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2024-03-28 14:37:56
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Oberwolfach Reports
Oberwolfach Rep.
OWR
1660-8933
1660-8941
General
10.4171/OWR
http://www.ems-ph.org/doi/10.4171/OWR
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© Mathematisches Forschungsinstitut Oberwolfach
6
2009
1
Toric Geometry
Klaus
Altmann
Freie Universität Berlin, BERLIN, GERMANY
Victor
Batyrev
Universität Tübingen, TÜBINGEN, GERMANY
Yael
Karshon
University of Toronto, TORONTO, ONTARIO, CANADA
Toric Geometry originated from investigations of torus actions on geometric and algebraic objects. It is addressed through algebraic geometry, symplectic geometry, equivariant topology, as well as the theory of convex polyhedra within discrete mathematics. In spite of using their own language these completely different disciplines often observe similar or even identical combinatorial phenomena. Thus toric geometry leads to a fascinating and fruitful interplay between these disciplines.
Algebraic geometry
General
5
74
10.4171/OWR/2009/01
http://www.ems-ph.org/doi/10.4171/OWR/2009/01
Discrete Differential Geometry
Alexander
Bobenko
Technische Universität Berlin, BERLIN, GERMANY
Richard
Kenyon
Brown University, PROVIDENCE, UNITED STATES
John
Sullivan
Technische Universität Berlin, BERLIN, GERMANY
Günter
Ziegler
Freie Universität Berlin, BERLIN, GERMANY
This is the collection of extended abstracts for the 26 lectures and the open problems session at the second Oberwolfach workshop on Discrete Differential Geometry.
Differential geometry
General
75
144
10.4171/OWR/2009/02
http://www.ems-ph.org/doi/10.4171/OWR/2009/02
Random Trees
Ellen
Baake
Universität Bielefeld, BIELEFELD, GERMANY
Donald
Dawson
Carleton University, OTTAWA , ONT., CANADA
Andreas
Greven
Universität Erlangen-Nünberg, ERLANGEN, GERMANY
Frank
den Hollander
Universiteit Leiden, LEIDEN, NETHERLANDS
The meeting was devoted to random trees, a central concept in mathematics that provides a key way of thinking about relationships between objects such as particles in a fluid, individuals in a population, or labels in a search algorithm. Particular emphasis was put on the role of random trees in physics, biology, and computer science.
Combinatorics
Probability theory and stochastic processes
Statistical mechanics, structure of matter
Biology and other natural sciences
145
190
10.4171/OWR/2009/03
http://www.ems-ph.org/doi/10.4171/OWR/2009/03
Numerical Techniques for Optimization Problems with PDE Constraints
Matthias
Heinkenschloss
Rice University, HOUSTON, UNITED STATES
Ronald
Hoppe
Universität Augsburg, AUGSBURG, GERMANY
Volker
Schulz
Universität Trier, TRIER, GERMANY
The development, analysis and implementation of efficient and robust numerical techniques for optimization problems associated with partial differential equations (PDEs) is of utmost importance for the optimal control of processes and the optimal design of structures and systems in modern technology. The successful realization of such techniques invokes a wide variety of challenging mathematical tasks and thus requires the application of adequate methodologies from various mathematical disciplines. During recent years, significant progress has been made in PDE constrained optimization both concerning optimization in function space according to the paradigm ’Optimize first, then discretize’ and with regard to the fast and reliable solution of the large-scale problems that typically arise from discretizations of the optimality conditions. The contributions at this Oberwolfach workshop impressively reflected the progress made in the field. In particular, new insights have been gained in the analysis of optimal control problems for PDEs that have led to vastly improved numerical solution methods. Likewise, breakthroughs have been made in the optimal design of structures and systems, for instance, by the socalled ’all-at-once’ approach featuring simultaneous optimization and solution of the underlying PDEs. Finally, new methodologies have been developed for the design of innovative materials and the identification of parameters in multi-scale physical and physiological processes.
