Mathematical billiards is a dynamical system in which a point mass travels inside a smooth, closed, convex plane curve and reflects off the boundary as light would. This setup extends to Riemannian manifolds with boundary (the mass instead travels along a geodesic), and with a bit more work, to Finsler manifolds (a physical example of a Finsler billiard trajectory is a particle moving in a weak magnetic field). Following Farber and Tabachnikov, who computed lower bounds for the number of periodic trajectories of a billiard in a smooth, closed, convex hypersurface in Euclidean space, we used Morse-Lusternik-Schnirelmann theory to compute lower bounds for the number of periodic trajectories for billiards on a Finsler manifold.

Outer (or dual) billiards is a dynamical system played outside a smooth, closed, convex plane curve. The outer billiard system was generalized to even-dimensional standard symplectic space by Tabachnikov. With Godoy and Salvai, we defined and studied an outer billiard system on the 4-dimensional space of oriented geodesics in a 3-dimensional space of constant curvature (for example, on the space of affine lines in 3-dimensional Euclidean space).