Speaker: Iosif Polterovich (University of Montreal)

Title: The spectral function and the remainder in
local Weyl's law: View from below

Abstract.
Consider the Laplace operator on a compact manifold.
How small is the error term in Weyl's law for the
distribution of eigenvalues? It is a fascinating
question with remarkable links to number
theory: in particular, on a flat square torus this
question is equivalent to the famous Gauss circle
problem. In order to understand which upper bounds on
the remainder one may hope for, it is useful to study
the estimates from below. The talk focuses on the
lower bounds for the spectral function of the
Laplacian and for the pointwise error term in  Weyl's
law. I will first discuss some general results and
then explain
how the techniques of thermodynamic formalism for
hyperbolic flows yield stronger estimates on
negatively curved manifolds. This is a joint work with
Dmitry Jakobson (McGill).