conduct extensive interviews with the laureates. The interviews were broadcast by Norwegian television; moreover, they have appeared in the membership journals of several mathematical societies. The interviews from the period 2003 – 2016 have now been collected in this edition. They highlight the mathematical achievements of the laureates in a historical perspective and they try to unravel the way in which the world’s most famous mathematicians conceive and judge their results, how they collaborate with peers and students, and how they perceive the importance of mathematics for society.
9
1
2017
978-3-03719-177-4
978-3-03719-677-9
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/177
http://www.ems-ph.org/doi/10.4171/177
Abel Prize 2003: Jean-Pierre Serre
Martin
Raussen
Aalborg Universitet, Denmark
Christian
Skau
Norwegian University of Science and Technology (NTNU), Trondheim, Norway
History and biography
General
1
10
1
10.4171/177-1/1
http://www.ems-ph.org/doi/10.4171/177-1/1
Abel Prize 2004: Sir Michael Francis Atiyah and Isadore M. Singer
Martin
Raussen
Aalborg Universitet, Denmark
Christian
Skau
Norwegian University of Science and Technology (NTNU), Trondheim, Norway
History and biography
General
11
29
1
10.4171/177-1/2
http://www.ems-ph.org/doi/10.4171/177-1/2
Abel Prize 2005: Peter D. Lax
Martin
Raussen
Aalborg Universitet, Denmark
Christian
Skau
Norwegian University of Science and Technology (NTNU), Trondheim, Norway
History and biography
General
31
46
1
10.4171/177-1/3
http://www.ems-ph.org/doi/10.4171/177-1/3
Abel Prize 2006: Lennart Carleson
Martin
Raussen
Aalborg Universitet, Denmark
Christian
Skau
Norwegian University of Science and Technology (NTNU), Trondheim, Norway
History and biography
General
47
60
1
10.4171/177-1/4
http://www.ems-ph.org/doi/10.4171/177-1/4
Abel Prize 2007: Srinivasa S. R. Varadhan
Martin
Raussen
Aalborg Universitet, Denmark
Christian
Skau
Norwegian University of Science and Technology (NTNU), Trondheim, Norway
History and biography
General
61
78
1
10.4171/177-1/5
http://www.ems-ph.org/doi/10.4171/177-1/5
Abel Prize 2008: John Griggs Thompson and Jacques Tits
Martin
Raussen
Aalborg Universitet, Denmark
Christian
Skau
Norwegian University of Science and Technology (NTNU), Trondheim, Norway
History and biography
General
79
96
1
10.4171/177-1/6
http://www.ems-ph.org/doi/10.4171/177-1/6
Abel Prize 2009: Mikhail Gromov
Martin
Raussen
Aalborg Universitet, Denmark
Christian
Skau
Norwegian University of Science and Technology (NTNU), Trondheim, Norway
History and biography
General
97
121
1
10.4171/177-1/7
http://www.ems-ph.org/doi/10.4171/177-1/7
Abel Prize 2010: John Tate
Martin
Raussen
Aalborg Universitet, Denmark
Christian
Skau
Norwegian University of Science and Technology (NTNU), Trondheim, Norway
History and biography
General
123
140
1
10.4171/177-1/8
http://www.ems-ph.org/doi/10.4171/177-1/8
Abel Prize 2011: John Milnor
Martin
Raussen
Aalborg Universitet, Denmark
Christian
Skau
Norwegian University of Science and Technology (NTNU), Trondheim, Norway
History and biography
General
141
159
1
10.4171/177-1/9
http://www.ems-ph.org/doi/10.4171/177-1/9
Abel Prize 2012: Endre Szemerédi
Martin
Raussen
Aalborg Universitet, Denmark
Christian
Skau
Norwegian University of Science and Technology (NTNU), Trondheim, Norway
History and biography
General
161
182
1
10.4171/177-1/10
http://www.ems-ph.org/doi/10.4171/177-1/10
Abel Prize 2013: Pierre Deligne
Martin
Raussen
Aalborg Universitet, Denmark
Christian
Skau
Norwegian University of Science and Technology (NTNU), Trondheim, Norway
History and biography
General
183
200
1
10.4171/177-1/11
http://www.ems-ph.org/doi/10.4171/177-1/11
Abel Prize 2014: Yakov G. Sinai
Martin
Raussen
Aalborg Universitet, Denmark
Christian
Skau
Norwegian University of Science and Technology (NTNU), Trondheim, Norway
History and biography
General
201
217
1
10.4171/177-1/12
http://www.ems-ph.org/doi/10.4171/177-1/12
Abel Prize 2015: John F. Nash, Jr. and Louis Nirenberg
Martin
Raussen
Aalborg Universitet, Denmark
Christian
Skau
Norwegian University of Science and Technology (NTNU), Trondheim, Norway
History and biography
General
219
243
1
10.4171/177-1/13
http://www.ems-ph.org/doi/10.4171/177-1/13
Abel Prize 2016: Sir Andrew J. Wiles
Martin
Raussen
Aalborg Universitet, Denmark
Christian
Skau
Norwegian University of Science and Technology (NTNU), Trondheim, Norway
History and biography
General
245
265
1
10.4171/177-1/14
http://www.ems-ph.org/doi/10.4171/177-1/14
An Imaginary Interview with Niels Henrik Abel
Martin
Raussen
Aalborg Universitet, Denmark
Christian
Skau
Norwegian University of Science and Technology (NTNU), Trondheim, Norway
History and biography
General
267
287
1
10.4171/177-1/15
http://www.ems-ph.org/doi/10.4171/177-1/15
European Congress of Mathematics
Berlin, July 18–22, 2016
Volker
Mehrmann
Technical University Berlin, Germany
Martin
Skutella
Technical University Berlin, Germany
General
00Bxx
Mathematics
Mathematics, European congress of mathematics
The European Congress of Mathematics, held every four years, is a well-established major international mathematical event. Following those in Paris (1992), Budapest (1996), Barcelona (2000), Stockholm (2004), Amsterdam (2008), and Kraków (2012), the Seventh European Congress of Mathematics (7ECM) took place in Berlin, Germany, July 18–22, 2016, with about 1100 participants from all over the world. Ten plenary, thirty-three invited and four 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 second time – the Otto Neugebauer Prize for a highly original and influential piece of work in the history of mathematics. The program was complemented by forty-three minisymposia with about 160 talks as well as contributed talks, spread over all areas of mathematics. Several panel discussions and meetings were organized, covering a variety of issues ranging from the future of mathematical publishing to public awareness of mathematics. These proceedings present extended versions of most of the plenary and invited lectures which were delivered during the congress, providing a permanent record of the best what mathematics offers today.
8
6
2018
978-3-03719-176-7
978-3-03719-676-2
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/176
http://www.ems-ph.org/doi/10.4171/176
How has one, and How could have one approached the diversity of mathematical cultures?
Karine
Chemla
Université Paris Diderot, Paris, France
History and biography
This contribution argues that history of mathematics should take as its object not only knowledge, but also ways of doing mathematics that are collectively shared (what I call “mathematical cultures”), and additionally the connections between the two. I provide evidence showing that there is a history of ways of doing mathematics, and this history suggests that mathematical knowledge takes shape at the same time as practices do. Indeed, ways of doing mathematics do not fall out of the sky. They are shaped and transformed by actors in the process of working out some problems and addressing some issues. They represent one of the outcomes of mathematical research. I further argue that attending to the mathematical culture in the context of which actors worked is essential for interpreting their writings.
1
61
1
10.4171/176-1/1
http://www.ems-ph.org/doi/10.4171/176-1/1
Flexible polyhedra and their volumes
Alexander
Gaifullin
Steklov Mathematical Institute, Moscow, Russia
General
We discuss some recent results on flexible polyhedra and the bellows conjecture, which claims that the volume of any flexible polyhedron is constant during the flexion. Also, we survey main methods and several open problems in this area.
63
83
1
10.4171/176-1/2
http://www.ems-ph.org/doi/10.4171/176-1/2
Boolean Functions: Influence, threshold and noise
Gil
Kalai
Hebrew University, Jerusalem, Israel
General
This lecture studies the analysis of Boolean functions and present a few ideas, results, proofs, and problems. We start with the wider picture of expansion in graphs and then concentrate on the graph of the $n$-dimensional discrete cube $\Omega_n$. Boolean functions are functions from $\Omega_n to {0,1}$. We consider the notion of the influence of variables on Boolean functions. The influence of a variable on a Boolean function is the probability that changing the value of the variable changes the value of the function. We then consider Fourier analysis of real functions on $\Omega_n$ and some applications of Fourier methods. We go on to discuss connections with sharp threshold phenomena, percolation, random graphs, extremal combinatorics, correlation inequalities, and more.
85
110
1
10.4171/176-1/3
http://www.ems-ph.org/doi/10.4171/176-1/3
Quantum Fields and Probability
Antti
Kupiainen
University of Helsinki, Finland
General
111
131
1
10.4171/176-1/4
http://www.ems-ph.org/doi/10.4171/176-1/4
Existence of knotted vortex structures in stationary solutions of the Euler equations
Alberto
Enciso
Consejo Superior de Investigaciones Científicas, Madrid, Spain
Daniel
Peralta-Salas
Consejo Superior de Investigaciones Científicas, Madrid, Spain
General
In this paper, we review recent research on certain geometric aspects of the vortex lines of stationary ideal fluids. We mainly focus on the study of knotted and linked vortex lines and vortex tubes, which is a topic that can be traced back to Lord Kelvin and was popularized by the works of Arnold and Moffatt on topological hydrodynamics in the 1960s. In this context, we provide a leisurely introduction to some recent results concerning the existence of stationary solutions to the Euler equations in Euclidean space with a prescribed set of vortex lines and vortex tubes of arbitrarily complicated topology. The content of this paper overlaps substantially with the one the authors published in the Newsletter of the European Mathematical Society in June 2015.
133
153
1
10.4171/176-1/5
http://www.ems-ph.org/doi/10.4171/176-1/5
Symplectic rigidity and quantum mechanics
Leonid
Polterovich
Tel-Aviv University, Israel
General
I present new links between Symplectic Topology and Quantum Mechanics which have been discovered in the framework of function theory on symplectic manifolds. Recent advances in this emerging theory highlight some rigidity features of the Poisson bracket, a fundamental operation governing the mathematical model of Classical Mechanics. Unexpectedly, the intuition behind this rigidity comes from Quantum Mechanics.
155
179
1
10.4171/176-1/6
http://www.ems-ph.org/doi/10.4171/176-1/6
The topology and geometry of automorphism groups of free groups
Karen
Vogtmann
University of Warwick, Coventry, UK
General
In the 1970s Stallings showed that one could learn a great deal about free groups and their automorphisms by viewing the free groups as fundamental groups of graphs and modeling their automorphisms as homotopy equivalences of graphs. Further impetus for using graphs to study automorphism groups of free groups came from the introduction of a space of graphs, now known as Outer space, on which the group Out$(F_n$)
acts nicely. The study of Outer space and its Out$(F_n$)
action continues to give new information about the structure of Out$(F_n$)
, but has also found surprising connections to many other groups, spaces and seemingly unrelated topics, from phylogenetic trees to cyclic operads and modular forms. In this talk I will highlight various ways these ideas are currently evolving.
181
202
1
10.4171/176-1/7
http://www.ems-ph.org/doi/10.4171/176-1/7
Spectral synthesis in Hilbert spaces of entire functions
Anton
Baranov
St. Petersburg State University, Russia
Yurii
Belov
St. Petersburg State University, Russia
General
We give a survey of recent advances in the theory of spaces of entire functions related to the notion of spectral synthesis. In particular, we discuss a solution of a longstanding problem about spectral synthesis for systems of exponentials in $L^2(-\pi, \pi)$
as well as its generalization to de Branges spaces of entire functions. In the de Branges space setting the problem can be related (via a functional model) to spectral theory of rank one perturbations of compact selfadjoint operators; this leads to unexpected examples of rank one perturbations which do not admit spectral synthesis. Related problems for Fock-type spaces are also considered.
