Mathematics Colloquium, Fall 2007

Ergodic Theory, Polynomials, and Combinatorics

Various results obtained in recent years indicate that dynamical systems exhibit regular behavior along polynomial times. While being of interest in their own right, the polynomial recurrence and convergence theorems also lead to strong applications in combinatorics and number theory. We will discuss some of these applications, including various polynomial extensions of Szemeredi's theorem on arithmetic progressions and the recent work of Tao and Ziegler on polynomial patterns in primes. We shall conclude by formulating and discussing some open problems and conjectures. The talk is intended for a general audience.