The Hodge bundle is the space whose points correspond to Riemann
surfaces equipped with holomorphic 1-forms. This space admits a
\(\operatorname{GL}(2, \mathbb{R})\) 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
\(\operatorname{GL}(2, \mathbb{R})\) 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.