Partial differential equations
Calculus of variations and optimal control; optimization
Numerical analysis
Operations research, mathematical programming
191
284
10.4171/OWR/2009/04
http://www.ems-ph.org/doi/10.4171/OWR/2009/04
The Arithmetic of Fields
Moshe
Jarden
Tel Aviv University, TEL AVIV, ISRAEL
Florian
Pop
University of Pennsylvania, PHILADELPHIA, UNITED STATES
Leila
Schneps
Université Pierre et Marie Curie, PARIS CEDEX 05, FRANCE
The workshop “The Arithmetic of Fields” focused on a series of problems concerning the interplay between number theory, arithmetic and algebraic geometry, Galois theory, and model theory, such as: the Galois theory of function fields / covers of varieties, rational points on varieties, Galois cohomology, local-global principles, lifting/specializing covers of curves, model theory of finitely generated fields, etc.
Field theory and polynomials
Calculus of variations and optimal control; optimization
Numerical analysis
Operations research, mathematical programming
285
354
10.4171/OWR/2009/05
http://www.ems-ph.org/doi/10.4171/OWR/2009/05
Low Eigenvalues of Laplace and Schrödinger Operators
Mark
Ashbaugh
University of Missouri, COLUMBIA, UNITED STATES
Rafael
Benguria
Pontificia Universidad Católica de Chile, SANTIAGO DE CHILE, CHILE
Richard
Laugesen
University of Illinois at Urbana-Champaign, URBANA, UNITED STATES
Timo
Weidl
Universität Stuttgart, STUTTGART, GERMANY
This workshop brought together researchers interested in eigenvalue problems for Laplace and Schr¨dinger operators. The main topics o of discussions and investigations covered Dirichlet and Neumann eigenvalue problems, inequalities for the spectral gap, isoperimertic problems and sharp Lieb–Thirring type inequalities. The focus included not only the analytic and geometric sides of the problems, but also related probabilistic and computational aspects.
Partial differential equations
Operator theory
Quantum theory
General
355
428
10.4171/OWR/2009/06
http://www.ems-ph.org/doi/10.4171/OWR/2009/06
Wave Motion
Adrian
Constantin
King's College London, LONDON, UNITED KINGDOM
Joachim
Escher
University of Hannover, HANNOVER, GERMANY
Robin
Johnson
University of Newcastle, NEWCASTLE UPON TYNE, GREAT BRITAIN
Walter
Strauss
Brown University, PROVIDENCE, UNITED STATES
This workshop was devoted to recent progress in the mathematical study of water waves, with special emphasis on nonlinear phenomena. Both aspects related to the governing equations (free boundary Euler equations) as well as aspects related to various model equations were of interest.
Partial differential equations
Fluid mechanics
General
429
462
10.4171/OWR/2009/07
http://www.ems-ph.org/doi/10.4171/OWR/2009/07
Mini-Workshop: Category Theory and Related Fields: History and Prospects
Ralf
Krömer
Université de Nancy II, NANCY, FRANCE
Colin
McLarty
Case Western Reserve University, CLEVELAND, UNITED STATES
Michael
Wright
Science and Philosophy, FOUGERES, FRANCE
he workshop concerned various topics in the history of category theory and related fields, paying attention to some extent also to open questions, present and possible future development.
History and biography
Algebraic geometry
Category theory; homological algebra
$K$-theory
463
492
10.4171/OWR/2009/08
http://www.ems-ph.org/doi/10.4171/OWR/2009/08
Mini-Workshop: Product Systems and Independence in Quantum Dynamics
B.V. Rajarama
Bhat
Indian Statistical Institute, BANGALORE, INDIA
Uwe
Franz
Université de Franche-Comté, BESANCON CEDEX, FRANCE
Michael
Skeide
Universita degli Studi del Molise, CAMPOBASSO, ITALY
Quantum dynamics, both reversible (i.e., closed quantum systems) and irreversible (i.e., open quantum systems), gives rise to product systems of Hilbert spaces or, more generally, of Hilbert modules. When we consider reversible dynamics that dilates an irreversible dynamics, then the product system of the latter is equal to the product system of the former (or is contained in a unique way). Whenever the dynamics is on a proper subalgebra of the algebra of all bounded operators on a Hilbert space, in particular, when the open system is classical (commutative) it is indispensable that we use Hilbert modules. The product system of a reversible dynamics is intimately related to a filtration of subalgebras that are independent in a state or conditionally independent in a conditional expectation of the reversible system. This has been illustrated in many concrete dilations that have been obtained with the help of quantum stochastic calculus. Here the underlying Fock space or module determines the sort of quantum independence underlying the reversible system. The mini-workshop brought together experts from quantum dynamics, product systems and quantum independence who have contributed to the general theory or who have studied intriguing examples. As the implications of the tight relationship between product systems and independence had so far been largely neglected, we expect from our mini-workshop a strong innovative impulse to this field.
