## Integrable systems and Lie algebras

### Prof. Gregorio Falqui

The aim of the couse is to provide the students with tools in the modern theory of Integrable Systems. Especially, I plan to center the course on those aspects related with the theory of Lie Algebras.

The course will cover the following topics:

- Geometry of Symplectic and Poisson manifolds. Liouville-Arnol'd theorem, action-angle variables and the Hamiton-Jacobi theory.
- Lie Algebras and Lie Poisson brackets. Universal enveloping algebras and the PBW theorem.
- Hamiltonian G-actions on Poisson manifolds. The Marsden-Weinstein and Marsden-Ratiu reduction theorems.
- Lax pairs: properties and applications. The AKS scheme.
- Spin chains: Heisenberg, Gaudin and Toda systems. Classical and quantum r-matrix theory.
- Loop algebras and Drinfel'd-Sokolov type integrable hierarchies of PDEs.