A comprehensive study of peri odic trajectories of the billiards within ellipsoids in the d-dimensional Euclidean space is presented. The novelty of the approach is based on a re lationship established between the periodic billiard trajectories and the extremal polynomials of the Chebyshev type on the systems of d intervals o n the real line. Classification of periodic trajectories is based on a ne w combinatorial object: billiard partitions.
\nThe case study of tra jectories 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 caust ics which generates d + 1-periodic trajectories. A complete catalog of bil liard trajectories with small periods is provided for d = 3.
\nThe talk is based on the following papers:
\nV. Dragović\, M. Radnović \, Periodic ellipsoidal billiard trajectories and extremal polynomials\, Communications Mathematical Physics\, 2019\, Vol. 372\, p. 183-211.
\n< p>G. Andrews\, V. Dragović\, M. Radnović\, Combinatorics of the periodic billiards within quadrics\, \narXiv: 1908.01026\, The Ramanujan Journal\, DOI: 10.1007/s11139-020-00346-y.\n LOCATION:https://researchseminars.org/talk/TQFT/22/ END:VEVENT END:VCALENDAR