THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES 20th ANNIVERSARY YEAR
	PUBLIC LECTURE September 20, 2012 at 4:00 p.m. Fields Insititue, Room 230 Stéphane Nonnenmacher Commissariat à l'énergie atomique, Saclay Counting stationary modes: a discrete view of geometry and dynamics Co-sponsored by the Fields Institute and Department of Mathematics, University of Toronto

Abstract: (presentation)

In this lecture I first plan to present the historical context leading to Hermann Weyl's first result on the high frequency eigenvalue counting for the Laplacian on a planar domain. I will then sketch the mathematical developments on that question, and various extensions of this result, including a semiclassical version useful in quantum mechanics, as well as the case of "fractal domains". Such spectral asymptotics can often reveal a lot of information on the geometry of the domain (or manifold) and the associated geodesic (or Hamiltonian) dynamics.

I will then switch to the study of (quantum) scattering systems. Such systems admit a discrete set of complex-valued "generalized eigenvalues", called resonances. Counting such resonances has proved a difficult task, mainly due to the nonselfadjoint nature of the problem.

Yet, I will present some resonance counting estimates, which may also reflect some dynamical features of the corresponding classical dynamics; this is the case, for instance, of the "fractal Weyl's law" expected to hold for chaotic scattering systems.