KAM theory in finite and infinite dimensional spaces

Abstract

Kolmogorov-Arnold-Moser (KAM) theory is one of the greatest mathematical achievements of the last century with great impact in science. In recent years, various problems with KAM arise in many branches of mathematics and physics, such as celestial mechanics, condensed matter physics, dynamical systems, Hamiltonian PDEs, mathematical and physical equations, and operator spectral theory. These problems cannot be solved easily by the classical KAM theory, and then motivated the further development of KAM theory. In this paper, we give a brief (not complete) survey on the recent development of both finite dimensional and infinite dimensional KAM theory, including the non-degeneracy condition, KAM for lower dimensional tori, and KAM for PDEs.