Spectral Geometry, Graphs and Semiclassics – GeRaSic

The aim of this project is to explore new developing trends in spectral geometry. The latter is understood in a very broad sense that encompasses the geometry of moduli spaces, the spectral theory in the semiclassical regime and its application to quantum chaos, the quantization of (hyperbolic) dynamical systems, as well as the spectral theory on graphs.

Spectral geometry aims at understanding how the spectrum of geometrically relevant objects depends on the underlying geometry. It is thus a natural question to try to follow spectral quantities when geometric parameters vary. We propose to perform a systematic study of spectrally relevant quantities viewed as functions defined over the moduli space or more generally over the manifold of metrics. Generic properties, averaged or randomized version of classical results are among our objectives.

Semiclassical analysis aims at understanding how quantum mechanics is influenced by the underlying classical dynamics. In these questions, semiclassical measures play a very special role. They are at the heart of the celebrated quantum ergodicity theorem which is a milestone for the quantum chaos investigations. Even if striking progress has been made in the last decade, several old questions remain unanswered and new ones have appeared. Among these we intend to look at systems satisfying only weak notions of hyperbolicity, open systems and Fractal Weyl Laws, and semiclassical methods for classical chaos.

Graphs, and in particular Schrödinger operators on graphs, form a privileged setting for spectral investigations. On non-compact graphs, the exact conditions under which a (possibly magnetic) Schrödinger operator is self-adjoint isn't completely described. This is an attractive challenge that will open the way to several classical spectral questions. Another direction we will address consists in pushing the analogy between hyperbolic regular trees and hyperbolic manifolds in particular to see if a suitable pseudodifferential calculus can be obtained in this setting.

Regular meetings, workshops, two international conferences will help develop these themes and collaborations between members and with other researchers through invitations. A post-doctoral fellowship will also participate to the success of this project.