The aim of this project is to study refined spectral, microlocal or semi-classical estimates for mainly non-selfadjoint operators and their applications to dynamical and evolution problems. This involves in particular resolvent type estimates, spectral and pseudospectral estimates, numerical simulations, Weyl law type estimates and resonances results. By evolution problems we mean scattering, diffusion, dissipation, damping, propagation or return to the equilibrium phenomena, arising in kinetic theory, relativity, superconductivity, oceanography and more generally mathematical physics. The central idea of the project is to help interplay between researchers working on estimates and researchers studying or modeling evolution problems.

The projet de recherche collaboratif ``Beyond KAM theory'' is a project in Mathematics. Its goal is the study of dynamical systems both in finite and infinite dimensions in view of applications to partial differential equations and spectral theory. More specifically, we will be interested in systems displaying quasi-periodic behaviors which means displaying quasi-periodic patterns in time or space. A fundamental tool in this approach is the so-called KAM theory (for Kolmogorov, Arnold, Moser) that allows to prove, for certain perturbations of integrable hamiltonian systems, the existence of invariant tori on which the dynamics of these systems is quasi-periodic. KAM theory is a powerful tool: its range of application goes from the study of one-dimensional dynamical systems (circle diffeomorphisms) to that of infinite dimensional hamiltonian systems such as hamiltonian partial differential equations. The domain of application of KAM theory is nevertheless hampered by three classical restrictions: KAM method generally applies to perturbations of simple model systems ; (b) small divisors phenomena impose quantitative non-resonance conditions; (c) the existence of resonances and their geometry often make necessary the introduction of parameters and non-degeneracy assumptions on the way these parameters control the system. One main goal of our project will be, whenever possible and for different types of systems, to go beyond these restrictions. The systems we will consider are finite dimensional hamiltonian systems, diffeomorphisms of the circle, of the disk and of the torus, quasiperiodic cocycles jointly with quasiperiodic Schrödinger operators and hamiltonian partial differential equations. The funding of this project by the Agence Nationale de Recherche, that will last 4 years, will allow the collaboration of mathematicians, with complementary skills (dynamical systems, small divisors, hamiltonian theory, partial differential equations, normal forms, cocycles) and that use in their research KAM theory as a fundamental tool. The partners of the project are: (a) Partner at Nantes, Lab. J. Leray, Univ. Nantes: Benoît Grébert (representative), Eric Paturel, Georgi Popov, Laurent Thomann; (b) Partner at Nice, Lab. J.A. Dieudonné, Univ. Nice Sophia Antipolis: Philippe Bolle, Claire Chavaudret, Laurent Stolovitch (representative); (c) Partner at Paris, Lab. de Prob. et Mod. Aléat., Univ. Pierre et Marie Curie: Artur Avila, Abed Bounemoura, Hakan Eliasson, Bassam Fayad, Jacques Féjoz, Sergei Kuksin, Raphaël Krikorian (coordinator), Laurent Niederman and Jean-Christophe Yoccoz. The funding will allow the organization of one international conference gathering international leading experts, four annual meetings where all the participants of the project will present their current works, and one summer or winter school. This funding will also permit deeper collaborations between the member of the project and, through various invitations, with other worldwide experts; it will allow the members of the project to participate to conferences in the field, and this way, to diffuse and deepen their ideas. The financial support of missions for the members of the project or for their PhD students (in particular to attend the summer or winter schools) is an important point. Finally, this financial support will make possible the hiring of two post-doctoral researchers each financed for one year. The amount of the requested funding is 305 keuros.