Jahrestagung 2011 Köln

Vom 19. bis zum 22. September

- Holger Dette, Bochum
- Irene Fonseca, Pittsburgh
- Bernhard Keller, Paris
- Matthias Kreck, HMI Bonn (Cantor-Medaille)
- Shrawan Kumar, Chapel Hill
- Ladislav Kvasz, Prag
- Christian Lubich, Tübingen
- Ken Ono, Atlanta
- Francisco Santos, Cantabria
- Simone Warzel, München (Noether-Vorlesung)

Montag, 19.9., 12:00 (Cantor-Medaille)

Although I have no idea of physics I will start with some basic ideas of classical mechanics and their quantum counterparts to motivate the concept of topological (or geometric or conformal) quantum field theories as suggested by Atiyah and Siegel. All this will be done very slowly. In the remaining time I want to report about some recent joint work with Hohnhold, Stolz and Teichner which gives a description of de Rahm cohomology in the language of quantum field theories. Stolz and Teichner have done something similar and much more complicated for K theory and conjecturally for elliptic cohomology. This sheds some light on some general principles of constructing certain cohomology theories and corresponding equivariant cohomology theories which will be addressed at the end.

Dienstag, 20.9., 9:00

The talk explains several relations between different areas of mathematics: Mathematical statistics, random matrices and special functions. We give a careful introduction in the theory of optimal designs, which are used to improve the accuracy of statistical inference without performing additional experiments. It is demonstrated that for certain regression models orthogonal polynomials play an important role in the construction of optimal designs. In the next step these results are connected with some classical facts from random matrix theory. In the third part of this talk we discusss some new results on special functions and random matrices. In particular we analyse random band matrices, which generalize the classical Gaußschen ensemble. We show that the random eigenvalues of such matrices behave similarly as the deterministic roots of matrix orthogonal polynomials with varying recurrence coefficients. We study the asymptotic zero distribution of such polynomials and demonstrate that these results can be used to find the asymptotic proporties of the spectrum of random band matrices.

Dienstag, 20.9., 10:20

Bei der Behandlung des Zusammenhanges von Geometrie und Malerei beschränkt sich die Fachliteratur häufig auf die Zeit der Renaissance; manchmal wird noch der Kubismus erwähnt. In dem Vortrag wird versucht, eine systematische Parallele zwischen diesen beiden Gebieten auszuarbeiten. Anlehnend an Wittgensteins Abbildtheorie der Bedeutung kann man die Entwicklung der Geometrie rekonstruieren. Wenn wir die Abbildtheorie auf Bilder anwenden, die in geometrischen Texten vorkommen und wenn wir die so gewonnene Theorie mit der Geschichte der Malerei in Beziehung bringen, zeigen sich uns eine Reihe interessanter Berührungspunkte. Bei den bekannten Bildern der Frührenaissance (Giotto, Lorenzetti, Massaccio), der Hochrenaissance (Alberti, Leonardo, Uccello), und der Spätrenaissance (Dürer, Holbein), des Manierismus (El Greco), des Barocks (Rembrandt, Velázquez, Pozzo), des Impressionismus (Renoir, Manet), des Postimpressionismus (Seurat, Cézanne), des Kubismus (Picasso, Braque) und der abstrakten Malerei (Kandinski) wird ihre geometrische Struktur untersucht. Es wird versucht, die Entwicklung der synthetischen Geometrie mit diesen Bildern in Beziehung zu bringen. Konkreter zeigen sich Verbindungen von projektiver Geometrie mit Dürer, nichteuklidischer Geometrie mit Velazquez und Pozzo, dem Erlanger Programm mit Manet und Seurat und der kombinatorischen Topologie mit Cézanne, Braque und Picasso.

Dienstag, 20.9., 11:40

One easily sees that

and so we say that there are 5 partitions of 4. The stuff of partitions seems like mere child's play. Professor Ono will explain how the simple task of adding and counting has fascinated many of the world's leading mathematicians: Euler, Ramanujan, Hardy, Rademacher, to name just a few. As is typical in number theory, many of the most fundamental (and simple to state) questions have remained unsolved. In 2010, Ono, with the support of the American Institute for Mathematics and the National Science Foundation, assembled an international team of distinguished researchers to attack some of these problems. He will announce their findings: new theories which solve some of these famous old questions.

