Ellipsoidal billiards, extremal polynomials, and combinatorics
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, is given. In particular, it is proven that all -periodic trajectories are contained in a coordinate-hyperplane and that for a given ellipsoid, there is a unique set of caustics which generates -periodic trajectories. A complete catalog of billiard trajectories with small periods is provided for .
The talk will be via Zoom at: https://utoronto.zoom.us/j/99576627828