Functional analysis
Associative rings and algebras
Probability theory and stochastic processes
Quantum theory
493
548
10.4171/OWR/2009/09
http://www.ems-ph.org/doi/10.4171/OWR/2009/09
Mini-Workshop: Support Varieties
Karin
Erdmann
Oxford University, OXFORD, UNITED KINGDOM
Henning
Krause
Universität Bielefeld, BIELEFELD, GERMANY
The notion of support is a fundamental concept which provides a geometric approach for studying various algebraic structures. The prototype for this has been Quillen’s description of the algebraic variety corresponding to the cohomology ring of a finite group, based on which Carlson introduced support varieties for modular representations. This has made it possible to apply methods of algebraic geometry to obtain representation theoretic information. Their work has inspired the development of analogous theories in various contexts, notably modules over commutative complete intersection rings, and over cocommutative Hopf algebras. The aim of this workshop has been to bring together experts from these fields and to stimulate interaction and exchange of ideas.
Associative rings and algebras
Commutative rings and algebras
Algebraic geometry
Category theory; homological algebra
549
592
10.4171/OWR/2009/10
http://www.ems-ph.org/doi/10.4171/OWR/2009/10
Control Theory: On the Way to New Application Fields
Frank
Allgöwer
Universität Stuttgart, STUTTGART, GERMANY
Uwe
Helmke
Universität Würzburg, WÜRZBURG, GERMANY
Eduardo
Sontag
Rutgers University, PISCATAWAY, UNITED STATES
Control theory is an interdisciplinary field that is located at the crossroads of pure and applied mathematics with systems engineering and the sciences. Recently, deep interactions are emerging with new application areas, such as systems biology, quantum control and information technology. In order to address the new challenges posed by the new application disciplines, a special focus of this workshop has been on the interaction between control theory and mathematical systems biology. To complement these more biology oriented focus, a series of lectures in this workshop was devoted to the control of networks of systems, fundamentals of nonlinear control systems, model reduction and identification, algorithmic aspects in control, as well as open problems in control.
Systems theory; control
General
593
680
10.4171/OWR/2009/11
http://www.ems-ph.org/doi/10.4171/OWR/2009/11
Mini-Workshop: Numerical Upscaling for Flow Problems: Theory and Applications
Achi
Brandt
Weizmann Institute of Science, REHOVOT, ISRAEL
Yalchin
Efendiev
Texas A&M University, COLLEGE STATION, UNITED STATES
Oleg
Iliev
ITWM, KAISERSLAUTERN, GERMANY
The objective of this workshop was to bring together researchers working in multiscale simulations with emphasis on multigrid methods and multiscale finite element methods, aiming at chieving of better understanding and synergy between these methods. The goal of multiscale finite element methods, as upscaling methods, is to compute coarse scale solutions of the underlying equations as accurately as possible. On the other hand, multigrid methods attempt to solve fine-scale equations rapidly using a hierarchy of coarse spaces. Multigrid methods need “good” coarse scale spaces for their efficiency. The discussions of this workshop partly focused on approximation properties of coarse scale spaces and multigrid convergence. Some other presentations were on upscaling, domain decomposition methods and nonlinear multiscale methods. Some researchers discussed applications of these methods to reservoir simulations, as well as to simulations of filtration, insulating materials, and turbulence.
