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Explore the mathematical dynamics of billiard ball motion within polygonal boundaries through this Members' Colloquium lecture from the Institute for Advanced Study. Delve into the behavior of point masses traveling in straight lines at constant speed within polygons, following elastic collision rules when hitting sides. Learn how trajectories behave differently depending on whether polygon angles are rational or irrational multiples of pi, with rational cases creating predictable invariant surfaces while irrational cases remain largely mysterious. Discover the groundbreaking work of Kerckhoff, Masur and Smillie proving ergodicity in certain irrational polygons, where typical trajectories spend time in regions proportional to their area or volume. Examine two significant results strengthening ergodicity to weak mixing: the existence of polygons where billiard flow exhibits weak mixing with respect to natural volume on the unit tangent bundle, and a complete classification of rational polygons demonstrating weak mixing on invariant surfaces. Gain foundational understanding of ergodic theory and weak mixing concepts while exploring connections between billiards in rational polygons and translation surfaces, with no prior knowledge of billiards, ergodic theory, or translation surfaces required.
