Mathematics

The main assumption of the classical implict function theorem in Banach spaces is that the linearized operator has a bounded inverse. This is sufficient for constructing bifurcation theory of periodic solutions of finite dimensional dynamical systems. On the other hand, there are several problems where this assumption is not satisfied, i.e. the linearized operator is unbounded, for example for the search of quasi-periodic solutions. To overcome this challenge, the Nash-Moser theory was developed. 

 

In this course we shall prove a Nash-Moser-Zehnder implicit function theorem for the search of zeros of a nonlinear operator acting on scales of Banach spaces, of either analityic or differentiable functions. As applications we shall prove the Siegel linearization theorem for an analytic map close to an elliptic fixed point and the classical KAM (Kolmogorov–Arnold–Moser) theorem concerning the persistence of quasi-periodic solutions for sufficiently small and smooth perturbations of completely integrable, non-degenerate, finite dimensional Hamiltonian systems.

  

Other applications shall be considered time permitting.