203
218
1
10.4171/176-1/8
http://www.ems-ph.org/doi/10.4171/176-1/8
Periodic waves in unsaturated porous media with hysteresis
Bettina
Detmann
Universität Duisburg-Essen, Germany
Pavel
Krejcí
Academy of Sciences, Prague, Czechia
Elisabetta
Rocca
Università degli Studi di Pavia, Italy
General
We consider a PDE system with degenerate hysteresis describing unsaturated flow in 3D porous media. Assuming that a time periodic forcing is prescribed on the boundary, we prove that a time periodic response exists as long as the amplitude of the forcing terms is small enough to keep the solution within the convexity domain of the hysteresis operator.
219
234
1
10.4171/176-1/9
http://www.ems-ph.org/doi/10.4171/176-1/9
2D Ising model: Correlation functions at criticality via Riemann-type boundary value problems
Dmitry
Chelkak
Ecole Normale Supérieure, Paris, France
General
In this note we overview recent convergence results for correlations in the critical planar nearest-neighbor Ising model. We start with a short discussion of the combinatorics of the model and a definition of fermionic and spinor observables. After that, we illustrate our approach to spin correlations by a derivation of two classical explicit formulae in the infinite-volume limit. Then we describe the convergence results (as the mesh size tends to zero, in arbitrary planar domains) for fermionic correlators [14], energy-density [18] and spin expectations [11]. Finally, we discuss scaling limits of mixed correlators involving spins, disorders and fermions, and the classical fusion rules for them.
235
256
1
10.4171/176-1/10
http://www.ems-ph.org/doi/10.4171/176-1/10
On perverse equivalences and rationality
Joseph
Chuang
City University of London, UK
Radha
Kessar
City University of London, UK
General
We show that perverse equivalences between module categories of finite-dimensional algebras preserve rationality. As an application, we give a connection between some famous conjectures from the modular representation theory of finite groups, namely Broué’s Abelian Defect Group conjecture and Donovan’s Finiteness conjectures.
257
262
1
10.4171/176-1/11
http://www.ems-ph.org/doi/10.4171/176-1/11
Torsion homology growth in arithmetic groups
Nicolas
Bergeron
Université Pierre et Marie Curie, Paris, France
General
Various recent works show that certain arithmetic groups -- that generalize the modular group -- can have `a lot' of torsion in their homology. Among these groups are the finite index (congruence) subgroups of $\mathrm{SL}_3 (\mathbb{Z})$ or $\mathrm{SL}_2 (\mathbb{Z} [i])$. In the latter case homology reduces to abelianization. In particular, for \[\textstyle \Gamma_0 (N) = \left\{ \left( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right) \in \mathrm{SL}_2 (\mathbb{Z} [i]) \; \big| \; N | c \right\} \quad (N \in \mathbb{Z}[i]), \] one may ask about the structure of the finitely generated $\mathbb{Z}$-module $\Gamma_0 (N) ^{\rm ab} = \Gamma_0 (N) / [\Gamma_0 (N) : \Gamma_0 (N)].$ It has a finite torsion part $\Gamma_0 (N)^{\rm ab}_{\rm tors}$ and Akshay Venkatesh and I have conjectured that, as $N$ tends to $\infty$ among primes, we have: \[ \frac{\log |\Gamma_0 (N)^{\rm ab}_{\rm tors}|}{|N|^2} \to \frac{\lambda }{18\pi}, \quad \lambda=L(2, \chi_{\mathbb{Q} (i)}) = 1- \frac{1}{9} + \frac{1}{25} - \frac{1}{49} + \ldots \] More generally one may ask: \emph{How does the amount of torsion in the homology of an arithmetic group grow with the level $N$?} We propose a conjectural partial answer. This contribution presents ideas for how to attack this conjecture and discusses recent progress towards it. This topic interacts with more classical questions of geometry (analytic torsion, Gromov--Thurston norm, (higher) cost, rank and deficiency gradient \dots) and number theory (BSD conjecture, ABC conjecture~\dots). A big motivation is provided by (one of) Peter Scholze's recent breakthrough(s): \emph{very roughly} a mod $p$ torsion class in $\Gamma_0 (N)^{\rm ab}_{\rm tors}$ parametrizes a field extension $K/\mathbb{Q} (i)$ whose Galois group is a subgroup of $\mathrm{GL}_2 (\overline{\mathbb{F}}_p)$. Moreover, it is anticipated that there is a corresponding 'torsion Langlands program'.
263
287
1
10.4171/176-1/12
http://www.ems-ph.org/doi/10.4171/176-1/12
Positivity and higher Teichmüller theory
Olivier
Guichard
Université de Strasbourg, France
Anna
Wienhard
Universität Heidelberg, Germany and Heidelberg Institute for Theoretical Studies, Heidelberg, Germany
General
We introduce $\Theta$-positivity, a new notion of positivity in real semisimple Lie groups. The notion of $\Theta$-positivity generalizes at the same time Lusztig’s total positivity in split real Lie groups as well as well known concepts of positivity in Lie groups of Hermitian type. We show that there are two other families of Lie groups, SO($p,q$)
for $p \neq q$ and a family of exceptional Lie groups, which admit a $\Theta$-positive structure. We describe key aspects of $\Theta$-positivity and make a connection with representations of surface groups and higher Teichmüller theory.
289
310
1
10.4171/176-1/13
http://www.ems-ph.org/doi/10.4171/176-1/13
Diffusion, optimal transport and Ricci curvature
Giuseppe
Savaré
Università di Pavia, Italy
General
Starting from the pioneering paper of Otto-Villani [73], the link between Optimal Transport and Ricci curvature in smooth Riemannian geometry has been deeply studied [35, 86]. Among the various functional and analytic applications, the point of view of Optimal Transport has played a crucial role in the Lott–Sturm–Villani [69, 84, 85, 87] formulation of a “synthetic” notion of lower Ricci curvature bound, which has been extended from the realm of smooth Riemannian manifold to the general framework of metric measure spaces $(X,\mathrm d,\mathfrak m)$
, i.e., (separable, complete) metric spaces endowed with a finite or locally finite Borel measure \mathfrak m. Lower Ricci curvature bounds can also be captured by the celebrated Bakry–Émery [21] approach based on Markov semigroups, diffusion operators and $\Gamma$-calculus for strongly local Dirichlet forms [22]. We will discuss a series of recent contributions [5, 7–9, 12, 38] showing the link of both the approaches with the metric-variational theory of gradient flows [6] and diffusion equations. As a byproduct, when the Cheeger energy on $(X,\mathrm d,\mathfrak m)$ is quadratic (or, equivalently, the Sobolev space $W^{1,2}(X,\mathrm d,\mathfrak m)$ is Hilbertian), we will show that the two approaches lead to essentially equivalent definitions and to a nice geometric framework suitable for deep analytic results.
311
331
1
10.4171/176-1/14
http://www.ems-ph.org/doi/10.4171/176-1/14
Non-discrete simple locally compact groups
Pierre-Emmanuel
Caprace
Université Catholique de Louvain, Louvain-la-Neuve, Belgium
General
Simple Lie groups and simple algebraic groups over local fields are the most prominent members of the class $\mathcal S$ of compactly generated non-discrete simple locally compact groups. We outline a new trend, which emerged in the past decade, whose purpose is the study of $\mathcal S$ as a whole.
333
354
1
10.4171/176-1/15
http://www.ems-ph.org/doi/10.4171/176-1/15
An invitation to circle actions
Leonor
Godinho
Instituto Superior Técnico, Lisboa, Portugal
Silvia
Sabatini
Universität Köln, Germany
General
The problem of determining whether a manifold admits symmetries has been widely studied in mathematics and physics. It is in general hard to determine whether, given a Lie group $G$ and a manifold $M$, there exists a nontrivial action of $G$ on $M$ that preserves a prescribed structure. When $M$ is symplectic, for instance when $M$ is the phase space of a particle, having one conserved quantity whose associated (Hamiltonian) flow on the manifold is periodic, is equivalent to having a Hamiltonian circle action. The following questions are therefore natural: Which symplectic manifolds admit symplectic circle actions? What are their topological properties? Here we discuss these problems when the fixed point set is discrete. In particular we review some of the consequences of the fact that the Chern number $c_1c_{n-1}[M]$ is completely determined by the fixed point data of the action. For example, this allows us to construct an algorithm which, in some cases where the fixed point set is minimal, determines the possible representations at the fixed points of Hamiltonian circle actions with discrete fixed point sets. In general it can be applied towards proving the symplectic Petrie conjecture on these actions. Another application is the obtainment of lower bounds and divisibility results for the number of fixed points on the broader category of almost complex manifolds. This lower bound problem is related to the Kosniowski conjecture, which has been open since 1979. Finally, we show how to extend the 12 and 24 Theorems on the number of lattice points of reflexive polytopes of dimensions 2 and 3 to Delzant reflexive polytopes of any dimension.
355
371
1
10.4171/176-1/16
http://www.ems-ph.org/doi/10.4171/176-1/16
Sampling and interpolating sequences in finite dimensional spaces
Joaquim
Ortega-Cerdà
Universitat de Barcelona, Spain
General
We illustrate how the use of techniques from optimal transport on the study of sampling and interpolation families in finite dimensional spaces provide alternative descriptions of these sequences.
373
387
1
10.4171/176-1/17
http://www.ems-ph.org/doi/10.4171/176-1/17
Weak hyperbolic structures and robust properties of diffeomorphisms and flows
Christian
Bonatti
Université de Bourgogne, Dijon, France
Adriana
da Luz
Universidad de la República, Montevideo, Uruguay, and Université de Bourgogne, Dijon, France
General
A property of a dynamical system is called $C^r$-robust if it holds on a $C^r$-open set of systems. For diffeomorphisms or for non-singular flows, there are many results relating $C^1$-robust properties and global structures of the dynamics, as hyperbolicity, partial hyperbolicity, dominated splitting. However, a difficulty appears when a robust property of a flow holds on a set containing recurrent orbits accumulating a singular point. This phenomenon is now mainly understood in dimension 3, but is still wide open in higher dimensions. Here, we propose a general procedure for adapting the usual hyperbolic structures to the singularities, opening the door for bypassing the difficulty of the coexistence of singular and regular orbits. In particular, this new definition allows us to adapt the proof in [15] to get a characterization of star flows on a $C^1$-open and dense set.
389
405
1
10.4171/176-1/18
http://www.ems-ph.org/doi/10.4171/176-1/18
Free group rings and derived functors
Roman
Mikhailov
Steklov Mathematical Institute, St. Petersburg, Russia
Inder Bir
Passi
Panjab University, Chandigarh, India
General
An approach to identify the normal subgroups determined by ideals in free group rings with the help of the derived functors of non-additive functors is explored. A similar approach, i.e., via derived functors, for computing limits of functors from the category of free presentations to the category of abelian groups, arising from commutator structure of free groups, is also discussed.
407
425
1
10.4171/176-1/19
http://www.ems-ph.org/doi/10.4171/176-1/19
Congested transport at microscopic and macroscopic scales
Bertrand
Maury
Université Paris-Sud, Orsay, France
General
This note addresses mathematical issues raised by congestion constraints in transport equations that arise in the modeling of crowd motion or more general active entities. We address in particular the differences between the microscopic and the macroscopic settings.
427
442
1
10.4171/176-1/20
http://www.ems-ph.org/doi/10.4171/176-1/20
Complex Brunn–Minkowski inequalities and their applications in geometry
Bo
Berndtsson
Chalmers University of Technology, Göteborg, Sweden
General
We survey some positivity results for direct images of line bundles, emphasising their similarities with the classical theorems of Brunn–Minkowski and Prékopa. We also give examples how these results can be applied in complex geometry.
443
457
1
10.4171/176-1/21
http://www.ems-ph.org/doi/10.4171/176-1/21
Counting Steiner Triple Systems
Peter
Keevash
University of Oxford, UK
General
We prove a conjecture of Wilson from 1974 on the number of Steiner Triple Systems. The proof illustrates our method of Randomised Algebraic Construction, which we developed recently to resolve a question of Steiner from 1853 on the existence of combinatorial designs.
