Flat surfaces: from billiards to hyperbolic geometry

Tuesday, 26-March-2024 

Abstract

Gluing opposite sides of a parallelogram together gives a torus. A translation surface is a surface obtained by identifying sides of more complicated polygons by translations. Some of these surfaces, called Veech, are more symmetric than the others. They arise naturally when exploring both billiards trajectories and geodesics in the moduli space of curves. I will give an introduction to these objects and explain these connections. I will then present a recent contribution to the classification of Veech surfaces in genus three.