General Events - Department of Mathematics

Title: Billiards, dynamics, and the moduli space of Riemann surfaces

Abstract: The Hodge bundle is the space whose points correspond to Riemann surfaces equipped with holomorphic 1-forms. 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. Building on work of Eskin and Mirzakhani, I will describe my work on a program to classify GL(2,R) orbit closures and derive consequences for deceptively simple sounding problems about billiards in polygons. Along the way, I will describe an application of Hurwitz spaces to realize a hope of McMullen of describing intricate orbit closures with finite combinatorial data.