459
481
1
10.4171/176-1/22
http://www.ems-ph.org/doi/10.4171/176-1/22
Amenability versus non amenability: An introduction to von Neumann algebras
Stefaan
Vaes
Katholieke Universiteit Leuven, Belgium
General
The theory of von Neumann algebras was initiated by Murray and von Neumann and has deep connections to several areas of mathematics, in particular group theory and ergodic theory. Amenable von Neumann algebras were completely classified by Connes and Haagerup, while numerous classification theorems in the non amenable case were obtained within Popa’s deformation/rigidity theory. This survey article provides an introduction to von Neumann algebras, written for non specialists and with the dichotomy between amenability and non amenability as our guide.
483
500
1
10.4171/176-1/23
http://www.ems-ph.org/doi/10.4171/176-1/23
Recent progress in nonlinear potential theory
Giuseppe
Mingione
Università di Parma, Italy
General
Nonlinear Potential Theory aims at reproducing, in the nonlinear setting, the classical results of potential theory concerning the fine and regularity properties of solutions to linear elliptic and parabolic equations. Potential estimates, integrability, differentiability and continuity properties of solutions are at the heart of the matter. Here we give a brief survey of a few recent results.
501
524
1
10.4171/176-1/24
http://www.ems-ph.org/doi/10.4171/176-1/24
Stochastic dynamics for adaptation and evolution of microorganisms
Sylvain
Billiard
Université des Sciences et Technologies de Lille, Villeneuve-d'Ascq, France
Pierre
Collet
École Polytechnique, Palaiseau, France
Régis
Ferrière
École Normale Supérieure, Paris, France
Sylvie
Méléard
Ecole Polytechnique, Palaiseau, France
Viet Chi
Tran
Université des Sciences et Technologies de Lille, Villeneuve-d'Ascq, France
General
We present a model for the dynamics of a population of bacteria with a continuum of traits, who compete for resources and exchange horizontally (transfer) an otherwise vertically inherited trait with possible mutations. Competition influences individual demographics, affecting population size, which feeds back on the dynamics of transfer. We consider a stochastic individual-based pure jump process taking values in the space of point measures, and whose jump events describe the individual reproduction, transfer and death mechanisms. In a large population scale, the stochastic process is proved to converge to the solution of a nonlinear integro-differential equation. When there are only two different traits and no mutation, this equation reduces to a non-standard two-dimensional dynamical system. We show how crucial the forms of the transfer rates are for the long-term behavior of its solutions. We describe the dynamics of invasion and fixation when one of the two traits is initially rare, and compute the invasion probabilities. Then, we study the process under the assumption of rare mutations. We prove that the stochastic process at the mutation time scale converges to a jump process which describes the successive invasions of successful mutants. We show that the horizontal transfer can have a major impact on the distribution of the successive mutational fixations, leading to dramatically different behaviors, from expected evolution scenarios to evolutionary suicide. Simulations are given to illustrate these phenomena.
525
550
1
10.4171/176-1/25
http://www.ems-ph.org/doi/10.4171/176-1/25
Learning and sparse control of multiagent systems
Massimo
Fornasier
TU München, Garching, Germany
General
In the past decade there has been a large scope of studies on mathematical models of social dynamics. Self-organization, i.e., the autonomous formation of patterns, has been so far the main driving concept. Usually first or second order models are considered with given predetermined nonlocal interaction potentials, tuned to reproduce, at least qualitatively, certain global patterns (such as flocks of birds, milling school of fish or line formations in pedestrian flows etc.). However, it is common experience that self-organization of a society does not always spontaneously occur. In the first part of this survey paper we address the question of whether it is possible to externally and parsimoniously influence the dynamics, to promote the formation of certain desired patterns. In particular we address the issue of finding the sparsest control strategy for finite agent models in order to lead the dynamics optimally towards a given outcome. In the second part of the paper we show the rigorous limit process connecting finite dimensional sparse optimal control problems with ODE constraints to an infinite dimensional sparse mean-field optimal control problem with a constraint given by a PDE of Vlasov-type, governing the dynamics of the probability distribution of the agent population. Moreover, often in practice we do not dispose of a precise knowledge of the governing dynamics. In the last part of this paper we present a variational and optimal transport framework leading to an algorithmic solution to the problem of learning the interaction potentials from the observation of the dynamics of a multiagent system.
551
581
1
10.4171/176-1/26
http://www.ems-ph.org/doi/10.4171/176-1/26
Variational modeling of dislocations in crystals in the line-tension limit
Pilar
Ariza
Universidad de Sevilla, Spain
Sergio
Conti
Universität Bonn, Germany
Adriana
Garroni
Università di Roma La Sapienza, Italy
Michael
Ortiz
California Institute of Technology, Pasadena, USA
General
Dislocations are line singularities in crystals which are crucial for the plastic deformation of materials. Mathematically they can be modeled as measures supported on rectifiable curves, or as vector-valued one-currents. They have a lattice-valued multiplicity, which is a conserved quantity, in the sense that the divergence of the measure (or the boundary of the current) vanishes. Dislocations are necessarily accompanied by large elastic strains, and indeed their energetics can be understood starting from the theory of elasticity, in an appropriate scaling regime. We discuss recent progress in the rigorous derivation of dislocation models in the line-tension regime from linear elasticity, and their application to specific problems in metals. We present numerical simulations on dislocations in bcc molybdenum which show how our line-tension model provides a simple and efficient description of dislocation structures.
583
598
1
10.4171/176-1/27
http://www.ems-ph.org/doi/10.4171/176-1/27
Phase transitions in discrete structures
Amin
Coja-Oghlan
J.W. Goethe-Universität, Frankfurt a.M., Germany
General
Many important parameters of random discrete structures such as random graphs, formulas and codes undergo phase transitions. While some phase transitions such as the emergence of a giant component in the Erd˝os-Rényi random graph are very well understood, other, less understood ones resemble phase transitions in statistical physics models of disordered systems. This paper gives an impression of what we know about this latter class with the particular example of the random graph colouring problem.
599
618
1
10.4171/176-1/28
http://www.ems-ph.org/doi/10.4171/176-1/28
Transverse stability issues in Hamiltonian PDE
Nikolay
Tzvetkov
Université de Cergy-Pontoise, France
General
We present results concerning the transverse stability of one dimensional solitary waves subject to periodic transverse perturbations in the context of the KP equations and the water-waves system.
601
639
1
10.4171/176-1/29
http://www.ems-ph.org/doi/10.4171/176-1/29
Digits of primes
James
Maynard
University of Oxford, UK
General
We discuss some different results on the digits of prime numbers, giving a simplified proof of weak forms of a result of Maynard and Mauduit-Rivat.
641
661
1
10.4171/176-1/30
http://www.ems-ph.org/doi/10.4171/176-1/30
The Hodge theory of the Hecke category
Geordie
Williamson
University of Sydney, Australia
General
Ideas from Hodge theory have found important applications in representation theory. We give a survey of joint work with Ben Elias which uncovers Hodge theoretic structure in the Hecke category (“Soergel bimodules”). We also outline similarities and differences to other combinatorial Hodge theories.
663
683
1
10.4171/176-1/31
http://www.ems-ph.org/doi/10.4171/176-1/31
Additive combinatorics and graph theory
Endre
Szemerédi
Hungarian Academy of Sciences, Budapest, Hungary
General
In this survey we mainly discuss the conditions for a set to contain long arithmetic progressions. We mention the ergodic, Fourier analytic and combinatorial approach to provide such conditions. In addition we describe defferent graph theoretic results with various applications in additive combinatorics and computer science. Finally, we mention some elementary results in discrete geometry.
685
716
1
10.4171/176-1/32
http://www.ems-ph.org/doi/10.4171/176-1/32
The arithmetic and topology of differential equations
Don
Zagier
Max-Planck-Institut für Mathematik, Bonn, Germany
General
This survey paper attempts to present in as elementary a way as possible a wide panorama of results concerning the relations between differential equations on the one hand and algebraic geometry, number theory and topology on the other. We use the famous Apéry numbers as a running example to illustrate the connections with, among other things, the theory of periods (Picard–Fuchs differential equations), the theory of modular forms (and the special values of their L-series), the theory of motives (starting with counting points on varieties over finite fields), and mirror symmetry (in particular, the Gamma Conjecture relating the asymptotics of solutions of differential equations to the multiplicative “Gamma class” of a variety). A number of relations to works by Friedrich Hirzebruch, in whose honor the lecture was given, are also described.
717
776
1
10.4171/176-1/33
http://www.ems-ph.org/doi/10.4171/176-1/33
Two-scale space-time methods for computational solid mechanics
Patrice
Hauret
Michelin Centre de Technologies de Ladoux, Clermont-Ferrand, France
Eric
Lignon
Michelin Centre de Technologies de Ladoux, Clermont-Ferrand, France
Nicole
Spillane
Ecole Polytechnique, Palaiseau, France
Benoît
Pouliot
Université Laval, Québec, Canada
General
The efficient, robust and accurate assessment of structures in large deformation simultaneously requires: i) the resolution of micro-scale states to avoid the use of empirical material laws and assess reliability, ii) the availability of sufficiently light models to enable optimal structure design and uncertainty quantification. The present work contributes to the first objective by the use of variational integrators, a non-conforming space discretization in the sense of mortar methods and the design of optimal coarse grids to enhance traditional domain decomposition methods. The second issue is handled by an homogenized problem iteratively improved by accurate subgrid models in space and time. Several aspects of the method are analyzed and some examples are displayed as an illustration.
777
794
1
10.4171/176-1/34
http://www.ems-ph.org/doi/10.4171/176-1/34
Living mathematics: Poincaré and Weyl in context
Jeremy
Gray
The Open University, Milton Keynes, UK
General
Henri Poincaré and Hermann Weyl enriched both mathematics and physics. Indeed, Poincaré and Weyl lived their mathematics, physics, and philosophy, and they reflected deeply on their work in their popular essays. By looking at their popular writings we can gain an intimate sense of what animated them, the different sets of values and aspirations that they had, and the ways they saw the significance of their work. Tradition is for the mathematician to create, change, even transcend – and surely Poincaré and Weyl transcended it – and for the historian to take apart, complicate, re-balance, even reject. But everyone is marked by their time and place.
795
812
1
10.4171/176-1/35
http://www.ems-ph.org/doi/10.4171/176-1/35
Mesoscopic models in biology
Vincent
Calvez
Université Claude Bernard Lyon 1, Villeurbanne, France
General
I review some mathematical problems which arose in the field of biological sciences, including microbiology, population dynamics and evolutionary biology. My focus is on partial differential equation (PDE) models with multiple levels of description. For each case study, I mean to describe the biological essence of the problem, and the mathematical questions associated with it. I also discuss how mathematical answers may enhance biological knowledge.
813
831
1
10.4171/176-1/36
http://www.ems-ph.org/doi/10.4171/176-1/36
On the singular part of measures constrained by linear PDEs and applications
Guido
De Philippis
SISSA, Trieste, Italy
General
The aim of this note is to present some recent results on the structure of the singular part of measures satisfying a PDE constraint, and to describe some applications.
833
845
1
10.4171/176-1/37
http://www.ems-ph.org/doi/10.4171/176-1/37
Recent progress on Bernoulli convolutions
Péter
Varjú
University of Cambridge, UK
General
The Bernoulli convolution with parameter $\lambda \in (0,1)$ is the measure on $\mathbf R$ that is the distribution of the random power series $\Sigma ± \lambda^n$, where ± are independent fair coin-tosses. This paper surveys recent progress on our understanding of the regularity properties of these measures.
