The moduli space of Riemann surfaces admits a Kobayashi hyperbolic metric called the Teichmuller 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 Teichmuller geodesics. I will describe work showing that for every (not just almost every!) flat surface the set of directions in which Teichmuller geodesic flow diverges on average - i.e. spends asymptotically zero percent of its time in any compact set - has Hausdorff dimension 1/2.          The lower bound is joint with H. Masur and the upper bound is joint with H. al-Saqban, A. Erchenko, O. Khalil, S. Mirzadeh, and C. Uyanik.

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