[Distinguished Lecture] Can you hear the shape of a drum? and deformational spectral rigidity
Time: 2023-08-25
Published By: Ying Hao
Speaker(s): Vadim Kaloshin (Institute of Science and Technology Austria)
Time: 16:00-17:00 August 25, 2023
Venue: Room 77201, Jingchunyuan 78, BICMR
Abstract: 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. Recently, jointly with C. Fiedrobe we made progress on non perturbative dynamical spectral rigidity
of generic axisymmetric domains.
Bio-sketch: Dr. Vadim Kaloshin is the Professor at the Institute of Science and Technology Austria (ISTA). After receiving his Ph.D. from the Princeton University in 2001, he was awarded the American Institute of Mathematics five year fellowship. He was an invited speaker for International Congress of Mathematicians (ICM) 2006 and a plenary speaker for International Congress on Mathematical Physics (ICMP) 2015. He was a recipient of Sloan fellowship (2004) and Simons fellowship (2016), besides, he was awarded a Moscow Mathematical Society Prize (2001) and the Barcelona Prize in Dynamical Systems (2019). Professor Kaloshin is a member of the Academia Europaea, a prestigious scientific society, and grantee of the European Research Council (ERC) Grant.
From 2007 to 2019 he was an editor of Inventiones mathematicae. He holds the editorial boards of Advances in Mathematics, Analysis & PDE, Dynamical Systems and Ergodic Theory, and Revista Matemática Iberoamericana.
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. Recently, jointly with C. Fiedrobe we made progress on non perturbative dynamical spectral rigidity
of generic axisymmetric domains.
Bio-sketch: Dr. Vadim Kaloshin is the Professor at the Institute of Science and Technology Austria (ISTA). After receiving his Ph.D. from the Princeton University in 2001, he was awarded the American Institute of Mathematics five year fellowship. He was an invited speaker for International Congress of Mathematicians (ICM) 2006 and a plenary speaker for International Congress on Mathematical Physics (ICMP) 2015. He was a recipient of Sloan fellowship (2004) and Simons fellowship (2016), besides, he was awarded a Moscow Mathematical Society Prize (2001) and the Barcelona Prize in Dynamical Systems (2019). Professor Kaloshin is a member of the Academia Europaea, a prestigious scientific society, and grantee of the European Research Council (ERC) Grant.
From 2007 to 2019 he was an editor of Inventiones mathematicae. He holds the editorial boards of Advances in Mathematics, Analysis & PDE, Dynamical Systems and Ergodic Theory, and Revista Matemática Iberoamericana.
