Cusps of caustics by reflection and Jacobi's last geometric statement
时间  Datetime
2025-10-30 14:00 — 18:00
地点  Venue
会议室(706)
报告人  Speaker
Gil Bor
单位  Affiliation
CIMAT, CIMAT, Guanajuato, Mexico （墨西哥国家数学研究中心，瓜纳华托）
报告摘要  Abstract

A light source is placed inside a closed strictly convex mirror (or ``billiard table"). The envelope of  the light rays  emanating from the light source after undergoing n reflections is called the n-th caustic by reflection. Theorem: for a generic light source the n-th caustic by reflection has at least 4 cusps (GB and S Tabachnikov, 2023). Conjecture: for an elliptical mirror and a non-focal light source there are  exactly 4 cusps. The subject belongs to a long list of results motivated by the celebrated 4-vertex theorem (S. Mukhopadhyaya, 1909), as well as  "Jacobi's last geometric statement" (1843). I will explain in detail this theorem and partial progress on the conjecture, as well as the generalization to other families of caustics (eg, Liouville surfaces and magnetic billiards). The mini-course is aimed at  "general audience", like graduate students or advanced undergraduates.
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FROM 202.120.11.*