Mathematics

Mathematical billiards arise naturally in the study of several problems in physics (such as the Lorentz gas or the Ehrenfest model) and are an idealization of the billiard game in special shaped "tables". We will focus on polygonal billiards and explain how their study is connected to the study of "flat" surfaces and their moduli space. The study of chaotic properties of billiard trajectories in rational polygons and the associated compact flat surfaces has been a topical and successful area of research for the past 30 years thanks to the beautiful connection with Teichmueller dynamics. Only very recently, though, there have been breakthroughs in our understanding of the Ehrenfest model, systems of Eaton lenses and other infinite periodic billiards and the associated infinite periodic flat surfaces. We will in particular survey some recent progress in this area.