From asymptotics to bounds in spectral geometry, ODEs and number theory

There are many situations when asymptotic behaviour of some multi-parameter-dependent quantity (when one of the parameters gets large) provides an upper or lower bound for this quantity for all values of parameters. A classical example of this phenomenon is Pólya’s conjecture in spectral geometry stating that the leading term of Weyl’s asymptotics of the eigenvalue counting function for the Laplacian in a bounded Euclidean domain is an upper bound (in the Dirichlet case) or a lower bound (in the Neumann case) for the counting function for all values of the parameter. This conjecture is still open in full generality: I’ll discuss its recent resolution for disks, balls, sectors and annuli, as well as other similar or related effects arising elsewhere.