Billiards and Hodge theory

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.