Can you hear the shape of a drum and deformational spectral rigidity


第二十一讲：Can you hear the shape of a drum and deformational spectral rigidity
报告题目(Title)：
Can you hear the shape of a drum and deformational spectral rigidity

报告人(Speaker)：
Vadim Yu. Kaloshin 院士 (Institute of Science and Technology Austria)

地点(Place)：
Zoom ID：98761001871, Passcode: 123456

时间(Time)：
2023年10月9日 15:00-16:00
报告摘要
M. Kac popularized the following question "Can one hear the shape of a drum?" Mathematically, consider a bounded planar domain Ω ? R2 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.
主讲人简介
Vadim Yu. Kaloshin，欧洲科学院院士，国际数学大会邀请报告人，2006-2018期间为Inventiones Mathematicae编委。在动力系统、天体力学、谱刚性等做出贡献，发表高水平学术论文七十余篇。
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