Billiards, dynamics, and the moduli space of Riemann surfaces

The Hodge bundle is the space whose points correspond to a Riemann surface equipped with a holomorphic 1-form. This space admits a GL(2, R) action whose dynamics governs the geometry of the moduli space of Riemann surfaces, an object of central importance in geometry, algebra, and physics. I will describe work, joint with Alex Wright, that classifies roughly half of all GL(2, R) orbit closures. I will also describe applications to deceptively simple sounding problems about billiards in polygons. Along the way I will highlight connections to algebraic geometry, homogeneous dynamics, and more.