Billiards, Arithmetic and Hodge Theory

Explore the fascinating connections between billiards, arithmetic, and Hodge theory in this captivating lecture by Curtis McMullen from Harvard University. Delve into the intriguing question of periodic billiard path slopes in regular polygons and discover their relationships to cusps of thin groups, curves on Hilbert modular varieties, heights from Jacobians with real multiplication, and spectral gaps for Galois orbits of triangles. Encounter issues of chaos and decidability, particularly in polygons with 7 and 12 sides, as you journey through this complex mathematical landscape. Gain insights into the intersections of various mathematical disciplines and their applications in this hour-long exploration of advanced mathematical concepts.