Emerging Interactions of Geometric and Variational Methods

Organizers: Vadim Kaloshin, Univ of Maryland and Marian Gidea, Yeshiva University

Participants: Marie-Claude Arnaud, Abed Bounemoura, Amadeu Delsham, Jacques Fejoz, Marcel Guardia, Rafael de la Llave, Jean Pierre Marco, Tere Seara, Ke Zhang

Summary: The last several years witnessed remarkable progress on the celebrated Arnold Diffusion Problem for nearly integrable Hamiltonian systems. It asserts that ‘typical’ integrable Hamiltonian systems, subjected to ‘generic’ perturbations, display trajectories that cover a significant region in the action space. The problem has been classified into two cases: the a priori unstable case, when the unperturbed Hamiltonian possesses invariant tori with hyperbolic invariant manifolds, and the a priori stable case, when the phase space of the unperturbed Hamiltonian is foliated by Lagrangean invariant tori. While in the former case a significant body of work has been produced during the last few decades, in the latter case breakthrough advancements have been registered only recently. A fundamental role in shaping the field, and in providing many ideas and inspiration was played by John Mather.

The state of the art is the following: for two-and-a-half/three degrees of freedom, smooth, integrable Hamiltonian systems that are convex and superlinear, applying sufficiently small perturbations selected from some cusp-residual set, yields the existence of trajectories that shadow any prescribed set of resonances. The most promising methodology for this problem seems to lie at interface between geometric and variational methods.

There are some outstanding questions that were addressed by the proposed working group:

1. Clarify the relation between the mechanisms of diffusion via variational methods and via geometric methods.

2. Investigate whether wider classes of nearly integrable Hamiltonians can exhibit diffusion.

3. Develop explicit mechanisms of diffusion that can be verified in concrete models, and provide quantitative estimates.

Working Group Report

A Delshams Lecture Arnold diffusion for `complete' families of perturbations with two or three independent harmonics

R de la Llave Lecture Some geometric mechanisms for Arnold diffusion

JP Marco Lecture Symplectic geometry of hyperbolic cylinders and their homoclinic intersections

T Serre Lecture A General Shadowing result for normally hyperbolic invariant manifolds and its application to Arnold diffusion

M Gidea Lecture Diffusion along chains of normally hyperbolic cylinders

M Guardia Lecture Growth of Sobolev norms for the cubic NLS near 1D quasi-periodic solutions