Topology of cyclic configuration spaces and periodic trajectories of multi-dimensional billiards

We give lower bounds on the number of periodic trajectories in strictly convex smooth billiards in R^m+1 for m ≥ 3. For plane billiards (when m = 1) such bounds were obtained by Birkhoff in the 1920s. Our proof is based on topological methods of calculus of variations - equivariant Morse and Lusternik-Schnirelman theories. We compute the equivariant cohomology ring of the cyclic configuration space of the sphere S^m, i.e., the space of n-tuples of points (x₁,...,x_n), where x_i ∈ S^m and x_i ≠ x_i+1 for i = 1,...,n.