Can you hear the shape of a drum? and deformational spectral rigidity
时间  Datetime
2023-12-01 16:00 — 17:00
地点  Venue
会议室(706)
报告人  Speaker
Prof. Vadim Kaloshin
单位  Affiliation
马里兰大学
邀请人  Host
吴文俊数学中心
备注  remarks
报告摘要  Abstract

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.
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