847
867
1
10.4171/176-1/38
http://www.ems-ph.org/doi/10.4171/176-1/38
Random currents expansion of the Ising model
Hugo
Duminil-Copin
Institut des Hautes Études Scientifiques (IHÉS), Bures-sur-Yvette, France and Université de Genève, Switzerland
General
Critical behavior at an order/disorder phase transition has been a central object of interest in statistical physics. In the past century, techniques borrowed from many different fields of mathematics (Algebra, Combinatorics, Probability, Complex Analysis, Spectral Theory, etc) have contributed to a more and more elaborate description of the possible critical behaviors for a large variety of models. The Ising model is maybe one of the most striking success of this cross-fertilization, for this model of ferromagnetism is now very well understood both physically and mathematically. In this article, we review an approach, initiated in [7, 24] and based on the notion of random currents, enabling a deep study of the model.
869
889
1
10.4171/176-1/39
http://www.ems-ph.org/doi/10.4171/176-1/39
Functional Analysis and Operator Theory for Quantum Physics
The 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
Quantum physics (quantum mechanics)
Differential equations
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
5
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
2523-515X
2523-5168
Relative partition function of Coulomb plus delta interaction
Sergio
Albeverio
Universität Bonn, Germany
Claudio
Cacciapuoti
Università dell'Insubria, Como, Italy
Mauro
Spreafico
Università del Salento & INFN, Lecce, Italy
Relative zeta function, relative partition function, relative spectral measures, Coulomb interaction, point interactions, zeta regularization, finite temperature quantum fields, Casimir effect, asymptotic expansions
Global analysis, analysis on manifolds
Number theory
Quantum theory
The relative partition function and the relative zeta function of the perturbation of the Laplace operator by a Coulomb potential plus a point interaction centered in the origin is discussed. Applications to the study of the Casimir effect are indicated.
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
Canonical anticommutation relations, fermion Fock space, inequivalent representation, mass, quantum Dirac field
Quantum theory
Operator theory
A family of irreducible representations of the canonical anticommutation relations over an abstract Hilbert space indexed by a set of bounded linear operators is presented and a theorem on the mutual equivalence of them is established. As an application of the theorem, it is proved that quantum Dirac fields of different masses are mutually inequivalent. Moreover, a new class of irreducible representations of the CAR over a Hilbert space, which includes, as a special case, time-zero quantum Dirac fields, is constructed.
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, Czech Republic
Olga
Rossi
University of Ostrava, Czech Republic
Spectral transition, discrete spectrum, eigenvalue estimates
Quantum theory
Partial differential equations
We show that the operator for $\lambda\frac{\pi^2}{4}$ its spectrum contains the real line. In the critical case $\lambda=\frac{\pi^2}{4}$ we prove that the spectrum coincides with the half line $[0, \infty)$.
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, Italy
Alessandro
Teta
Università di Roma La Sapienza, Italy
Point interactions, self-adjoint extensions, unitary gas, Thomas effect
Quantum theory
Functional analysis
Operator theory
In this note we discuss the quantum mechanical three-body problem with pairwise zero-range interactions in dimension three. We review the state of the art concerning the construction of the corresponding Hamiltonian as a self-adjoint operator in the bosonic and in the fermionic case. Exploiting a quadratic form method, we also prove self-adjointness and boundedness from below in the case of three identical bosons when the Hilbert space is suitably restricted, i.e., excluding the "s-wave“ subspace.
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, Austria
Fritz
Gesztesy
Baylor University, Waco, USA
Helge
Holden
University of Trondheim, Norway
Roger
Nichols
The University of Tennessee at Chattanooga, USA
Factorization of operator-valued analytic functions, multiplicity of eigenvalues, index computations for finitely meromorphic operator-valued functions, Birman–Schwinger operators, dual pairs
Operator theory
We revisit and connect several notions of algebraic multiplicities of zeros of analytic operator-valued functions and discuss the concept of the index of meromorphic operator-valued functions in complex, separable Hilbert spaces. Applications to abstract perturbation theory and associated Birman–Schwinger-type operators and to the operator-valued Weyl–Titchmarsh functions associated to closed extensions of dual pairs of closed operators are provided.
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, Austria
Matthias
Langer
University of Strathclyde, Glasgow, UK
Vladimir
Lotoreichik
Nuclear Physics Institute, Řež - Prague, Czech Republic
Trace formula, delta interaction, Schrödinger operator, singular potential
Partial differential equations
Quantum theory
Operator theory
Let $\Sigma\subset\mathbb R^d$ be a $C^\infty$-smooth closed compact hypersurface, which splits the Euclidean space $\mathbb R^d$ into two domains $\Omega_\pm$. In this note self-adjoint Schrödinger operators with $\delta$ and $\delta'$-interactions supported on $\Sigma$ are studied. For large enough $m\in\mathbb N$ the difference of $m$th powers of resolvents of such a Schrödinger operator and the free Laplacian is known to belong to the trace class. We prove trace formulae, in which the trace of the resolvent power difference in $L^2(\mathbb R^d)$ is written in terms of Neumann-to-Dirichlet maps on the boundary space $L^2(\Sigma)$.
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, USA
Brezis–Nirenberg problem, hyperbolic space, nonexistence of solutions, Pohozaev identity, Hardy inequality
Partial differential equations
Using a Rellich–Pohozaev argument and Hardy's inequality, we derive an improved bound on the nonlinear eigenvalue for the non existence of radial solutions of a Brezis–Nirenberg problem, with Dirichlet boundary conditions, on a geodesic ball of $\mathbb{H}^n$, for $2
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, La Garde, France
Hiba
Hammedi
Université de Toulon et du Var, La Garde, France
Waveguide, mixed boundary conditions, twisting
Quantum theory
Operator theory
This paper is concerned with the study of the existence/non-existence of the discrete spectrum of the Laplace operator on a domain of $\mathbb R ^3$ which consists in a twisted tube. This operator is defined by means of mixed boundary conditions. Here we impose Neumann Boundary conditions on a bounded open subset of the boundary of the domain (the Neumann window) and Dirichlet boundary conditions elsewhere.
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, Germany
Periodic waveguides, spectral gaps, asymptotic analysis
Partial differential equations
Operator theory
In this note we investigate spectral properties of a periodic waveguide $\Omega^\varepsilon$ ($\varepsilon$ is a small parameter) obtained from a straight strip by attaching an array of $\varepsilon$-periodically distributed identical protuberances having "room-and-passage" geometry. In the current work we consider the operator $\mathcal{A}^\varepsilon =-\rho^\varepsilon\Delta_{\Omega^\varepsilon}$, where $\Delta_{\Omega^\varepsilon}$ is the Neumann Laplacian in $\Omega^\varepsilon$, the weight $\rho^\varepsilon$ is equal to $1$ everywhere except the union of the „rooms". We will prove that the spectrum of $\mathcal{A}^\varepsilon$ has at least one gap as $\varepsilon$ is small enough provided certain conditions on the weight $\rho^\varepsilon$ and the sizes of attached protuberances hold.
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, Italy
Michele
Correggi
Università degli Studi Roma Tre, Italy
Rodolfo
Figari
Università degli Studi di Napoli Federico II, Italy
Partial differential equations
Quantum theory
We study the time-evolution of a quantum particle subjected to time-dependent zero-range forces in two dimensions. After establishing a conceivable ansatz for the solution to the Schrödinger equation, we prove that the wave packet time-evolution is completely specified by the solutions of a system of Volterra-type equations – the charge equations – involving the coefficients of the singular part of the wave function, thus extending to the two-dimensional case known results in one and three dimensions.
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, USA
Peter
Kuchment
Texas A&M University, College Station, USA
Beng
Ong
CGG, Houston, USA
Quantum graph, spectral gap, resonator
Quantum theory
Partial differential equations
Global analysis, analysis on manifolds
In this brief paper we present some results on creating and manipulating spectral gaps for a (regular) quantum graph by inserting appropriate internal structures into its vertices. Complete proofs and extensions of the results are planned for another publication.
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, USA
Partial differential equations
Quantum theory
We derive an adiabatic theory for a stochastic differential equation, $$ \varepsilon\, \d X(s) = L_1(s) X(s)\, \d s + \sqrt{\varepsilon} L_2(s) X(s) \, \d B_s, $$ under a condition that instantaneous stationary states of $L_1(s)$ are also stationary states of $L_2(s)$. We use our results to derive the full statistics of tunneling for a driven stochastic Schrödinger equation describing a dephasing process.
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
We consider Schrödinger operators in $\mathbb R^d$ with complex potentials supported on a hyperplane and show that all eigenvalues lie in a disk in the complex plane with radius bounded in terms of the $L^p$ norm of the potential with $d-1 < p \leq d$. We also prove bounds on sums of powers of eigenvalues.
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, Portugal
David
Krejčiřík
Czech Technical University in Prague, Prague, Czech Republic
Dirichlet Laplacian in tubes, quantum waveguides, quantum layers, ground-state energy
Number theory
Algebraic geometry
We derive a lower bound to the spectral threshold of the Dirichlet Laplacian in tubular neighbourhoods of constant radius about complete surfaces. This lower bound is given by the lowest eigenvalue of a one-dimensional operator depending on the radius and principal curvatures of the reference surface. Moreover, we show that it is optimal if the reference surface is non-negatively curved.
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
Vector-valued Sturm–Liouville operator, point interaction, spectrum, absolutely and singular continuous spectrum, eigenvalues, boundary triplets, Weyl function
Ordinary differential equations
Operator theory
The paper is devoted to the spectral theory of vector-valued Sturm–Liouville operators on the half-line with a summable potential and a finite number of point interactions. It is shown that the positive spectrum is purely absolutely continuous and of constant multiplicity. The negative spectrum is either finite or discrete with the only accumulation point at zero. Our approach relies on the thorough investigation of the corresponding Weyl functions and involves technique elaborated in our previous papers.
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, Germany
Timo
Weidl
Universität Stuttgart, Germany
Laplacian, Neumann window, spectral asymptotics, Dirichlet-to-Neumann operator, Birman–Schwinger analysis
Partial differential equations
Quantum theory
In the present article we will give a new proof of the ground state asymptotics of the Dirichlet Laplacian with a Neumann window acting on functions which are defined on a two-dimensional infinite strip or a three-dimensional infinite layer. The proof is based on the analysis of the corresponding Dirichlet-to-Neumann operator as a first order classical pseudo-differential operator. Using the explicit representation of its symbol we prove an asymptotic expansion as the window length decreases.
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, France
Rola
Kiwan
American University in Dubai, United Arab Emirates
Courant theorem, nodal lines, nodal domains, Dirichlet, cube
Partial differential equations
Global analysis, analysis on manifolds
This paper is devoted to the refined analysis of Courant's theorem for the Dirichlet Laplacian in a bounded open set. Starting from the work by A. Pleijel in 1956, many papers have investigated in which cases the inequality in Courant's theorem is an equality. All these results were established for open sets in $\mathbb R^2$ or for surfaces like $\mathbb S^2$ or $\mathbb T^2$. The aim of the current paper is to look for the case of the cube in $\mathbb R^3$. We will prove that the only eigenvalues of the Dirichlet Laplacian which are Courant sharp are the two first eigenvalues.
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, Japan
Electron-phonon interaction, Carleman operator, infrared catastrophe
Quantum theory
This paper is dedicated to Pavel Exner on the occasion of his 70th birthday by introducing a part of our attempts in material science, that is, a mathematical modeling of electron-phonon interaction for small wave numbers close to zero. We extrapolate the electron-phonon interaction on the presupposition that the phonon dispersion relation and the electron-phonon coupling function are estimated using some experimental data.
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, Japan
Trotter product formula, Trotter–Kato product formula, Zeno product formula, unitary groups, resolvents, form sum of selfadjoint operators, Feynman path integral
Operator theory
Quantum theory
Modified unitary form-sum Trotter–Kato product formula and Zeno product formula, which replace their short-time unitary groups appearing as factors in their products by their resolvents, are proved by Kato's method.
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, Poland
Schrödinger operator with delta potential, separation of variables, eigenvalues asymptotics
Partial differential equations
We consider two dimensional system governed by the Hamiltonian with delta interaction supported by two concentric circles separated by distance $d$. We analyze the asymptotics of the discrete eigenvalues for $d \to 0$ as well as for $d\to \infty$.