Mittwoch, 21.9., 9:00 (Noether-Vorlesung)

Motivated by the quest for a theory of quantum transport in disordered media, in 1958 the physicist P.W. Anderson came up with a model for a quantum particle in a random energy landscape. Among its interesting features is a conjectured sharp transition from a regime of localized eigenstates to one of diffusive transport. Until today it remains a mathematical challenge to establish these features in the framework of random Schrödinger operators. In this talk, I will give an introduction into the mathematical results and tools in the field, which combines spectral analysis with probability theory. I will also describe some recently discovered surprising effects of disorder on the spectra of Schr"odinger operators in a tree-graph geometry.

Mittwoch, 21.9., 10:20

Several questions in applied analysis motivated by issues in computer vision, physics, materials sciences and other areas of engineering may be treated variationally leading to higher order problems and to models involving lower dimension density measures. Their study often requires state-of-the-art techniques, new ideas, and the introduction of innovative tools in partial differential equations, geometric measure theory, and the calculus of variations. In this talk it will be shown how some of these questions may be reduced to well understood first order problems, while in others the higher order terms play a fundamental role. Applications to phase transitions, to the equilibrium of foams under the action of surfactants, imaging, micromagnetics, thin films, and quantum dots will be addressed.

Mittwoch, 21.9., 11:40

The talk relates different areas of mathematics: differential equations, matrix manifolds and numerical linear algebra. It requires no deep understanding of any of these, but combines them in a surprising way. When studying the ε-pseudospectrum of a matrix, one is often interested in computing the extremal points having maximum real part or modulus. This is a crucial step, for example, when computing the distance to instability of a stable system. Using the key property that the pseudospectrum is determined via perturbations by rank-1 matrices, we derive two different continuous dynamical systems leading to the critical rank-1 perturbations associated with the extremal points of (locally) maximum real part and modulus. This approach also allows us to track the boundary contour of the pseudospectrum in a neighbourhood of the extremal points. The technique we propose is related to an idea recently developed by Guglielmi and Overton, who derived discrete dynamical systems instead of the continuous ones we present. The method appears promising in dealing with large-size, sparse problems. The talk is based on joint work with Nicola Guglielmi.

Donnerstag, 22.9., 9:00

The classical Hermitian eigenvalue problem addresses the following question:
What are the possible eigenvalues of the sum A+B of two Hermitian matrices A and B,
provided we fix the eigenvalues of A and B. A systematic study of this problem was
initiated by H. Weyl (1912). By virtue of contributions from a long list of mathematicians,
notably Weyl (1912), Horn (1962), Klyachko (1998) and Knutson-Tao (1999), the problem
is finally settled. The solution asserts that the eigenvalues of A+B are given in terms of
certain system of linear inequalities in the eigenvalues of A and B. These inequalities can
be given explicitly and are related to a classical problem in Schubert calculus.

Belkale (2001) gave an optimal set of inequalities for the problem in this case.
The Hermitian eigenvalue problem has been extended by Berenstein-Sjamaar (2000) and
Kapovich-Leeb-Millson (2005) for any semisimple complex algebraic group. Their solution
is again in terms of a system of linear inequalities. Again these inequalities can
be given explicitly and they are related to a classical problem in Schubert calculus.
However, their solution is far from being optimal. In a joint work with P. Belkale, we define
a deformation of the cup product used for the Schubert calculus and use this new product
to generate our system of inequalities which solves the problem for any
semisimple complex algebraic group optimally (as shown by Ressayre).

Donnerstag, 22.9., 10:20

The Hirsch conjecture, stated in 1957, said that if a polyhedron is defined by n linear inequalities in d variables then its combinatorial diameter should be at most n-d. That is, it should be possible to travel from any vertex to any other vertex in at most n-d steps (traversing an edge at each step). The unbounded case was disproved by Klee and Walkup in 1967. In this talk I describe my construction of the first counter-examples to the bounded case (polytopes). The conjecture was posed and is relevant in connection to linear programming since the simplex method, one of the mathematical algorithms with the greatest impact in science and engineering, solves linear programming problems by traversing the graph of the feasibility polyhedron. In the first half of the talk we will explain this connection.

Donnerstag, 22.9., 11:40

Sergey Fomin and Andrei Zelevinsky invented cluster algebras at the beginning of the last decade as a tool for studying dual canonical bases and total positivity in semisimple Lie groups. It soon turned out that the algebraic/combinatorial framework they created is also relevant in a large array of other subjects including Teichmüller theory, Poisson geometry, quiver representations and the study of Donaldson-Thomas invariants in algebraic geometry. In this talk, I will give a concise introduction to cluster algebras and sketch two significant applications: one in Lie theory and one in the study of certain discrete dynamical systems.