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Explore the deep connections between billiard dynamics and the geometry of Riemann surfaces in this mathematical lecture that examines the Hodge bundle and its GL(2, R) action. Delve into how the dynamics of this action governs the geometry of the moduli space of Riemann surfaces, a fundamental object in geometry, algebra, and physics. Learn about ongoing research building on the foundational work of Eskin and Mirzakhani to classify GL(2, R) orbit closures and understand their implications for seemingly simple problems involving billiards in polygons. Discover how Hurwitz spaces can be applied to realize McMullen's vision of describing complex orbit closures using finite combinatorial data, bridging abstract mathematical theory with concrete geometric problems.
