University of California, Irvine

Title: Recent developments on Kac’s problem “Can one hear the shape of a drum?”

Abstract: In 1966 Mark Kac popularized the inverse spectral problem for planar domains by asking whether one can determine the shape of a plane domain from the eigenvalues of the Laplacian with Dirichlet or Neumann boundary conditions. Recently there has been great progress in this topic motivated by the works of Avila, Kaloshin, Sorrentino, de Simoi, and Wei, in the field of dynamical systems.  I will report on my joint work with Steve Zelditch where we prove that nearly circular ellipses are determined by their spectrum among all smooth domains. We use the heat trace and wave trace techniques combined with some dynamical systems results for billiard tables.