Geometry and Topology Seminar

Polya's conjecture in spectral geometry is a hypothesized uniform bound for eigenvalues of the Dirichlet Laplacian on a domain in Euclidean space. Although proven sixty years ago for domains which tile the plane, the conjecture remains completely open for non-tiling domains. Recently, we have proven Polya's conjecture for Euclidean balls. Our results make use of some novel bounds for zeroes of Bessel functions and their derivatives, as well as lattice point counting arguments. This is joint work with N. Filonov (St. Petersburg), M. Levitin (Reading), and I. Polterovich (Montreal).