Special Year Learning Seminar

KAM theory and modern symplectic dynamics emphasize, at first sight, rather different features of Hamiltonian motion. On the one hand, the Kolmogorov–Arnold–Moser theorem shows that, for nearly integrable Hamiltonian systems satisfying suitable non-degeneracy conditions, a large-measure set of invariant tori filled with quasi-periodic solutions survives perturbation. On the other hand, periodic orbits play a central role in global symplectic dynamics, despite forming a set of zero measure in phase space.

The aim of this seminar is to pose the question of whether and how the KAM viewpoint can be integrated into modern symplectic dynamics. I will begin by recalling the notion of quasi-periodic functions and integrable Hamiltonian systems, and then present the KAM theorem and the role of small divisors.

A related theme concerns Hamiltonian PDEs, where global variational methods have been successfully applied to construct periodic solutions as critical points of strongly indefinite action functionals. Notably, Rabinowitz’s celebrated 1980 existence result for periodic trajectories on star-shaped energy hypersurfaces stemmed from his work on the existence of periodic solutions of nonlinear wave equations.