## Seminar:

Fall 2009, Thursdays, Milner 216, 4:00-4:50 PM
__ Date:__ ** September 17 **

**" Billiards in polygons and Teichmuller theory "**

**Prof. Yaroslav Vorobets**

__Abstract__ Billiard is a dynamical system that describes free motion of a mass-point (billiard ball) in a domain (billiard table) subject to elastic collisions with the boundary (cushion). The billiard is a simple model of mechanical systems with impacts. It was introduced almost a century ago by Birkhoff whose goal was to find a simple setup for studying complex dynamical phenomena. The model turned out to be very useful as the billiards exhibit all kinds of dynamical behavior, from very regular to extremely chaotic.

Mathematical theory of billiards is a fascinating subject which is both a rich source of new problems and a testing ground for conjectures in dynamics, geometry, and mathematical physics.

I will begin the talk with a brief general overview of the billiard dynamics. Then I will describe in detail one class of billiards, billiards in polygons. The polygonal billiards offer even simpler setup than general billiards. Nevertheless, it is enough for challenging problems, beautiful results and advanced techniques. In addition to usual analytic tools, the study of billiards in polygons involves more algebra and geometry. The theory of polygonal billiards has been rapidly growing in the last 10 years attracting many mathematicians from different areas.

I will explain how polygonal billiards are related to geodesic flows on flat surfaces and how the study of those billiards led to the progress in the Teichmuller theory, a sophisticated theory that deals with Riemann surfaces, moduli spaces and quasi-conformal maps.