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, Italy
Bartosch
Ruszkowski
Universität Stuttgart, Germany
Timo
Weidl
Universität Stuttgart, Germany
Heisenberg Laplacian, Berezin–Li–Yau inequality
Operator theory
Difference and functional equations
We study Riesz means of eigenvalues of the Heisenberg Laplacian with Dirichlet boundary conditions on a cylinder in dimension three. We obtain an inequality with a sharp leading term and an additional lower order term.
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, France
Singular Schrödinger operator, leaky graph, $\delta$-potential, star graph
Ordinary differential equations
We provide a purely variational proof of the existence of eigenvalues below the bottom of the essential spectrum for the Schrödinger operator with an attractive $\delta$-potential supported by a star graph, i.e. by a finite union of rays emanating from the same point. In contrast to the previous works, the construction is valid without any additional assumption on the number or the relative position of the rays. The approach is used to obtain an upper bound for the lowest eigenvalue.
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
Green function, ground state, Liouville theorem, positive solution, principal eigenvalue, small perturbation
Partial differential equations
Operator theory
For a given second-order linear elliptic operator $L$ which admits a positive minimal Green function, and a given positive weight function $W$, we introduce a family of weighted Lebesgue spaces $L^p(\phi_p)$ with their dual spaces, where $1\leq p\leq \infty$. We study some fundamental properties of the corresponding (weighted) Green operators on these spaces. In particular, we prove that these Green operators are bounded on $L^p(\phi_p)$ for any $1\leq p\leq \infty$ with a uniform bound. We study the existence of a principal eigenfunction for these operators in these spaces, and the simplicity of the corresponding principal eigenvalue. We also show that such a Green operator is a resolvent of a densely defined closed operator which is equal to $(-W^{-1})L$ on $C_0^\infty$, and that this closed operator generates a strongly continuous contraction semigroup. Finally, we prove that if $W$ is a (semi)small perturbation of $L$, then for any $1\leq p\leq \infty$, the associated Green operator is compact on $L^p(\phi_p)$, and the corresponding spectrum is $p$-independent.
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, Germany
Abstract boundary value problems, Dirichlet-to-Neumann operator, graph Laplacians, coupled spaces
Operator theory
Combinatorics
Dynamical systems and ergodic theory
Global analysis, analysis on manifolds
In this note we present some abstract ideas how one can construct spaces from building blocks according to a graph. The coupling is expressed via boundary pairs, and can be applied to very different spaces such as discrete graphs, quantum graphs or graph-like manifolds. We show a spectral analysis of graph-like spaces, and consider as a special case vector-valued quantum graphs. Moreover, we provide a prototype of a convergence theorem for shrinking graph-like spaces with Dirichlet boundary conditions.
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, USA
Periodic Jacobi matrices, discriminant, magic formula
Operator theory
Functions of a complex variable
Let $K$ be a doubly infinite, self-adjoint matrix which is finite band (i.e. $K_{jk} = 0$ if $|j-k| > m$) and periodic ($KS^n = S^nK$ for some $n$ where $(Su)_j = u_{j+1}$) and non-degenerate (i.e. $K_{j j+m} \ne 0$ for all $j$). Then, there is a polynomial, $p(x,y)$, in two variables with $p(K,S^n) = 0$. This generalizes the tridiagonal case where $p(x,y) = y^2 - y \Delta(x) + 1$ where $\Delta$ is the discriminant. I hope Pavel Exner will enjoy this birthday bouquet.
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, Japan
Taksu
Cheon
Kochi University of Technology, Japan
Adiabatic theorem, eigenspace, homotopy, covering space
Quantum theory
Algebraic geometry
An adiabatic time evolution of a closed quantum system connects eigenspaces of initial and final Hermitian Hamiltonians for slowly driven systems, or, unitary Floquet operators for slowly modulated driven systems. We show that the connection of eigenspaces depends on a topological property of the adiabatic paths for given initial and final points. An example in slowly modulated periodically driven systems is shown. These analysis are based on the topological analysis of the exotic quantum holonomy in adiabatic closed paths.
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 graph, vertex coupling, spectral filtering, quantum control
Quantum theory
We examine transmission through a quantum graph vertex to which auxiliary edges with constant potentials are attached. We find a characterization of vertex couplings for which the transmission probability from a given „input" line to a given „output" line shows a flat passband. The bandwidth is controlled directly by the potential on the auxiliary edges. Vertices with such couplings can thus serve as controllable band-pass filters. The paper extends earlier works on the topic. The result also demonstrates the effectivity of the $ST$-form of boundary conditions for a study of scattering in quantum graph vertices.
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, France
Chernoff lemma, semigroup theory, product formula, convergence rate
Operator theory
The Chernoff $\sqrt{n}$-lemma is revised. This concerns two aspects: an improvement of the Chernoff estimate in the strong operator topology and an operator-norm estimate for quasi-sectorial contractions. Applications to the Lie–Trotter product formula approximation for semigroups is presented.
565
573
1
10.4171/175-1/28
http://www.ems-ph.org/doi/10.4171/175-1/28
Frobenius Algebras II
Tilted and Hochschild Extension Algebras
Andrzej
Skowroński
Nicolaus Copernicus University, Toruń, Poland
Kunio
Yamagata
Tokyo University of Agriculture and Technology, Japan
Associative rings and algebras
Commutative rings and algebras
Linear and multilinear algebra; matrix theory
Category theory; homological algebra
(primary; secondary): 16-01; 13E10, 15A63, 15A69, 16Dxx, 16E10, 16E30, 16E40, 16G10, 16G20, 16G60, 16G70, 16S50, 16S70, 18A25, 18E30, 18G15
Mathematics
Algebra
Algebra, module, bimodule, representation, quiver, ideal, radical, simple module, semisimple module, uniserial module, projective module, injective module, tilting module, hereditary algebra, tilted algebra, Frobenius algebra, symmetric algebra, selfinjective algebra, Hochschild extension algebra, category, functor, torsion pair, projective dimension, injective dimension, global dimension, Euler form, Grothendieck group, irreducible homomorphism, almost split sequence, Auslander-Reiten translation, Auslander-Reiten quiver, stable equivalence, syzygy module, duality bimodule, Hochschild extension
This is the second of three volumes which will provide a comprehensive introduction to the modern representation theory of Frobenius algebras. The first part of the book is devoted to fundamental results of the representation theory of finite dimensional hereditary algebras and their tilted algebras, which allow to describe the representation theory of prominent classes of Frobenius algebras. The second part is devoted to basic classical and recent results concerning the Hochschild extensions of finite dimensional algebras by duality bimodules and their module categories. Moreover, the shapes of connected components of the stable Auslander-Reiten quivers of Frobenius algebras are described. The only prerequisite in this volume is a basic knowledge of linear algebra and some results of the fi rst volume. 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 fi elds.
5
12
2017
978-3-03719-174-3
978-3-03719-674-8
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/174
http://www.ems-ph.org/doi/10.4171/174
EMS Textbooks in Mathematics
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
Probability + statistics
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
10
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
Differential equations
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
23
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
2523-5176
2523-5184
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 geometry
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
12
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
2523-515X
2523-5168
Symmetric superspaces: slices, radial parts, and invariant functions
Alexander
Alldridge
Universität Köln, Germany
Chevalley restriction theorem, differential operator, Harish-Chandra homomorphism, Lie superalgebra, radial part, Riemannian symmetric superspace
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, Italy
Quiver Grassmannians, Dynkin quivers, cluster algebras
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, Germany
Reductive groups, spherical varieties
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, France
Michela
Varagnolo
Université de Cergy-Pontoise, France
Éric
Vasserot
Université Paris Diderot Paris 7, France
Finite reductive groups, Deligne–Lusztig theory, higher representation theory
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, Germany
Homological mirror symmetry, singularities, strange duality, invertible polynomials, derived categories, weighted projective lines, Coxeter-Dynkin diagrams, group action, orbifold E-function, Burnside ring, unimodal, bimodal
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, Germany
Catharina
Stroppel
Universität Bonn, Germany
Orthosymplectic supergroup, finite dimensional representations, Brauer algebra, Deligne category
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, USA
Noah
Snyder
Indiana University, Bloomington, United States
Geordie
Williamson
Max-Planck-Institut für Mathematik, Bonn, Germany
Frobenius extensions, diagrammatic algebra, Soergel bimodules
$K$-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, Germany
Ghislain
Fourier
University of Glasgow, UK
Peter
Littelmann
Universität Köln, Germany
Flag varieties, spherical varieties, cluster algebras, toric degenerations
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, Germany
Restricted representations, critical level, Kac–Moody algebras, Feigin–Frenkel conjecture
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, Sweden
Christoph
Schweigert
Universität Hamburg, Germany
Topological field theory, tensor categories, categorification
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, UK
Triangulated categories, quasi-hereditary algebras, exceptional sequences, recollements, gentle algebras, derived equivalences, derived composition series, derived Jordan–Hölder property
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, Germany
Dominant dimension, representation dimension, Schur algebras, gendosymmetric algebras
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, Germany
Highest weight category, strict polynomial functor, polynomial representation, divided power, Schur algebra, quasi-hereditary algebra, Ringel duality
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, Germany
BGG category $\mathcal O$, bocs, Eilenberg–Moore category, exact Borel subalgebra, Kleisli category, $q$-Schur algebras, quasi-hereditary algebras, reduction algorithm, representation type, Schur algebras, tame, wild
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, Israel
Algebra of quaternion type, 2-CY-tilted algebra, Brauer graph algebra, derived equivalence, Jacobian algebra, marked surface, periodic modules, quiver with potential, ribbon graph, ribbon quiver, triangulation quiver, triangulation algebra, symmetric algebra
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, UK
Moment graphs, semi-infinite order, character formulae
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
TU Kaiserslautern, Germany
Local-global conjectures, McKay conjecture, Alperin–McKay conjecture, Alperin weight conjecture, Brauer's height zero conjecture, Dade conjecture, reduction theorems
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, Germany
Infinite dimensional Lie group, host algebra, semibounded representation, bounded representation, holomorphic induction
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, Germany
Alexey
Petukhov
The University of Manchester, UK
Primitive ideals, finitary Lie algebras, highest weight modules, $\mathfrak{osp}$-duality
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, Germany
Algebraic groups, spherical varieties, representation theory, Kac–Moody groups
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, Germany
Quiver moduli, GIT quotients, small desingularizations
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, Germany
Valdemar
Tsanov
Georg-August Universität Göttingen, Germany
Flag variety, geometric invariant theory, principal SL(2)-subgroup, branching cone, Mori dream space
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, Germany
Reduction theorems, character triples, counting conjectures
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, Germany
Rational Cherednik algebras, Calogero–Moser spaces, reflection groups
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, Germany
infinite dimensional Lie group, smooth vector, analytic vector, semibounded representation, continuous representation
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
Differential equations
Differential + Riemannian geometry
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
1
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
Differential equations
Mathematical logic
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
1
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
Differential equations
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
2523-5176
2523-5184
Lines in positive genus: An introduction to flat surfaces
Diana
Davis
Williams College, Williamstown, USA
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, USA
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, USA
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
Complex analysis
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
12
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
Groups + group theory
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
Winner of the 2016 EMS Monograph Award! 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
Functional analysis
Groups + group theory
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
2523-5230
2523-5249
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, 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, USA
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, USA
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
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, USA
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, USA
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, 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, 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, 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, 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, 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, Austria
Johannes
Wallner
Technische Universität 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, 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, Germany
Ezra
Tampubolon
Technische Universität 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, 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, 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, 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
Differential + Riemannian geometry
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
2523-5176
2523-5184
Introduction to geodesics in sub-Riemannian geometry
Andrei
Agrachev
SISSA, Trieste, Italy
Davide
Barilari
Université Paris 7, Denis Diderot, Paris, France
Ugo
Boscain
Ecole Polytechnique, Palaiseau, 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
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, Germany
Paul
Gassiat
Université Paris-Dauphine, Paris, 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, 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
Differential + Riemannian geometry
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
2523-5176
2523-5184
Some topics of geometric measure theory in Carnot groups
Francesco
Serra Cassano
Università di Trento, 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, 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, USA
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
Université de Strasbourg, France
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
Functional analysis
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
2523-5133
2523-5141
27
Introduction to Teichmüller theory, old and new, VI
Athanase
Papadopoulos
Université de Strasbourg, France
General
1
29
1
10.4171/161-1/1
http://www.ems-ph.org/doi/10.4171/161-1/1
Alexander Grothendieck
Valentin
Poénaru
Université Paris-Sud 11, Orsay, 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, Switzerland
Lizhen
Ji
University of Michigan, Ann Arbor, USA
Athanase
Papadopoulos
Université de Strasbourg, 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, France
Hideki
Miyachi
Osaka University, Japan
Ken’ichi
Ohshika
Osaka University Graduate School of Science, 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, Spain
