﻿ On convex planar billiards, Birkhoff Conjecture and whispering galleries-东北师范大学数学与统计学院

On convex planar billiards, Birkhoff Conjecture and whispering galleries

A mathematical billiard is a system describing the inertial motion of a point mass inside a domain with elastic reflections at the boundary. In the case of convex planar domains, this model was first introduced and studied by G.D. Birkhoff, as a paradigmatic example of a low dimensional conservative dynamical system.

A very interesting aspect is represented by the presence of `caustics', namely curves inside the domain Ω with the property that a trajectory, once tangent to it, stays tangent after every reflection (as on the right Figure). Besides their mathematical interest, these objects can explain a fascinating acoustic phenomenon, known as "whispering galleries", which can be sometimes noticed beneath a dome or a vault.

The classical Birkhoff conjecture states that the only integrable billiard, i.e., the one having a region filled with caustics, is the billiard inside an ellipse. We show that this conjecture holds near ellipses. This is based on a joint work with Avila-De Smoi, with Sorrentino and with Huang-Sorretino.

Vadim Kaloshin, 美国马里兰大学-帕克分校数学系Brin首席教授、奥地利科技学院教授, 曾获得美国科学院院士提名、西蒙斯数学奖等荣誉, 现担任Adv. Math., Ergodic Theory Dynam. Systems等杂志编委, 主要从事动力系统领域的研究, 在国际上最顶尖的四大综合性数学期刊Acta Math., Ann. of Math., J. Amer. Math. Soc., Invent. Math.上公开发表高质量学术论文8篇, 在Duke Math. J., Geom. Funct. Anal., J. Eur. Math. Soc. (JEMS), Comm. Pure Appl. Math., Arch. Ration. Mech. Anal.等国际权威期刊上公开发表高水平学术论文65篇.