Math Colloquium

Fall 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-fall , 2007-spring

29.10.2007, 12:15

Aner Shalev, The Hebrew University of Jerusalem.

Quantitative representation theory, Waring problems, and further applications.

ABSTRACT: Quantitative representation theory deals with counting irreducible representations of various (finite and infinite) groups and bounding the values of the associated characters. In recent years there is growing interest in this subject with diverse applications to other fields; these include random walks, commutator maps, algorithmic questions, and Waring type problems (establishing non-commutative analogues of the celebrated Waring problem in number theory).
I will present some new results in some of these fields, and outline the role of representation theory in the proofs.

5.11.2007, 12:15

Zeev Rudnick, Tel Aviv University.

The Work of Atle Selberg on prime numbers and on closed geodesics

ABSTRACT: Atle Selberg, which passed away this summer at the age of 90, was one of the world's greatest analytic number theorists. Among his many honors are the 1950 Fields medal and the Wolf prize in 1986. To pay tribute to his memory, I will explain some of his work on the Riemann zeta function, the theory of primes and on the surprisingly related theory of closed geodesics on Riemann surfaces.

12.11.2007, 12:15

Stephan Gelbart, Weizmann Institute of Science.

Atle Selberg and the theory of automorphic forms

ABSTRACT: Atle Selberg gave birth to the modern theory of automorphic forms. In particular, I have in mind Selberg's Eigenvalue Conjecture, Selberg's theory of Eisenstein series, and Selberg's Trace Formula.

I shall briefly discuss these theories, with emphasis on the subsequent coming of Langlands's Program.

19.11.2007, 12:15

Tamar Ziegler, Technion.

Polynomial progressions in primes

ABSTRACT: In 1975 Szemeredi proved that any subset of the integers of positive density contains arbitrarily long arithmetic progressions. A couple of years later Furstenberg gave an ergodic theoretic proof for Szemeredi's theorem. At around the same time, Furstenberg and Sarkozy independently proved that any subset of the integers of positive density contains a perfect square difference, namely elements x,y with x-y=n2 for some positive integer n.

In 1995, Bergelson and Leibman proved, using ergodic theoretic methods, a vast generalization of both Szemeredi's theorem and the Furstenberg-Sarkozy theorem, establishing the existence of arbitrarily long polynomial progression in subsets of the integers of positive density.

The ergodic theoretic methods are limited, to this day, to handling sets of positive density. However, in 2004 Green and Tao proved that the question of finding arithmetic progressions in some special subsets of the integers of zero density - for example the prime numbers - can be reduced to that of finding arithmetic progressions in subsets of positive density. In recent work with T. Tao we show that one can make a similar reduction for polynomial progressions, thus establishing, through the Bergelson-Leibman theorem, the existence of arbitrarily long polynomial progressions in the prime numbers.

26.11.2007, 13:00-14:00

Alex Lubotzky, The Hebrew University of Jerusalem.

From the Selberg eigenvalues theorem to combinatorics and geometry

ABSTRACT: A seminal theorem (and conjecture) of Selberg gives a bound on the eigenvalues of the Laplacian acting on functions over arithmetic surfaces. This theorem (and a related conjecture) led to a lot of mathematics. In the talk we will describe some topics in combinatorics (expanders, Ramanujan graphs etc.) and in geometry (Thurston conjecture on hyperbolic manifolds, Heegard splitting of 3-manifolds etc.) which were influenced by that theorem. Most of these applications come via property 'tau'- a weak form of Kazhdan property T. (All notions will be explained as needed).

3.12.2007, 12:15

Tsachik Gelander, The Hebrew University of Jerusalem.

Uniform Independence for groups of matrices.

ABSTRACT: In his celebrated 1972 paper, J.Tits proved a fundamental dichotomy for linear groups, known today as the Tits alternative. Jointly with E. Breuillard we established several results improving those of Tits. For instance we showed that for any finitely generated non-virtually solvable linear group G, there is a constant m=m(G) such that for any generating set S of G, one can find generators of a free group F2 in the S-ball of radius m around the identity. This has many applications, e.g.:
1. Growth:   (A) Eskin-Mozes-Oh theorem about uniform exponential growth, as well as some improvements, e.g.
                   (B) Uniformity of Cheeger constants - a uniform positive expansion constant.
2. Dynamic: (C) Non-amenable linear groups are uniformly non-amenable.
                   (D) Connes-Sullivan conjecture on amenable actions (originally proved by Zimmer).
3. Riemannian foliations: (E) Carrier conjecture about exponential vs. polynomial leaves growth.
I will also explain the topological version of Tits alternative whose connected (Archimedean) case follows from the uniform version (but proved earlier), and some applications to finite group theory.

