The Founder

GÖRAN GUSTAFSSON Lectures in Mathematics


June 4 - 8, 2010
KTH, Stockholm

Jean Bourgain
Institute for Advanced Study,Princeton

Abstracts:

Lecture 1: Expander graphs,spectral gaps and some applications

 This is a non-technical introductory talk with emphasis on the expander properties of SL^2(p) Cayley graphs and spectral gaps of Hecke operators acting on SU(d).'Classical' applications such as the Solovay-Kitaev algorithm in quantum-computation and aperiodic tilings will be discussed

Lecture 2: Expansion in SL^d and prime number sieving

An extension of the work of A.Selberg on congruence subgroups of SL^2(Z) to the setting of arbitrary non-elementary subgroups of SL^2 is described.Combined with methods introduced by Lax-Phillips and Lalley on hyperbolic lattice point counting,good asymptotics are obtained for the size of 'balls' in those groups,providing the necessary input for the sieving axioms in number theory.Applications include the Apollonian circle packings.

Lecture 3: Survey of the methods

The main ingredients belong to 'arithmetic combinatorics',representation theory and the theory of random matrix products.In particular,we will explain the role of 'approximate groups' which is a key notion in this scheme,the 'discretized ring theorem' and the problem of 'escape from subgroups'.

Sponsored by the Göran Gustafsson Foundation

2010-05-16