Juan
Souto
Université de Rennes 1, 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, 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, USA
Ian
McIntosh
University of York, UK
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, 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, Switzerland
Lizhen
Ji
University of Michigan, Ann Arbor, USA
Athanase
Papadopoulos
Université de Strasbourg, 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, 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, Switzerland
Lizhen
Ji
University of Michigan, Ann Arbor, USA
Athanase
Papadopoulos
Université de Strasbourg, 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, 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, France
Athanase
Papadopoulos
Université de Strasbourg, 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, Switzerland
Athanase
Papadopoulos
Université de Strasbourg, 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, 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
Functional analysis
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
2523-5133
2523-5141
26
Introduction to Teichmüller theory, old and new, V
Athanase
Papadopoulos
Université de Strasbourg, 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, USA
Ser Peow
Tan
National University of Singapore, Singapore
Hyperbolic manifolds, identities, orthogeodesic, ortholength, orthospec- trum, simple geodesics, geodesic flow
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
Teichmüller space, Thurston metric
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
The signature cocycle, Meyer function, local signature
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, Japan
Yusuke
Kuno
Tsuda College, Tokyo, Japan
Mapping class group, Johnson homomorphism, Goldman-Turaev Lie bial- gebra, Dehn twist
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, Japan
Automrophism groups of free groups, IA-automorphism groups, Johnson homomorphisms, Magnus representations
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
Character variety, rigidity and flexibility, bounded cohomology
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, USA
Satake compactification, Anosov subgroup, reduction theory, coarse fundamental domain, Borel-Serre compactification, geometrically finite space, locally symmetric space
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, Canada
Symplectic geometry, representations, fundamental groups
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, France
Athanase
Papadopoulos
Université de Strasbourg, 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, Switzerland
Norbert
A’Campo
Universität Basel, Switzerland
Vincent
Alberge
Université de Strasbourg, France
Athanase
Papadopoulos
Université de Strasbourg, 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
Real analysis
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
Differential + Riemannian geometry
Probability + statistics
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
2523-5133
2523-5141
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
Combinatorics + graph theory
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, 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, 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, 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
Instituto Nacional de Matemática Pura e Aplicada, 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, 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
Universität 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, 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
Topology
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
2523-5176
2523-5184
Tempered Homogeneous Function Spaces
Hans
Triebel
University of Jena, Germany
Functional analysis
Fourier analysis
Functional 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
2523-5176
2523-5184
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
Groups + group theory
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
2523-5176
2523-5184
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
Differential + Riemannian geometry
Algebraic topology
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
2523-5133
2523-5141
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
Statistical physics
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
Calculus + mathematical analysis
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
Differential equations
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
Fields + rings
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
2523-515X
2523-5168
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, Sas Nagar, Punjab, India
Sanjeev
Kumar
Panjab University, Chandigarh, India
Valued fields, non-Archimedean valued fields, irreducible polynomials
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, Israel
Arno
Fehm
University of Konstanz, Germany
Totally $\mathfrak{S}$-adic numbers, Hilbertian fields
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
Defect extensions, valued rational function fields, dependent, independent Artin-Schreier defect extensions
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
Université Paris 6, France
Charles
Favre
École Polytechnique, Palaiseau, France
Mattias
Jonsson
University of Michigan, Ann Arbor, USA
Izumi's theorem, divisorial valuation, quasimonomial valuations, volume of a valuation, toroidal embeddings, dual complexes
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, France
Anatoly
Libgober
University of Illinois at Chicago, USA
Plane curve singularities, multivariable Alexander polynomial, faces of quasi-adjunction, spectrum of singularity, Newton trees, log-canonical thresholds
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, France
Mickaël
Matusinski
Université Bordeaux 1, Talence, France
Guillermo
Moreno-Socías
CNRS/UVSQ, Versailles, France
Dicritical divisors, Rees valuations, horizontal divisors, pencil of curves
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, France
Olivier
Piltant
Université de Versailles Saint-Quentin, Versailles, France
Ana
Reguera
Universidad de Valladolid, Spain
Rational surface singularity, divisorial valuation, Hilbert-Samuel function
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, France
$C$-minimality, algebraically closed valued fields, cell decomposition
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 Dale
Cutkosky
University of Missouri, Columbia, United States
Valuation, Noetherian local ring, semigroup, generating sequence, defect
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, Turkey
Additive polynomials, valued fields
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, France
Formally real fields, real valuations, valuation fans, $\mathbb{R}$-places, Henselian fields, model theory of fields, real algebraic varieties, abstract spaces of orderings
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, Spain
Miguel Ángel
Olalla Acosta
Universidad de Sevilla, Spain
Mark
Spivakovsky
Université Paul Sabatier, Toulouse, France
Bernard
Teissier
UMR 7586 du CNRS, Paris, France
Extensions of valuations, formal completion, excellent ring
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, Spain
M.
Martínez
Universidad de Valladolid, Spain
C.
Rodríguez
Universidad de Léon, Spain
Valuation, parameterized tree, real rank, factorization
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, United Kingdom
Whitney stratifications, Henselian valued fields, isometries, semi-algebraic sets
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
Valued fields, imaginary elements
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, Canada
Asim
Naseem
GC University, Lahore, Pakistan
Valued function field, defect, completion, Abhyankar valuation
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, France
Hardy fields, generalized series fields and exponential-logarithmic series fields with derivations, asymptotic integration, integration
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, France
Jet schemes, embedded Nash problem, rational double point singularities, Hilbert-Poincaré series
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
Valuative tree, non-metric tree, valuations centered at a local domain
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, France
Local uniformization, rank one valuations, valuations centered at a local ring
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
Large fields, ultraproducts, PAC, pseudo closed fields, Henselian pairs, elementary equivalence, algebraic varieties, rational points, function fields, (inverse) Galois theory, embedding problems, model theory, rational connectedness, extremal fields
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
Quasi-valuations, approximation theorem
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, France
Toric geometry, valuations, uniformization, key polynomials
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, USA
Valuation theory, pro-$\ell$ Galois groups, abelian-by-central, local theory
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, USA
Hahn fields, truncation
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, ergodicity, $P$-adic spheres, Van der Put series
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, USA
Igor
Zhukov
St. Petersburg University, Russian Federation
Complete discrete valuation field, higher local field, imperfect residue field, wild ramification, Swan conductor, Artin conductor
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
History of mathematics
Geometry
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
2523-5133
2523-5141
23
Sophus Lie, a giant in mathematics
Lizhen
Ji
University of Michigan, Ann Arbor, USA
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, USA
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, UK
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, 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, Switzerland
Athanase
Papadopoulos
Université de Strasbourg, 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, 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, Switzerland
Athanase
Papadopoulos
Université de Strasbourg, 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, 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, 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, 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, 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.
Differential + Riemannian geometry
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
2523-5133
2523-5141
22
Weak Minkowski spaces
Athanase
Papadopoulos
Université de Strasbourg, France
Marc
Troyanov
Ecole Polytechnique Fédérale de Lausanne, Switzerland
Weak Minkowski space, Minkowski geometry, norm, Hilbert geometry, Funk geometry, weak norm, Mazur–Ulam theorem, Desarguesian space, Busemann G-space
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, France
Marc
Troyanov
Ecole Polytechnique Fédérale de Lausanne, Switzerland
Funk metric, convexity, Hilbert metric, Busemann's methods
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, Switzerland
Hilbert geometry, Funk geometry, Finlser metric, flag curvature, projective metric
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, France
Hilbert geometry, Finsler geometry, metric spaces, normed vector spaces, Lipschitz distance
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, France
Horofunction boundary, Busemann function, detour cost, Hilbert isometry
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, USA
Hilbert geometry, hyperbolic geometry, Finsler structure
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, France
Flow completion, Burgers equation, manifolds of mappings
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, Chile
Geodesic flows, Finsler geometry, Hilbert geometry, hyperbolic dynamics, Lyapunov exponents, entropy
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, Switzerland
Hilbert metric, non-expansive maps
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, USA
Birkhoff's version of Hilbert's metric, Birkhoff's contraction coefficient, Denjoy–Wolff type theorems, dynamics of non-expansive mappings, isometric embeddings, nonlinear mappings on cones
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, France
Convex real projective structure, geodesic flow, deformation space, hyperbolic structure, geodesic current, topological entropy, volume entropy, Busemann cocycle, Patterson–Sullivan measure
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, Japan
Ken’ichi
Ohshika
Osaka University Graduate School of Science, Japan
Sumio
Yamada
Gakushuin University, Tokyo, Japan
Teichmüller space, Weil–Petersson metric, Funk metric, convex geometry
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, France
Sumio
Yamada
Gakushuin University, Tokyo, Japan
Hilbert metric, Funk metric, constant curvature, trigonometry
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, Switzerland
David Hilbert , Hilbert geometry
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, France
Hilbert problems, Busemann geometry, Hilbert's Problem IV, Crofton formula, Hilbert metric, Desarguesian space, projective metric
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
History of mathematics
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
2523-5214
2523-5222
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
History of mathematics
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
2523-5214
2523-5222
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
History of mathematics
Number theory
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
2523-5214
2523-5222
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
Combinatorics + graph theory
Lattice theory
Fields + rings
Linear algebra
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
2523-5133
2523-5141
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, 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, France
Dominique
Manchon
Université Blaise Pascal, Aubière, France
Proalgebraic groups, Hopf algebras, operads
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, USA
Luis
Duffaut Espinosa
University of NSW at the Australian Defence Force Academy, Canberra, Australia
Nonlinear control systems, nonlinear operators, Hopf algebras
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, Norway
Hans
Munthe-Kaas
University of Bergen, Norway
Geometric numerical integration, Butcher series, Lie{Butcher series, combinatorial Hopf algebras
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, France
Ordinary differential operators, trees, moulds, arborification, Hopf algebras
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, USA
confinement, dynamical chiral symmetry breaking, Dyson-Schwinger equations, hadron spectrum, hadron elastic and transition form factors, in-hadron condensates, parton distribution functions, $U_A$(1)-problem
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
Mathematics
Number theory
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
2523-5176
2523-5184
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
Calculus + mathematical analysis
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
2523-5176
2523-5184
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
Calculus + mathematical analysis
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
Winner of the 2014 EMS Monograph Award! 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
Groups + group theory
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
Geometry
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
Mathematical foundations
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
2523-5087
2523-5095
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
Differential + Riemannian geometry
Differential equations
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
2523-5192
2523-5206
Foundations of Rigid Geometry I
Kazuhiro
Fujiwara
Nagoya University, Japan
Fumiharu
Kato
Tokyo Institute of Technology, Japan
Number theory
Order, lattices, ordered algebraic structures
Commutative rings and algebras
Algebraic geometry
Primary: 11G99, Secondary: 06E99, 13F30, 13J07, 14A15, 14A20
Number theory
Rigid geometry, formal geometry, birational geometry
Rigid geometry is one of the modern branches of algebraic and arithmetic geometry. It has its historical origin in J. Tate’s rigid analytic geometry, which aimed at developing an analytic geometry over non-archimedean valued fields. Nowadays, rigid geometry is a discipline in its own right and has acquired vast and rich structures, based on discoveries of its relationship with birational and formal geometries. In this research monograph, foundational aspects of rigid geometry are discussed, putting emphasis on birational and topological features of rigid spaces. Besides the rigid geometry itself, topics include the general theory of formal schemes and formal algebraic spaces, based on a theory of complete rings which are not necessarily Noetherian. Also included is a discussion on the relationship with Tate‘s original rigid analytic geometry, V.G. Berkovich‘s analytic geometry and R. Huber‘s adic spaces. As a model example of applications, a proof of Nagata‘s compactification theorem for schemes is given in the appendix. The book is encyclopedic and almost self-contained.
