Mathematics

In the first part of this course, I will present the basic elements of calculus in infinite dimensional spaces, holomorphic functions, and the classical implicit function theorem in Banach spaces. As applications we will consider some result about the bifurcation theory of periodic solutions. In the second part I will present the Nash-Moser hard implicit function theorem in scales of Banach spaces with different applications, as, for example, the Siegel linear conjugacy theorem or the KAM (Kolmogorov-Arnold-Moser) theorem concerning quasi-periodic solutions of nearly integrable Hamiltonian and reversible dynamical systems and PDEs. The final part will be a more advanced course.