Birkhoff's theorem for classical Hamiltonian systems. A model problem: the nonlinear wave equation. Birkhoff normal form for completely resonant PDEs, dynamical consequences: exponentially stable periodic solutions. Birkhoff normal form for nonresonant PDEs, dynamical consequences: approximatively invariant tori, almost global existence.

1. Basic principles of classical mechanics: Newtonian mechanics, Lagrangian mechanics, Hamiltonian mechanics
2. The N body problem: The 2 body problem, Collision and regularization, Final motions in the 3 body problem, Restricted 3 body problem, Ergodic theorems of classical mechanics.

1. Variational theory: We want to discuss some extensions of Aubry-Mather theory to systems on lattices and PDE's. Models considered: Frenkel-Kontorova Models in several dimensions. Elliptic equations. Ginzburg-Landau equations Minimal surfaces.
2. KAM theory. We want to study the existence of smooth quasi-periodic solitions. We will present versions of the KAM theory in situations where there is a Lagrangian theory without any Hamiltonian counterpart.
3. Hyperbolic theory.
4. Reduction methods

In this course, I will first introduce some of the fundamental methods in nonlinear partial differential equations such as variational methods, the maximum principle, etc.

Then, I will discuss two types of interesting solutions: the spike-layer solutions which arise in the analysis of the shadow system of a biological pattern formation model (the Gierier-Meinhardt system); and the transition layer solutions which play an important role in the study of phase transition via the Allen-Cahn equation and its counterpart in system of equations.

For the spike-layer solutions, I will emphasize the new variational methods for the existence of higher energy solutions with multiple concentration. Regarding transition layer solutions, I will talk about the symmetry of solutions in entire spaces including De Giorgi conjecture, the existence of triple, quadruple junction solutions, a new Hamiltonian type equality and its application, etc. The topics are related to current research interests.

We will give an introduction to dislocations dynamics. Dislocations are curves defects in a crystal. When a stress is applied on the crystal, these curves can move with a dynamics given by the normal velocity depending on the whole shape of the curves and on the interactions with the other defects in the crystal.

Mathematically, this dynamics is described by non-local Hamilton-Jacobi equations in the framework of viscosity solutions.

After presenting classical results on homogenization, we will give in particular some results about the homogenization of the dynamics of self-interacting dislocations. In the limit we recover an effective plastic law which involves a fractional Levy operator. This will be an opportunity to present the introduction of recent tools on homogenization.