In the first part of the course I will present general tools of nonlinear analysis and the classical implicit function theorem (IFT) in Banach spaces with some applications to bifurcation of periodic solutions.

Then we shall consider problems where the IFT fails due to "losses of derivatives” produced by small divisors, as for example fo the search of quasi-periodic solutions.

We shall first prove a Nash-Moser implicit function theorem in scales of analytic functions, focusing on the application to the Siegel conjugacy linerization problem.

Then we shall prove a Nash-Moser-Hormander implicit function theorem in scales of functions with finite differentiability.

In the final part of the course we shall discuss the existence of quasi-periodic solutions for PDEs, in particular for nonlinear wave equations on $T^d$, generalizing the multiscale approach of Bourgain.