Mathematics

The main focus of the course will be on the problem of reducibility of linear time dependent PDEs, namely the problem of finding a coordinate transformation conjugating the equation to a time independent one. The course can be considered as an introduction to KAM theory and its use in PDEs. I will start by presenting a related topic, namely Poincare theory for the persistence of periodic orbits, which is one of the theory in which elementary reducibility theory finds application. Then I will develop elementary reducibility theory for periodic ODEs, namely Floquet theory. I will also give some applications of Poincare theory to a few interesting problems. Then I will move to the main topic of the course, namely KAM theory for the reducibility problem for the Schrodinger equation. I will present physical motivations and prove a KAM type theorem for a particular case of this equation. Then the course can take different directions depending on the interest of the students.I can present some recent results in which the theory of pseudodifferential operators is used in connection with KAM theory in order to prove reducibility results for systems with unbounded perturbations. In particular these new results are promising for the development of KAM theory in PDEs. Alternatively I can develop some normal form or Nekhoroshev type results for PDEs.