Sergei Tabachnikov (Penn State)

Multi-dimensional Birkhoff theorem: periodic trajectories in smooth convex billiards.

Abstract. The classical Birkhoff theorem concerns periodic trajectories in smooth strictly convex plane billiards; it asserts that for every n and $r \leq n/2$ there exist two distinct n-periodic billiard trajectories with rotation number r. In a recent series of papers, M. Farber and myself found generalizations of this result in multi-dimensional setting. One of the main results reads: for a generic smooth strictly convex billiard table in m-dimensional space, there exist at least (n₁)(m₁) distinct n-periodic billiard trajectories. The work is based on a topological study of the cyclic configuration space of the sphere and Morse-Lusternik-Schnirelman theory.