Determining the shape of a billiard table from its bounces

Resumen: Consider a billiard table shaped as a Euclidean polygon with labeled sides (with no restrictions on the angles). A ball moving around on the table determines a bi-infinite “bounce sequence” by recording the labels of the sides it bounces off, and the set of all possible bounce sequences gives the bounce spectrum of the table. In this talk I will explain why the bounce spectrum essentially determines the shape of the table: with the exception of a very small family (right-angled tables), if two tables have the same bounce spectrum then they have to be related by a Euclidean similarity. In the language of Euclidean cone surfaces, this can be phrased as another rigidity statement: the metric is (generically) determined by the endpoints of its non-singular geodesics in its universal cover. Time allowing, I will discuss some ongoing work generalizing these results to billiards with obstacles. This is joint work with Moon Duchin, Chris Leininger, and Chandrika Sadanand.