“Cusps of caustics by reflection”
Mercoledì 13 Maggio 2026, ore 13:00 - Aula 2AB40 - Sergei Tabachnikov (Pennsylvania State University)
Abstract
The “Last Geometric Statement of Jacobi” claims that the conjugate locus of a non-umbilic point on a triaxial ellipsoid has exactly four cusps. This theorem was proved only in this century and, conjecturally, the loci of the second, third, etc., conjugate points also have exactly four cusps. I shall discuss the billiard version of this problem: Nth caustic by reflection in a convex billiard table is the envelope of the 1-parameter family of the billiard trajectories, starting at a point inside the billiard table and reflected N times. For every oval, every N, and a generic choice of the point, Nth caustic by reflection has at least four cusps and, conjecturally, for an ellipse it has exactly four cusps. This problem has many extensions, for example, to Finsler billiards associated with a projective Finsler metric (the subject of Hilbert’s 4th Problem) and to magnetic billiards.