Numerical analysis
General
681
724
10.4171/OWR/2009/12
http://www.ems-ph.org/doi/10.4171/OWR/2009/12
Mini-Workshop: The Pisot Conjecture — From Substitution Dynamical Systems to Rauzy Fractals and Meyer Sets
Valerie
Berthe
Université de Montpellier II, MONTPELLIER CEDEX 5, FRANCE
David
Damanik
Rice University, HOUSTON, UNITED STATES
Daniel
Lenz
Friedrich-Schiller-Universität Jena, JENA, GERMANY
This mini-workshop brought together researchers with diverse backgrounds and a common interest in facets of the Pisot conjecture, which relates certain properties of a substitution to dynamical properties of the associated subshift.
Dynamical systems and ergodic theory
Measure and integration
Convex and discrete geometry
General
725
766
10.4171/OWR/2009/13
http://www.ems-ph.org/doi/10.4171/OWR/2009/13
Mini-Workshop: Non-Negativity is a Quantum Phenomenon
Stephane
Launois
University of Kent, CANTERBURY, KENT, UNITED KINGDOM
Thomas
Lenagan
University of Edinburgh, EDINBURGH, UNITED KINGDOM
In recent publications, the same combinatorial description has arisen for three separate objects of interest: non-negative cells in the real grassmannian (Postnikov, Williams); torus orbits of symplectic leaves in the classical grassmannian (Brown, Goodearl and Yakimov); and, torus invariant prime ideals in the quantum grassmannian (Launois, Lenagan and Rigal). The aim of this meeting was to explore the reasons for this coincidence in matrices and the grassmannian in particular, and to explore similar ideas in more general settings.
Combinatorics
Algebraic geometry
Associative rings and algebras
Nonassociative rings and algebras
767
800
10.4171/OWR/2009/14
http://www.ems-ph.org/doi/10.4171/OWR/2009/14
Enveloping Algebras and Geometric Representation Theory
Shrawan
Kumar
University of North Carolina at Chapel Hill, CHAPEL HILL, UNITED STATES
Peter
Littelmann
Universität Köln, KÖLN, GERMANY
Wolfgang
Soergel
Universität Freiburg, FREIBURG, GERMANY
The meeting brought together experts investigating Lie theory from the geometric, algebraic and combinatorial points of view to discuss recent progress and bring forward the research in this area by fostering scientific interaction.
Nonassociative rings and algebras
Algebraic geometry
Group theory and generalizations
General
801
866
10.4171/OWR/2009/15
http://www.ems-ph.org/doi/10.4171/OWR/2009/15
Sparse Recovery Problems in High Dimensions: Statistical Inference and Learning Theory
Peter
Bartlett
University of California, BERKELEY, UNITED STATES
Vladimir
Koltchinskii
Georgia Institute of Technology, ATLANTA, UNITED STATES
Alexandre
Tsybakov
Timbre J340, MALAKOFF CEDEX, FRANCE
Sara
van de Geer
ETH Zentrum, ZÜRICH, SWITZERLAND
The statistical analysis of high dimensional data requires new techniques, extending results from nonparametric statistics, analysis, probability, approximation theory, and theoretical computer science. The main problem is how to unveil, (or to mimic performance of) sparse models for the data. Sparsity is generally meant in terms of the number of variables included, but may also be described in terms of smoothness, entropy, or geometric structures. A key objective is to adapt to unknown sparsity, yet keeping computational feasibility.
Statistics
General
867
916
10.4171/OWR/2009/16
http://www.ems-ph.org/doi/10.4171/OWR/2009/16
Representations of Finite Groups
Joseph
Chuang
The City University, LONDON, UNITED KINGDOM
Markus
Linckelmann
City University, LONDON, UNITED KINGDOM
Gunter
Malle
Universität Kaiserslautern, KAISERSLAUTERN, GERMANY
Jeremy
Rickard
University of Bristol, BRISTOL, UNITED KINGDOM
The workshop Representations of Finite Groups was organised by Joseph Chuang (London), Markus Linckelmann (Aberdeen), Gunter Malle (Kaiserslautern) and Jeremy Rickard (Bristol). It covered a wide variety of aspects of the representation theory of finite groups and related objects like Hecke algebras. A particular focus was placed on the rapidly evolving area of fusion systems.
Group theory and generalizations
General
917
970
10.4171/OWR/2009/17
http://www.ems-ph.org/doi/10.4171/OWR/2009/17