1) In the first part of the course I will provide a complete presentation and a detailed and self-contained proof of the classical KAM –Kolmogorov-Arnold-Moser– theorem concerning persistence of quasi-periodic solutions for sufficiently small and smooth perturbations of completely integrable, non-degenerate, finite dimensional Hamiltonian systems. I will prove the KAM theorem in the setting of sufficiently many times differentiable or $C^\infty$ Hamiltonians. The proof will reduce the problem to the application of a Nash-Moser-Zehnder implicit function Theorem for the search of zeros of a nonlinear operator acting on scales of Banach spaces.

2) The second part of the course will be devoted to pseudo-differential calculus and the nonlinear para-differential calculus, with applications to the local existence theory for partial differential equations of evolutionary type. The theory will be developed from the beginning from the Paley-Littlewood decomposition of functions.