Subscribe to Events

Abstract: Small denominator problems appear in various areas of analysis, PDE, and dynamical systems, including spectral theory of quasiperiodic Schro?dinger operators, non-linear Schro?dinger equations, and non-linear wave equations. These problems have traditionally been approached by KAM-type constructions. We will discuss the new methods, originally developed in the  spectral theory of quasiperiodic Schro?dinger operators, that are both considerably simpler and lead to results completely unattainable through KAM techniques. For quasiperiodic operators, these methods have enabled precise treatment of various types of resonances and their combinations, leading to .proofs of sharp (arithmetic) spectral transitions, the ten martini problem, and the discovery of universal hierarchical structures of eigenfunctions.