Joint seminar in Algebraic Geometry and Number Theory/KASS

Abstract: Polygonal billiards: Imagine playing billiards where the table is a finite sided Euclidean polygon (simply connected, but not necessarily convex). If we label the sides and record every sequence we can obtain by a trajectory of the billiard ball (which we call its bounce sequence), can we determine the shape of the table? In this talk I will show that the answer is yes, in the sense if two tables have the same bounce sequence, then they must differ by a similarity (outside of a highly non-generic set of tables). We prove this by studying Euclidean cone surfaces where we obtain another rigidity result: if the universal covers of two Euclidean cone surfaces have the same set of endpoints of their non-singular geodesics (those that avoid the cone points) then they are the same surfaces.