U. Haifa Topology & Geometry seminar: Sunday, June 12, 2022. Speaker: Gil Bor (CIMAT, Mexico). Title: “Cusps of caustics by reflection in a convex billiard table”.
Time: 17:00 (Israel Time)
Location: In Room 614 of the Science & Education Building, University of Haifa and on Zoom
We place a light source inside a smooth convex billiard table (or mirror). The n-th caustic by reflection is the envelope of the light rays after n reflections. Theorem: each of these caustics, for a generic point light source, has at least 4 cusps. This is a billiard version of “Jacobi’s Last Geometric Statement”, concerning the number of cusps of the conjugate locus of a point on a convex surface, proved so far only in the n=1 case. I will show various proofs, using different ideas, including the curve shortening flow and Legendrian knot theory. I will also show computer experiments supporting the conjecture that for an elliptical billiard table and any position of the light source, other than the foci, the n-th caustic by reflection has exactly 4 cusps.
(Joint work with Serge Tabachnikov of Penn State)
Meeting ID: 849 8759 1026