Billiards and Hodge theory

Analysis Math-Physics Seminar
Topic:Billiards and Hodge theory
Speaker:Simion Filip
Affiliation:Harvard University
Date:Wednesday, April 19
Time/Room:2:45pm - 3:45pm/S-101
Video Link:https://video.ias.edu/analysis/2017/0419-SimionFilip

A polygon with rational angles can be unfolded and glued into a finite genus Riemann surface equipped with a flat metric and some singularities. The moduli space of all such structures carries an action of the group $\mathrm{PSL}(2,\mathbb R)$ and this can be viewed as a renormalization of the billiard flow in the initial polygon. After introducing the basics, I will explain how Hodge theory can give information on the $\mathrm{PSL}(2,\mathbb R)$ dynamics, in particular on the Lyapunov exponents and orbit closures. I will also discuss some questions in Hodge theory suggested by dynamics.