The moduli space of Riemann surfaces admits a Kobayashi hyperbolic metric called the Teichmueller metric. The geodesic flow in this metric can be concretely understand in terms of a linear action on flat surfaces represented as polygons in the plane. In this talk, we will study the dynamics of this geodesic flow using the geometry of flat surfaces.

Given such a flat surface there is a circle of directions in which one might travel along Teichmueller geodesics. We will describe work showing that for every (not just almost every!) flat surface the set of directions in which Teichmueller geodesic flow diverges on average - i.e. spends asymptotically zero percent of its time in any compact set - is 1/2.

In the first part of the talk, we will recall work of Masur, which connects divergence of Teichmueller geodesic flow with the dynamics of straight line flow on flat surfaces.

In the second part of the talk, we will describe the lower bound (joint with H. Masur) and how it uses flat geometry to prove a quantitative recurrence result for Teichmueller geodesic flow.

In the third and final part of the talk, we will describe the upper bound (joint with H. al-Saqban, A. Erchenko, O. Khalil, S. Mirzadeh, and C. Uyanik), which adapts the work of Kadyrov, Kleinbock, Lindenstrauss, and Margulis to the Teichmueller geodesic flow setting using Margulis functions. Speaker(s): Paul Apisa (Yale)