10.12.2007, 12:15

Edriss S. Titi, Weizmann Institute of Science and Univeristy of California Irvine.

Global Regularity for Three-dimensional Navier-Stokes Equations and Other Relevant Geophysical Models

ABSTRACT: The basic problem faced in geophysical fluid dynamics is that a mathematical description based only on fundamental physical principles, the so-called "Primitive Equations", is often prohibitively expensive computationally, and hard to study analytically. In this talk I will survey the main obstacles in proving the global regularity for the three-dimensional Navier-Stokes equations and their geophysical counterparts. Even though the Primitive Equations look as if they are more difficult to study analytically than the three-dimensional Navier-Stokes equations I will show in this talk that they have a unique global (in time) regular solution for all initial data.

This is a joint work with Chongshen Cao.

17.12.2007: No colloquium

20.12.2007: (* Thursday ) 12:15 NOTE SPECIAL DAY *

Jake Solomon, Princeton University.

Open Gromov-Witten theory and the structure of real enumerative geometry.

ABSTRACT: I plan to illustrate the relationship between open Gromov-Witten theory and real enumerative geometry through examples in the real projective plane. A particularly illuminating example is the intersection theoretic interpretation of the number of real lines through two points in the projective plane. More generally, I will consider the problem of enumerating real rational plane curves of degree d through 3d-1 points introduced by Welschinger. Open Gromov-Witten theory explicitly relates the real enumerative problem with its classical complex analog, simultaneously solving both problems. The solution is most naturally expressed as a PDE very similar to the WDVV equation. Moreover, like the WDVV equation, the PDE of open Gromov-Witten theory holds for arbitrary target manifolds.

24.12.2007, 12:15

Uri Bader, Technion

Boundary Theory for Groups.

ABSTRACT: A fruitful idea in (infinite) group theory is to associate a "boundary" to a group, which reflects its "behavior at infinity". This idea is no orphan.
Mother idea: Regard a Group as a Geometric Object.
Father idea: Attach to a Geometric Object a Boundary.
In my talk I will survey this family of ideas, and try to convince that having boundaries is for the groups' own good (unless she/he is amenable), and convenient to group theorists too.

31.12.2007, 12:15

George Zaslavsky, N.Y.U.

Multidimentional Stochastic Webs and Pentagonal Symmetry

ABSTRACT: Diffusion along stochastic webs with symmetry or quasi-symmetry is much faster than along the Arnold web. The symmetry of the web can be used to generate structures of crystal or quasi-crystal type in space. This provides a new way to use the dynamical systems approach to study geometrical properties of quasi-crystal objects from one side, and transport properties in multi-dimensional systems from another side.
Author's pictures from Alhambra and Cordoba ornaments with quasi-crystal symmetry will be attached.

7.1.2008, 12:15

Hillel Furstenberg, The Hebrew University of Jerusalem.

The Dynamics of (3/2)n mod 1

ABSTRACT: A number of problems in diophantine approximation have been successfully attacked by displaying a dynamical system related to the problem and analyzing orbit closures. We'll discuss quickly some of the successes and move on to the open question of whether for n>0 the fractional part of (3/2)n can be made arbitrarily small. We'll describe a particular dynamical system which may hold the answer to this question, and raise a number of questions - interesting in their own right – that arise in this quest.

14.1.2008, 12:15

Simon Gindikin, Rutgers University.

Harmonic Analysis on Symmetric Spaces and Multidimensional Complex Analysis.

ABSTRACT: E.Cartan and H.Weyl developed two classical approaches to harmonic analysis on complex semisimple Lie groups and, more generally, on complex symmetric spaces - algebraic and analytic (transcendental). In the focus of the last one was "the unitary trick" - the reduction to maximal compact forms where it is possible to apply real analysis. It looks that a possibility to apply directly complex analysis was not considered. I will discuss this possibility which gives some new constructions and results: the integral Cauchy formula on complex groups and symmetric spaces, a complex version of Poisson's integral for spherical polynomials, etc. Another subject which I plan to discuss is an integral geometry on real symmetric spaces which operates with complex horospheres. The real horospherical transform of Gelfand-Graev has a kernel corresponding to discrete series. The appeal to complex horospheres corrects this collision.