1
22
2018
978-3-03719-135-4
978-3-03719-635-9
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/135
http://www.ems-ph.org/doi/10.4171/135
EMS Monographs in Mathematics
2523-5192
2523-5206
Compactness and Stability for Nonlinear Elliptic Equations
Emmanuel
Hebey
Université de Cergy-Pontoise, France
Global analysis, analysis on manifolds
Partial differential equations
58J05, 35J15
Differential equations
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
Probability + statistics
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
Complex analysis
Differential + Riemannian geometry
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
2523-5133
2523-5141
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
Differential equations
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
2523-5176
2523-5184
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
Groups + group theory
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
Differential equations
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
Calculus + mathematical analysis
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
2523-5176
2523-5184
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
Differential + Riemannian geometry
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
Mathematics
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
2523-5214
2523-5222
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
Fields + rings
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
2523-515X
2523-5168
Infinite dimensional tilting theory
Lidia
Angeleri Hügel
Università degli Studi di Verona, Italy
Tilting module, torsion pair, cotorsion pair, resolving subcategory, universal localization, Gabriel topology, quasi-coherent sheaves over a weighted projective line
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, UK
Modular representations, elementary abelian $p$-groups, constant Jordan type, vector bundles, Chern classes
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, Germany
Multiplicative basis, ray-category, coverings, Brauer–Thrall conjectures
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, USA
Quiver Hecke algebras, Quantum groups
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, China
Mutation, left mutation, exchange graph, ordered exchange graph
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
Donaldson–Thomas invariants, quivers with potentials, wall-crossing formulas
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, Japan
Cluster algebras, perverse sheaves, singular supports
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, USA
Modular Lie algebra, finite group scheme, support variety, rank variety, Jordan type, one-parameter subgroup, vector bundles
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, UK
Pure-injective module, superdecomposable, width, pointed module, pp formula, pp-type, modular lattice, tubular algebra, Krull–Gabriel dimension
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, Czech Republic
Monoidal model category, exact category, derived category, cotorsion pair, weak factorization system, quasi-coherent sheaf
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
Calculus + mathematical analysis
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
Functional analysis
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
Calculus + mathematical analysis
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
History of mathematics
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
Max-Planck-Institut für Wissenschaftsgeschichte, Berlin, Germany
Correspondence principle, Erwin Schrödinger, equivalence principle, matrix mechanics, optical dispersion, optical-mechanical analogy, Schrödinger equation, quantum mechanics, wave mechanics, Werner Heisenberg
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, Switzerland
Baptiste
Schubnel
ETH Zürich, Switzerland
Quantum mechanics, operator algebras, quantum probability theory, quantum theory of experiments, determinism, Schwinger–Wigner “Master Formula” for frequencies of histories, decoherence
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, USA
Schroedinger’s cat, quantum mechanics, decoherence, macrorealism
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, Austria
Peter
Zoller
Universität Innsbruck, Austria
Quantum simulation, quantum computer
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, Austria
Markus
Aspelmeyer
Universität Wien, Austria
Quantum optics, quantum optomechanics
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, Denmark
Erwin Schrödinger, Niels Bohr, Arthur Eddington, atomic models, interpretation of quantum mechanics, complementarity, Schrödinger’s cat
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, Netherlands
Wave optics, wave mechanics, geometrical optics, mechanics
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, France
Louis de Broglie, matter waves, Erwin Schrödinger
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
Mathematics and science
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, United Kingdom
Water waves, irrotational flow, particle trajectories
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, Switzerland
László
Székelyhidi Jr.
Universität Leipzig, Germany
Euler equations, anomalous dissipation, $h$-principle, Onsager's conjecture
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, USA
Algebraic topology, homology groups, distance, stability, algorithms, scale, shape analysis, topology repair, high-dimensional data
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
Institut des Hautes Études Scientifiques, Bures-Sur-Yvette, France
Shannon–Kolmogorov–Sinai entropy, Boltzmann formula, von Neumann entropy, strong subadditivity
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, USA
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, 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), France
Coulomb gas, log gases, Ginzburg–Landau, superconductivity, vortices, Ohta–Kawasaki, random matrices, Abrikosov lattice, renormalized energy.
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, Israel
Model theory, classification theory, dependent classes, dependent theories, the recounting theorem
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, France
Stochastic processes, Gaussian processes, upper and lower bounds
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, Switzerland
Duflo isomorphism, Kashiwara–Vergne conjecture, Drinfeld associators
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, Switzerland
Smoluchowski coagulation equation, random configuration model, Galton–Watson tree
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, France
Cremona group, geometric group theory, holomorphic dynamical systems, Tits alternative
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
Image inpainting, variational models, exemplar-based, non-local methods
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', Italy
KAM theory, stability, invariant tori
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
Galois representations, $L$-functions, modular forms, $p$-adic Hodge theory
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, UK
Alessio
Corti
Imperial College London, UK
Sergey
Galkin
The University of Tokyo, Chiba, Japan
Vasily
Golyshev
Russian Academy of Sciences, Moscow, Russian Federation
Alexander
Kasprzyk
Imperial College London, UK
Fano manifolds, mirror symmetry, variations of Hodge structure, Landau{ Ginzburg models
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, 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
Realisation of cycles, permutahedron, small cover, Coxeter group, hyperbolic manifold, simplicial volume
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, France
Navier–Stokes equations, Cauchy problem
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
Science, statistical inference
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, United Kingdom
Linear Schrödinger equation, exponential splittings, free Lie algebras, Krylov subspace methods
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, USA
Non-archimedean groups, Fraïssé theory, Ramsey theory, ample genericity, automatic continuity, unique ergodicity, spatial realizations, unitary representations
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, France
Cluster algebra, cluster monomial, triangulated category
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
Complex Monge–Ampère operator, plurisubharmonic function, Kähler–Einstein metric, Kähler–Ricci flow
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
Linearly reinforced random walk, random walk in random environment, partial exchangeability, Pólya urn
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, France
Wave equation, characteristic point, blow-up set
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
Commuting differential operators
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, USA
Fractional Brownian motion. Itô's formula. Empirical quantiles
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, Israel
Reconstruction of signals, Paley–Wiener spaces, complete systems and frames, cyclic vectors
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, UK
Periodic and almost-periodic operators, integrated density of states, Bethe–Sommerfeld Conjecture
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 Zürich, Switzerland
Many body quantum evolution, mean field limit, Hartree dynamics
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, Poland
representations of symmetric groups, Young diagrams, asymptotic representation theory, free cumulants, Kerov polynomials
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, USA
Vladimír
Šverák
University of Minnesota, Minneapolis, USA
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, Canada
Ramsey spaces, Unconditional Sequences
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
Columbia University, New York, United States
Minimal surfaces, constant mean curvature surfaces, Lawson conjecture, Schwarzschild manifold
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
ETH Zürich, Switzerland
Isoperimetric inequalities, Gagliardo–Nirenberg–Sobolev inequalities, Log-HLS inequality, stability, long-time asymptotic, Keller–Segel equation
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, USA
Von Neumann algebra, II1 factor, measure preserving action, group measure space construction, Cartan subalgebra, Bernoulli action, W*-superrigidity
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, France
Nonlinear analysis, variational methods, quantum mechanics, Dirac operator, renormalization, quantum electrodynamics, vacuum polarization
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, USA
Heegaard Floer homology, knot, 3-manifold, 4-manifold, grid diagram
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, France
Random maps, random trees, Brownian map, stable maps, $O(n)$ model
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, UK
Freiman's theorem, sumsets
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, UK
Parabolic flows, shearing, mixing, time-changes, horocycle flow, area preserving flow, locally Hamiltonian flows, Heisenberg nilflows
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), France
Optimal control, Pontryagin Maximum Principle, conjugate points, numerical methods, shooting method, orbit transfer, continuation method, dynamical systems
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, Netherlands
Islamic mathematics, tilings, pentagon, heptagon
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, 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, UK
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, Poland
Maciej
Wojtkowski
University of Warmia and Mazury in Olsztyn, Poland
Toral automorphisms, Markov partitions, Berg partitions, Sturmian sequences, tilings, substitution
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
Analytic geometry
Algebraic number theory
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
2523-515X
2523-5168
Nef divisors on $\M_{0,n}$ from GIT
Valery
Alexeev
University of Georgia, Athens, United States
David
Swinarski
Fordham University, Bronx, USA
Moduli spaces of curves, Geometric Invariant Theory, GIT
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, Germany
Fabrizio
Catanese
Universität Bayreuth, Germany
Moduli of surfaces, surfaces with pg = 0, group actions, topological methods
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, France
Rationality questions, unirational varieties, Cremona group
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, Italy
Flaminio
Flamini
Università di Roma Tor Vergata, Italy
Brill–Noether theory of vector bundles, Hilbert schemes of scrolls, moduli
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, Spain
Remke
Kloosterman
Humboldt-Universität zu Berlin, Germany
Zariski pairs and triples, Alexander polynomial, cuspidal curves, Mordell–Weil groups
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, France
Bas
Edixhoven
Universiteit Leiden, Netherlands
Galois representations, modular curves, Ramanujan tau-function, inverse Jacobi problem
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
Generic syzygy varieties, vector bundles, curves of genus 7, Green’s conjecture
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, USA
Torelli map, nef cone, moduli space of curves
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, USA
Singular moduli, generalized Jacobian, Heegner points
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
Universität Bonn, Germany
Shimura varieties, L-functions, motives
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, Germany
Coherent sheaves on a curve, moduli stack, étale cohomology
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, USA
Kristin
Lauter
Theory Group, Redmond, USA
Curve, rational point, zeta function, Weil bound, Serre bound, Oesterlé bound, Birch and Swinnerton-Dyer conjecture
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
Wild quotient singularities, surfaces of general type, Artin–Schreier coverings
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, Japan
Supersingular K3 surface, Artin invariant, characteristic 2, Néron–Severi group, generalized Kummer surface
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, USA
Du Bois singularities
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, Germany
Peter
Newstead
University of Liverpool, UK
Algebraic curve, stable vector bundle, Clifford index, K3 surface
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, Switzerland
A.
