This talk is intended as a prologue to the subsequent GATSBY lecture, which will be delivered by Jon Chaika.
I will begin by describing how playing billiards in a polygon with all angles rational multiples of pi, naturally leads to two distinct, but related, dynamical systems - the dynamics of straight line flow on a flat surface and the dynamics of Teichmuller geodesic flow on the moduli space of holomorphic one-forms on Riemann surfaces.
Special attention will be given to the connection between the non-unique-ergodicity of straight line flow and the divergence of Teichmuller geodesics. The talk will conclude with an example of a minimal non-uniquely ergodic foliation of a flat surface.
No background will be assumed for the talk.