5 January to 7 July 2000

**Organisers**: A Katok (*Penn State*), G Margulis (*Yale*), M Pollicott (*Manchester*)

The central scientific theme of this programme is the recent development of applications of ergodic theory to other areas of mathematics. In particular, the connections with geometry, group actions and rigidity, and number theory.

The potential of ergodic theory as a tool in number theory was revealed by Furstenberg's proof of Szemerdi's theorem on arithmetic progressions. Foremost amongst the recent contributions to number theory is the solution of the Oppenheim Conjecture, a problem on quadratic forms which had been open since 1929, and the Baker-Spindzuk conjectures in the metric theory of diophantine approximations.

Of equal importance is the role of ergodic theory in geometry and the rigidity of actions. The seminal result in this direction is the Mostow rigidity theorm. In recent years there have been diverse results, including rigidity results for higher rank abelian groups, and results on the classification of geodesic flows on manifolds of non-positive curvature.

This is a quickly evolving area of research. The program will explore these, and other, emerging applications of ergodic theory. It will bring together both national and international experts in ergodic theory and related disciplines, as well as others from the wider UK mathematical community.