Gil Bor (CIMAT)
This talk is concerned with a billiard version of Jacobi’s Last Geometric Statement and its generalizations. Given a point O inside an oval billiard table (or mirror), one considers the family of rays emanating from O and the caustic (or envelope) of the reflected family of rays after n reflections off the walls of the table. I will describe two related statements:
(1) Theorem: for a generic O this caustic has at least 4 cusps for each positive integer n.
(2) Conjecture: for an elliptic table there are exactly four (ordinary) cusps.
I will describe a proof of (1) and partial results concerning (2).
This is joint work with Mark Spivakovsky (Toulouse) and Serge Tabachnikov (Penn State).
References:
* https://arxiv.org/abs/2112.07852
* https://arxiv.org/abs/2406.11074.
