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 $T$, $d ≤ T ≤ 2d$ 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 will be via Zoom at:
https://utoronto.zoom.us/j/99576627828
Passcode: 448487
V.Dragović, M.Radnović, Periodic ellipsoidal billiard trajectories and extremal polynomials,
Communications. Mathematical Physics, 2019, Vol. 372, p. 183-211
and
G.Andrews, V.Dragović, M.Radnović, Combinatorics of the periodic billiards within quadrics, arXiv:
1908.01026, The Ramanujan Journal, DOI:10.1007/s11139-020-00346-y.