9/28/2005

Zeev Rudnick
Tel Aviv

Eigenvalue statistics and lattice points

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, with the position of the window chosen at random and the window size 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. In various regimes this case has been intensively studied since the early 1990's by Heath-Brown, Bleher, Dyson, Lebowitz, Sinai, Sarnak, Eskin, Mozes, Margulis and others. I will explain some recent progress, by Hughes and myself and by Wigman. Time permitting, I will also discuss the case of the modular surface.