Pixton
Princeton University, USA
Stable pairs, local curves, stationary descendents, Hilbert scheme of points, equivariant Chow ring
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
Modular operads, graphical enumeration, tensor species
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, Italy
Group schemes, modular curves, algebraic number fields
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
Collège de France, Paris, France
Unramified cohomology, Griffiths group, Deligne cycle class
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, USA
Étale cohomology, Poincaré duality, perfect pairing
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
Algebraic geometry
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
2523-5133
2523-5141
20
Optical caustics and their modelling as singularities
Alain
Joets
Université Paris Sud-XI, Orsay, France
Caustics, light focalization, caustic points, singularity theory
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, Germany
Equisingularity
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, Japan
Akiyoshi
Sannai
Nagoya University, Japan
Kei-ichi
Watanabe
College of Humanities and Sciences, Tokyo, Japan
Singularities, Jet schemes, $F$-regular singularities
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
General relativity, singularity theory, partial differential equations
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
Riemann surface, degenerating family, stable reduction, Teichmüller space, moduli space, orbifold, compactification
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
$b$-function, algebraic local cohomology, semiquasihomogeneous isolated singularities
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
Equivariant Chern class, MacPherson transformation, algebraic stacks
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, Japan
Mixed weighted homogeneous, polar action, degree
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, Japan
Splice quotient singularity, geometric genus, multiplicity, embedding dimension
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, Germany
Cyclic quotient surface singularity, affine toric variety, (versal) deformation of surface singularities, Artin component, monodromy covering
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, Germany
Characteristic classes, hypersurfaces, singularities, Milnor fiber, nearby cycles, vanishing cycles, Hodge theory, mixed Hodge modules, motivic Grothendieck group, intersection homology, knot theory
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
Singular distributions, localization of characteristic classes, Chern residues, Atiyah residues, Riemann–Roch theorem for embeddings
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
Birational invariant, statistical learning theory, singularity
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, Japan
Frobenius morphism, noncommutative resolution, alteration
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, Japan
Fulton–MacPherson’s bivariant theory, motivic Hirzebruch class, relative Grothendieck group, zeta function
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, Japan
Hyperplane arrangements, minimality, local system cohomology
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
Complex analysis
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
2523-5133
2523-5141
19
Introduction to Teichmüller theory, old and new, IV
Athanase
Papadopoulos
Université de Strasbourg, France
Teichmüller space, moduli space, Weil–Petersson metric, Weil–Petersson Funk metric, Thurston metric, operad, infinite-dimensional Teichmüller space, holomorphic dynamics, holomorphic family, Riemann surface, Teichmüller curve, universal curve, affine structure, deformation space, Teichmüller theory, Toledo invariant, curve complex, arc complex, Tits building, compactification, Thurston boundary, horofunction boundary, Bers boundary, reduced Bers boundary, Lipschitz algebra, quasiconformal mapp
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
Teichmüller space, Weil–Petersson metric, harmonic maps, deformation theory, convex geometry, CAT(0) geometry
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, Switzerland
Teichmüller spaces, simple closed geodesics
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, USA
Tits building, curve complex, spherical building, Euclidean building, hyperbolic building, arithmetic group, mapping class group, symmetric space, Teichmüller space, proper action, classifying space, universal space, locally symmetric space, Mostow strong rigidity, moduli space, ending lamination conjecture, simplicial volume, compactification, boundary, duality group, cohomological dimension, asymptotic cone, Novikov conjecture, quasi-isometry, Heegaard splitting
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, Japan
Teichmüller space, extremal length, flat structure, geodesic currents
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, Japan
Teichmüller space, compactification, mapping class group
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
Foliations, moduli spaces, operads, Sullivan PROP, string topology, string field theory, Deligne's conjecture, cyclic Deligne conjecture, cacti, stabilization, semi-simple algebras, $E_k$ operads, $E_{\infty}$ operad, little cubes
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, France
Horoboundary, Teichmüller space, Lipschitz metric, stretch metric, isometries
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
Teichmüller space, Teichmüller metric, horofunction, compactification
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, Japan
Teichmüller space, Lipschitz algebra, Banach algebra, Gardiner–Masur compactification, extremal length
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
Teichmüller theory, quasi-conformal mappings, extremal problem of quasi-conformal mappings, geodesic geometry of Busemann, Finsler geometry
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, Japan
Teichmüller space, mapping class group
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, Germany
Flat affine structure, locally homogeneous structure, surface, two-torus, development map, deformation space, moduli space, holonomy map, stratification
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 Zürich, Switzerland
Alessandra
Iozzi
ETH Zentrum, Zürich, Switzerland
Anna
Wienhard
Universität Heidelberg, Germany
Hermitian symmetric spaces, bounded Kähler class, bounded Euler class, Fricke–Klein space, Milnor–Wood inequality, maximal representation, bounded cohomology, Shilov boundary, totally geodesic embedding, tight homomorphism, tube type domain, hyperbolic surface
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
Quasiconformal, quasiregular, nonlinear analysis, conformal geometry, conformal invariant, moduli, geometric function theory
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
Riemann surfaces of infinite topological type, quasiconformal mapping class groups, Teichmüller modular groups, limit set, region of discontinuity, moduli spaces, asymptotic Teichmüller spaces
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, France
Guizhen
Cui
Chinese Academy of Sciences, Beijing, China
Lei
Tan
Université d'Angers, France
Thurston theorem, postcritically finite map, moduli space of rational maps, Teichmüller space of rational maps, Julia set, holomorphic dynamics
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, USA
Shear coordinate, quantum Teichmüller space, Kashaev coordinates, generalized Kashaev algebra
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
Complex structure, Teichmüller space, moduli, Teichmüller curve, universal curve, families of Riemann surfaces
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, Switzerland
Norbert
A’Campo
Universität Basel, Switzerland
Lizhen
Ji
University of Michigan, Ann Arbor, USA
Athanase
Papadopoulos
Université de Strasbourg, France
Teichmüller space, complex structure, period map, universal Teichmüller curve, Grothendieck–Teichmüller theory, Teichmüller functor
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
Numerical analysis
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 geometry
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
2523-515X
2523-5168
Categorical representability and intermediate Jacobians of Fano threefolds
Marcello
Bernardara
Université Paul Sabatier, Toulouse, France
Michele
Bolognesi
Université de Montpellier, France
Derived categories, semiorthogonal decompositions, intermediate Jacobian, algebraic cycles, Fano threefolds
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, Italy
Paolo
Stellari
Università degli Studi di Milano, Italy
Derived categories, Fourier–Mukai functors
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
The University of British Columbia, Vancouver, Canada
Flops, derived categories of coherent sheaves, derived equivalences, higher representation theory
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, Japan
Kazushi
Ueda
Osaka University, Japan
Dderived category, Fermat variety, dg category, orbit category
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
Loop space, Homology, Segal machine
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, France
Cluster algebra, quantum cluster algebra, derived category
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, Japan
AS-regular algebras, Fano algebras, McKay correspondence
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, USA
Abelian variety, derived category, Lagrangian-invariant sheaves, Lagrangian correspondence, Heisenberg groupoid
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
Northwestern University, Evanston, USA
Derived categories, perverse coherent sheaves, generic vanishing
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, USA
Derived category, derived equivalence, Calabi–Yau threefold, fundamental group, (1,8)-polarized abelian surface
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
Donaldson–Thomas invariants, derived category
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, Belgium
Deformation theory, abelian categories
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
Algebraic geometry
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
2523-515X
2523-5168
The influence of Oscar Zariski on algebraic geometry
Piotr
Blass
Ulam University, Boynton Beach, USA
Algebraic geometry, algebraic curves, algebraic surfaces, linear systems, desingularization, normalization, Zariski’s main theorem, history of mathematics
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, Germany
Nathan
Ilten
Bonn, Germany
Lars
Petersen
Freie Universität Berlin, Germany
Hendrik
Süß
Brandenburgische Technische Universität Cottbus, Germany
Robert
Vollmert
Freie Universität Berlin, Germany
Torus action, T-variety, polyhedral divisor
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, USA
Equivariant cohomology, localization, Grassmannian, Schubert variety, Schur function
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, Italy
Susan
Cooper
University of Nebraska, Lincoln, USA
Sandra
Di Rocco
Royal Institute of Technology, Stockholm, Sweden
Marcin
Dumnicki
Jagiellonian University, Krakow, Poland
Brian
Harbourne
University of Nebraska, Lincoln, USA
Anders
Lindquist
Royal Institute of Technology, Stockholm, Sweden
Hans
Munthe-Kaas
University of Bergen, Norway
Alex
Küronya
Budapest University of Technology & Economics, 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, Krakow, Poland
Zach
Teitler
Boise State University, USA
Linear series, divisors, curves, positivity, Seshadri constants, symbolic powers, multiplier ideals
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, UK
Singularities, equivariant localization, multidegree, hyperbolicity
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, UK
Vladimir
Lazić
Universität Bayreuth, Germany
Birational geometry, Minimal Model Program, canonical ring
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, Krakww, Poland
Sławomir
Rams
Jagiellonian University, Krakow, Poland
Hodge numbers, logarithmic differential forms, defect
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, Germany
Friedrich Prym, Friedrich Schottky, Prym variety, Schottky–Jung relations, moduli space, syzygy, Prym–Green conjecture, Nikulin surface, canonical singularity
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, Italy
Inna
Scherbak
Tel Aviv University, Israel
Wroński determinant, Grassmann bundle, linear system, linear ODE, ramification locus, Schubert Calculus
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, Ireland
Masha
Vlasenko
Trinity College, Dublin, Ireland
Bloch group, Grassmannian complexes
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, Germany
Frederik
Witt
Universität Münster, Germany
Toric stacks, symplectic quotients
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, Germany
Stefan
Kebekus
Universität Freiburg, Germany
Deformation of morphism, complex-symplectic manifold
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, Krakow, Poland
Toric degenerations, adjoint varieties, K3 surfaces, Calabi–Yau threefolds, geometric transitions
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, Krakow, Poland
Differential forms, reflexive differentials, log canonical
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, Japan
Enriques surface, K3 surface, Mathieu group, period
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
Log canonical threshold, F-pure threshold, graded sequence of ideals
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, Poland
Thom polynomial, singularity class, singularity, global singularity theory, cotangent map, degeneracy locus, $\mathcal{P}$-ideal, Schur function, resultant, recursion, Pascal staircase
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
Field extension, norm
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, Krakow, Poland
Seshadri constant, toric manifold, symplectic manifold, symplectic packing
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
History of mathematics
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
2523-5214
2523-5222
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
Geometry
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
Complex analysis
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
Combinatorics + graph theory
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
2523-5176
2523-5184
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
Probability + statistics
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
Algebra
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
Functional analysis
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
2523-5176
2523-5184
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
Differential equations
Differential + Riemannian geometry
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
2523-5192
2523-5206
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
Geometry
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
2523-5133
2523-5141
18
Notes on non-Euclidean geometry
Norbert
A’Campo
Universität Basel, Switzerland
Athanase
Papadopoulos
Université de Strasbourg, France
Hyperbolic geometry, neutral geometry, absolute geometry, spherical geometry, elliptic geometry, Euclid, axiom, parallel postulate, trirectangular quadrilateral, Khayyam–Saccheri quadrilateral, angular deficit, area, dissection, hyperbolic trigonometry, parallelism, parabolic transformations, Lobachevsky angle of parallelism function, model, transitional geometry
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, France
Geodesics, horocycles, hyperbolic geometry, diophantine approximations
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, Germany
Translation surface, origami, Teichmüller curve, Veech group, moduli space
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, Russian Federation
Sergey
Matveev
Chelyabinsk State University, Russian Federation
3-manifold, TV-invariants, special spine, JSJ-decomposition, normal surface, algorithmic classification
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, Germany
Symmetric space, higher rank Lie group, geometric boundary
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
Representation of groups, Chern–Simons theory, geometric quantization
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, Italy
Hyperbolic geometry, 3-manifold, knot, link, graph
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, Switzerland
Gromov hyperbolic space, large scale geometry
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
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
2523-5214
2523-5222
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
Complex analysis
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
2523-5133
2523-5141
17
Introduction to Teichmüller theory, old and new, III
Athanase
Papadopoulos
Université de Strasbourg, France
Teichmülller theory, moduli spaces, mapping class groups, 3-manifolds
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, France
Beltrami equation, uniformization, quasiconformal map
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, USA
Finite earthquakes, earthquakes, earthquake measures, quasisymmetric and symmetric homeomorphisms, earthquake curves, infinitesimal earthquakes, cross-ratio distortions and Zygmund bounded functions
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, UK
Line of minima, Teichmüller space, geodesic lamination, earthquake, Teichmüller geodesic, quasifuchsian group
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, USA
Arithmetic groups, mapping class groups, symmetric spaces, Teichmüller spaces, Lie groups, transformation groups, proper actions, classifying spaces, locally symmetric spaces, moduli spaces, Riemann surfaces, reduction theories, fundamental domains, compactifications, boundaries, universal spaces, duality groups, curve complexes, Tits buildings, hyperbolic groups, Coxeter groups, outer aut