Mathematics

• Definition of an integrable system in finite dimension.
• Lax pair.
• Infinite dimensional integrable systems.
• An example: the Toda lattice equation. Soliton solution and periodic solutions.
• Toda lattice and Hermitian random matrices.
• The beta-ensamble (non integrable).

The goal of this course is to give an introduction to the theory of Poisson vertex algebras and its applications to the theory of integrable systems. Then we will study in detail the Drinfeld-Sokolov Hamiltonian reduction construction of classical W-algebras and the corresponding integrable hierarchies of Hamiltonian equations.