Abstract

The mathematical billiard concerns the motion of a particle on a billiard table (i.e. a bounded domain in R^2. In the interior of the domain, the particle moves along straight lines, with elastic reflections at the boundary. In the case where the boundary is sufficiently smooth and strictly convex, Lazutkin's landmark theorem from 1973 almost entirely classifies the dynamics for small reflection angles: most trajectories are predictable and lie on invariant curves, while those that do not are confined to small regions. What happens if we relax the assumptions of the theorem? Naturally we expect the invariant curves to disappear, but what do we see in their absence? One phenomenon that may occur is the appearance of glancing trajectories: these are trajectories along which the reflection angle tends to 0. Although intuitively improbable from a geometric viewpoint, we prove that these trajectories exist if we relax the regularity, convexity, or dimensional assumptions. Finally, we will consider a modification of the classical billiard known as the coin billiard. Recent results will be presented, which give partial answers to open questions posed by Misha Bialy. The talk is aimed at a general mathematical audience.