**KAM for PDEs**- Periodic and Quasi-periodic solutions of Hamiltonian PDEs
- Nonlinear wave and Schroedinger equations
- Reversible KAM theory
- KAM for unbounded perturbations: quasi linear KdV, derivative wave equations
- Water waves equations
- Birkhoff Lewis periodic orbits
- Almost periodic solutions

**Bifurcation Theory and Nash-Moser Implicit Function Theorems****Birkhoff normal forms**- Long time existence results for quasi-linear Hamiltonian PDEs
- Paradifferential normal forms
**Variational and Topological Methods in the study of Hamiltonian systems**- Variational methods for periodic solutions
- Homoclinic and heteteoclinic solutions

**Dynamical systems**- Arnold Diffusion
- Chaotic dynamics
- Perturbation and Nekhoroshev Theory
- 3 body problem

**Benjamin-Feir and modulational instability**

## Spectral theory of deterministic and disordered systems

March 11-22 2024

The first week (possibly going into the second week as well) will provide an introduction to localization.

The proposed topics are:

## Linear and nonlinear bifurcation problems (Topics in Ad. Analysis 2)

After introducing the theory of analytic functions between Banach spaces, we shall present perturbative results for the spectrum of linear operators, in particular for separated eigenvalues of closed operators, with applications to the stability of traveling water waves. Then we shall present bifurcation results of periodic and quasi-periodic solutions of nonlinear dynamical systems as well as homoclinic solutions to hyperbolic equilibria of Hamiltonian systems.

## Weak turbulence and wave kinetic equation

## Water waves

The water waves equations were introduced by Euler in the 18th century to describe the motion of a mass of water under the influence of gravity and with a free surface. The unknown of the problem are two time dependent functions describing how the velocity field of the fluid and the profile of the free surface (giving the shape of the waves) evolve. The mathematical analysis of the water waves equations is particularly challenging due to their quasilinear nature, and it has been (and still is) a central research line in fluid dynamics.

## Periodic Orbits of Hamiltonian systems through Variational Methods

Hamiltonian systems give a very good description of those physical phenomena where the energy is (approximately) conserved: from planetary orbits to the motion of particles.

Typically, however, the dynamics is highly sensitive to the initial conditions and therefore it is difficult to find specific orbits in the systems such as those connecting two subsets of phase space or those which are periodic.