A comprehensive study of periodic trajectories of the billiards within ellipsoids in the $d$-dimensional Euclidean space is presented. The novelty of the approach is based on a relationship established between the periodic billiard trajectories and the extremal polynomials of the Chebyshev type on the systems of $d$ intervals on the real line. Classification of periodic trajectories is based on a new combinatorial object: billiard partitions.

The case study of trajectories of small periods $T$, $d \leq T \leq 2d$, is given. In particular, it is proven that all $d$-periodic trajectories are contained in a coordinate-hyperplane and that for a given ellipsoid, there is a unique set of caustics which generates $d + 1$-periodic trajectories. A complete catalog of billiard trajectories with small periods is provided for $d = 3$.

The talk is based on the following papers:

• V. Dragović, M. Radnović, Periodic ellipsoidal billiard trajectories and extremal polynomials, Communications Mathematical Physics, 2019, Vol. 372, p. 183-211.
• G. Andrews, V. Dragović, M. Radnović, Combinatorics of the periodic billiards within quadrics, The Ramanujan Journal, DOI: 10.1007/s11139-020-00346-y.