Mathematics Colloquium, 2002-3

Spring 2003

 3.3.2003 Noga Alon, Tel-Aviv University

Modern Discrete Mathematics: Methods, Applications & Challenges.

 10.3.2003 Alex Lubotzky, Hebrew University

Title: Counting Congruence Subgroups.

                                     Abstract
                                    ********
        ``Subgroup growth'' deals with counting finite index
        subgroups of a group. This theory led to counting
        congruence subgroups in arithmetic groups. The latter
        counting is a kind of ``non-commutative analytic number
        theory'' where ``counting primes'' on one hand and
        delicate finite group theory, on the other hand, are
        combined.

        We will present the main counting results, applications
        to group theory and connections with the congruence
        subgroup problem and the structure of the fundamental
        groups of hyperbolic manifolds.


 17.3.2003 Inna Scherbak, Tel-Aviv University

Title:  Rational Functions with Prescribed Critical Points & Related Topics.

 24.3.2003 Viktor Ginzburg, UC Santa Cruz, USA.

Title: Existence & non-Existence of Periodic Orbits.

 31.3.2003 Victor Palamodov, Tel-Aviv University

Title: One Century of Pioneering Minkowski's Paper.

 7.4.2003 Barak Weiss, Ben-Gurion University

Title: Dynamics on Parameter Spaces.
                                    
                                    Abstract
                                    ********

The talk will introduce the topic of dynamics of Lie group actions on
parameter spaces. The two central examples are the subgroup action on a
homogeneous space such as SL(n,R)/SL(n,Z), and the action of SL(2,R) or
its subgroups on the moduli space of quadratic differentials over complex
structures on a surface. The first space has been extensively studied in
connection with problems in diophantine approximation, and the second, in
connection with dynamics of interval exchange transformations and rational
billiards. Over the last several years many interesting analogies between
these two topics have been discovered, and I will present several new
lines in the "dictionary".

 28.4.2003 Michael Farber, Tel-Aviv University

Title: Topology & Robotics.

                                     Abstract
                                    ********
I shall discuss some topological problems arising in robotics.

I plan to focus on a new topological approach to the motion planning problem which
reveals surprising relations between the classical problems of robotics and methods
of the algebraic topology. In particular I shall explain how one may predict
the character of instabilities of robot motion knowing the cohomology algebra
of the configuration space of the robot.

I shall also mention the unknotting problem for the robot arm.
The latter is motivated by the global tasks of the molecular biology.

 19.5.2003 Yakov Eliashberg, Stanford University

Title: Contact Manifolds & Commuting Differential Operators.

 2.6.2003 Itai Benjamini, Weizmann Institute

Title: Random Walks on Infinite Graphs.

                                     Abstract
                                    ********
The classic simple random walk on Euclidean lattices was extended in two natural ways.
To simple random walk on  other graphs and to other more complicated random processes
such as self interacting random walks. We will review some of the qualitative aspects of
random processes in these two extensions.

Fall 2002

Geometric & Measurable Group Theory.

Normal Families, Shared Values & Omitted Functions.

Abstract:
This talk surveys recent progress in the theory of normal families of meromorphic functions on plane domains, with an emphasis on the role played by conditions involving such notions as shared values and omitted functions.

Singularity-Dominated Strong Fluctuations.

Minimal Representations - Spherical Vectors & Automorphic Functional

On the Polynomial Moment Problem

Abstract:
In the recent series of papers of M. Briskin, J.-P. Francoise and Y. Yomdin the following ``polynomial moment problem" was proposed as an infinitesimal version of the center problem for the Abel differential equation in the complex domain: for a complex polynomial   P(z)  and distinct  a,b \in \C  such that  P(a)=P(b) to describe polynomials   q(z)  orthogonal to all degrees of  P(z)  on the segment  [a,b].  In the lecture we present some new results concerning the polynomial moment problem and discuss some related topics.

Flavours of Asymptotic Geometric Analysis.

Non-conventional Ergodic Theorems, Ergodic Geometry & Nilpotent Groups.

 23.12.2002 Marc Teboulle, Tel-Aviv University

Convex Optimization & Duality Methods for Hard Quadratic Problems

Abstract:

Quadratic problems and their relatives, convex conic optimization problems, are currently one of the most active areas of research in optimization. These problems arise in a broad range of fields from mathematics, engineering and control to hard combinatorial problems. This talk will present some instances of quadratic problems, their connections, and their analysis from the perspective of convex dual methods. The main theme is to show that convex duality is a general technique  that can be used :to detect "hidden" convexity in seemingly hard problems, to demonstrate that well known conic relaxations of combinatorial problems (e.g., max-cut, stable set) are particular instances of the dual approach, to show that  dual approximations to these problems are  somehow  best computationally tractable bounds, and to formulate some open and challenging questions.

The talk is intended to a wide audience. We will assume (almost) no prior knowledge in continuous optimization, convex duality, and on the aforementioned topics.

 30.12.2002 Boris Solomyak, University of Washington

Spectra of Cantor-Lebesgue measures as multipliers in L^p on the circle.

Abstract:
Consider the convolution with a Cantor-Lebesgue type measure, with a constant dissection ratio $\theta$, as an operator in $L^p$ on the circle. We show that its spectrum is countable (and does not depend on $1<p<\infty$) when $\theta$ is a Pisot number, that is, an algebraic integer whose conjugates are less than one in modulus. This question was raised by Sarnak who obtained the result for integers $\theta$ in 1980. When $\theta$ is not a Pisot number, the statement follows from a classical result of Salem. This is a joint work with Nikita Sidorov.

 6.1.2003 Nati Linial, Hebrew University

Finite metric spaces: Combinatorics, Geometry & Algorithms.