- Variational Methods
- Perturbative Methods in Critical Point Theory
- Elliptic Equations on R^n and Nonlinear Schroedinger Equation
- H-Surfaces
- Singularly Perturbed Problems
- Geometric PDEs
- Hamiltonian systems
- Chaotic Dynamics and Arnold Diffusion
- KAM Theory
- Periodic Solutions of Infinite Dimensional Systems

Research Group:

## Crystalline anisotropic mean curvature flow

- Mean curvature flow of hypersurfaces. Introduction to the problem.
- First variation of the perimeter.
- Uniqueness of smooth flows. Short time existence.
- Examples of singularities: Grayson example.
- Fattening of the crossing.
- Anisotropic mean curvature flow.
- Finsler metrics and their duals.
- The Wulff shape. Duality mappings.
- Relations between the Minkowski content (or anisotropic perimeter) and the Hausdorff measure.

## Nonlinear Analysis and Dynamical Systems

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.

## Nonlinear Analysis and Bifurcation Theory

The goal of the course is to cover the following topics:

- Basic calculus in Banach spaces
- Local inversion theorems and implicit function theory
- Properties of bifurcation points
- Bifurcation from the simple eigenvalue
- Bifurcation from multiple eigenvalues
- Examples and applications

Reference:

*Ambrosetti-Prodi*, A primer in Nonlinear Analysis

## Topics in Nonlinear Analysis

In this course 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 systems.