Abstract:

One of the more challenging problems in spectral theory and mathematical physics today is to understand the statistical distribution of eigenvalues of the Laplacian on a compact manifold. Among the most studied quantities is the counting function for eigenvalues in a window [E,E+S], with the position E of the window chosen at random and the window size S=S(E) depending on its position. I will describe what is known about the statistics of this counting function for the very simple case of the flat torus, where the problem reduces to counting lattice points in annuli.