Math Colloquium

Spring 2007

Math colloquium meets on Mondays at 12:15 in Schreiber 006, Tel Aviv University.

Previous talks: 2002-2003 , 2003-2004 , 2004-2005 , 2005-2006 , 2006-2007

05.03.2007, 12:15
Ofer Zeitouni, Technion, Haifa
Probabilistic techniques for studying the asymptotics of spherical integrals.

ABSTACT: The spherical integrals in the title involve the integration, over the unitary/orthogonal groups, of the exponent of trace(UAU^*B), where A,B are (fixed) diagonal matrices and U is Haar distributed. These integrals, and their logarithmic asymptotics, appear in matrix models, in the enumeration of maps, and in communication theory. When the integration is over the unitary group, the integral can be reduced to a determinant by a formula of Harish-Chandra (and Itzykson-Zuber). I will describe joint work with Alice Guionnet, in which probabilisitic techniques were applied to yield expressions for the logarithmic asymptotics of these integrals.

19.03.2007, 12:15
Daniel Sternheimer, Universit\'e de Bourgogne Dijon, France and Keio University Yokogama, Japan
Quantization is deformation

ABSTACT: After an epistemological introduction we look at quantization as a typical example of Flato's deformation philosophy. Then we briefly explain the mathematical and physical background and survey some main points in deformation quantization and in the many developments it has triggered, which include Kontsevich's formality, noncommutative geometry, quantum groups and quantized spaces. We end by a presenting an ongoing program according to which a constant creation of matter could come from "quantized anti de Sitter black holes" at the edge of our universe.

26.03.2007, 12:15
Amit Singer, Yale University, USA; amit.singer@yale.edu
Random Computed Tomography and Protein Structuring

ABSTACT: In this talk we present a reconstruction algorithm for the 3D atomic structure of randomly oriented proteins using modified graph laplacians. The 2003 Nobel Prize in Chemistry was co-awarded to R. MacKinnon who was the first to structure a protein channel (the potassium channel) in 1998 by crystallizing the protein and using the classical X-ray computed tomography (CT). However, most membrane proteins cannot be crystallized and the classical CT cannot be used. In our experimental setup, the data consists of thousands of real noisy 2D projections (electron microscope images) of the protein given in random unknown directions, because the proteins are randomly oriented rather than being aligned as in MacKinnon's setup. Still, we show that the reconstruction is made possible by a certain modification of the images' graph laplacian coupled with the 3D Fourier slice theorem. The reconstruction is a particular case of a more general spectral non-linear independent component analysis (ICA) algorithm that combines local principal component analysis (PCA) with the graph laplacian. No prior knowledge of graph laplacians, tomography and proteins is needed for this talk. This is a joint work with Ronald Coifman, Yoel Shkolnisky and Fred Sigworth (Yale University).

30.04.2007, 12:15
Jon Keating , University of Bristol, UK
Eigenfunctions in open quantum chaotic systems

ABSTACT: When the Laplacian acts on a bounded domain, with boundary conditions that ensure self-adjointness, it has a discrete spectrum. Nearly 100 years ago, Weyl determined the leading order asymptotics of the eigenvalue counting function, which depends on the dimension and the volume of the domain. We also now know a good deal about the corresponding eigenfunctions, through work over the past 30 years inspired by results due to Schnirelman, Zelditch and Colin de Verdiere. Remarkably, when holes are made in the boundary of the domain, destroying self-adjointness, we still know almost nothing about the corresponding eigenvalue problem. I will describe some remarkable recent conjectures concerning the analogue of Weyl's asymptotic counting formula and the morphology of the corresponding eigenfunctions. I will also describe one model system in which these conjectures have recently been proved.

