摘要：M. Kac popularized the following question "Can one hear the shape of a drum?" Mathematically, consider a bounded planar domain Ω ⊆ *R*2 with a smooth boundary and the associated Dirichlet problem Δu + λu=0, u|∂Ω=0. The set of λ's for which this equation has a solution is called the Laplace spectrum of Ω. Does the Laplace spectrum determine Ω up to isometry? In general, the answer is negative. Consider the billiard problem inside Ω. Call the length spectrum the closure of the set of perimeters of all periodic orbits of the billiard inside Ω. Due to deep properties of the wave trace function, generically, the Laplace spectrum determines the length spectrum. Jointly with J. De Simoi and Q. Wei we show that an axially symmetric domain close to the circle is dynamically spectrally rigid, i.e. cannot be deformed without changing the length spectrum. This partially answers a question of P. Sarnak. We shall also talk about a recent result of K. Callis about the existence of absolute periodic orbits for convex billiards and its relation with Ivrii's conjecture.

嘉宾简介：Vadim Kaloshin是美国马里兰大学-帕克分校数学系Brin首席教授、奥地利科技学院讲席教授，欧洲科学院院士，曾获得美国科学院院士提名、西蒙斯奖等荣誉。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.上公开发表学术论文9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学术论文65篇。