Analysis & Applied Math

The celebrated Pólya conjecture (1954) in spectral geometry states that the eigenvalue counting functions of the Dirichlet and Neumann Laplacians on a bounded Euclidean domain can be estimated from above and below, respectively, by the leading term of Weyl's asymptotics. The conjecture is known to hold for domains that tile the Euclidean space but remains largely open in general. In this talk, we will explain the motivation behind the conjecture and discuss some recent advances, particularly for balls and annuli. These developments, in turn, are linked to new results on lattice point counting and zeros of Bessel functions. The talk is based on joint work with Nikolay Filonov, Michael Levitin, and David Sher.