!The lecture will be given May 9, 16.00-17.30

Dan David building, TAU, room 106

Sackler Distinguished Lecture

Toshiyuki Kobayashi, University of Tokyo

Branching Problems of Unitary Representations

ABSTACT: Branching problems ask how an irreducible representation of a group decomposes when restricted to a subgroup. In these Lectures, I plan to give a survey on new aspects on branching problems of unitary representations of reductive Lie groups. The first half will be based on representation theoretic viewpoints. Following an observation of wild feature of branching problems with respect to non-compact subgroups in a general setting, I will introduce the notion of admissible restrictions as a nice framework that enjoys two properties: finiteness of multiplicities and discreteness of spectrum. A criterion for finite multiplicity and discretely decomposable restrictions will be presented. The idea of our proof stems from microlocal analysis and algebraic geometry. Furthermore, an exclusive law of discrete spectrum is formalized for inductions and restrictions. The second half deals with applications. Admissible restrictions may give tools for the study of representations for bigger groups via restrictions, and for the construction of irreducibles of smaller groups via branching laws. Once we know that restrictions are discretely decomposable, we may employ an algebraic approach to branching problems. In this direction, new branching formulas have been recently obtained in various settings, of which I plan to illustrate by examples. Finally, I will discuss some applications of discretely decomposable branching laws to other areas of mathematics. The topics include (1) topological properties of modular varieties in locally symmetric spaces, (2) a construction of new discrete series representations for non-Riemannian non-symmetric homogeneous spaces. If time allows, I may mention also a mystery between tessellation of (and discontinuous groups for) non-Riemannian homogeneous spaces and branching problems of unitary representations.

The lecture will be accessible for a broad audience.

Special lecture
14.05.2007, 16:00
Stephen Smale, University of California, Berkeley, 2007 Wolf Prize Laureate
On the Mathematics of Vision

Place: Weizmann Institute, Ebner Auditorium

A reception in honor of Prof. Smale will be held after the lecture

21.05.2007, 12:15
David Harbater, University of Pennsylvania, USA 1995 Cole Prize Laureate
Patching and Galois theory

ABSTACT: Abstract: Many questions in Galois theory can be interpreted geometrically, in terms of fundamental groups and branched covers. Using patching methods and related approaches, the speaker and others have obtained results about such questions even in situations that are not "geometric" in the classical sense. For example, this point of view shows that every finite group is a Galois group over the field F(t) of rational functions over an algebraically closed field F, and -- more strongly -- that the absolute Galois group of the field F(t) is free. In characteristic p, fundamental groups are much larger than they are over the complex numbers; and finite covering groups of the characteristic p analog of Riemann surfaces have been classified by Raynaud and the speaker, using patching methods. This talk will discuss the analogy between Galois groups and covering spaces; how this analogy can be exploited; and how patching methods can be used to obtain results along these lines.

28.05.2007, 12:25
Amos Nevo, Technion
On matrices with prime entries, lattice points, and ergodic theorems

ABSTACT: I will discuss variants of the following two questions:

1) Are there infinitely many 2 x 2 integral matrices of fixed determinant all of whose elements are prime ?

2) If so, how many such matrices one can expect to find in ball of radius T in the space of 2 x 2 matrices ?

In the talk, I will explain an approach to this and more general problems, via non-Euclidean lattice point problems, ergodic theorems on Lie groups and sieve methods. The results are based on joint work with P. Sarnak and with A. Gorodnik.

4.06.2007, 12:15
Yehuda Pinchover, Haifa University
On positive solutions and Liouville type theorems for p-Laplacian-type equations

ABSTACT: We shall discuss relations between the existence and properties of positive solutions of p-Laplacian-type equations and functional-analytic properties of the corresponding energy functional. In particular, we study the existence of a ground state and a positive Green-type function for such quasilinear equations. As an application we shall present a Liouville comparison principle for such equations. This is a joint work with K. Tintarev and A. Tertikas.

11.06.2007, 12:15
Dan Haran, Tel Aviv University
Algebraic construction of absolute Galois groups

ABSTACT: For the last 10 years the method of Algebraic Patching --initiated in a collaboration with H. Voelklein and further developed in a collaboration with M. Jarden-- has been used to produce Galois groups over certain fields. We will describe the basic ideas of this method and discuss a recent development due to Elad Paran (TAU).

Organizer: Victor Palamodov , e-mail: palamodo AT post DOT tau.ac.il