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. 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.
Biography: 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 is 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.