## Hamiltonian integrable systems and Lie algebras

### Prof. Gregorio Falqui

The aim of the couse is to provide the Ph. D. 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 discuss both integrable systems of PDEs in 1 + 1 dimensions, and (classical and quantum) spin chain-like systems.

Contents:

- Geometry of Symplectic and Poisson manifolds. Lie Algebras and Lie Poisson brackets. Bihamiltonian manifolds and their properties.
- Hamiltonian G-actions on Poisson manifolds. The Marsden-Weinstein, Marsden-Ratiu, and Dirac reduction theorems.
- Lax pairs: properties and applications. The AKS scheme.
- Loop algebras, Drinfel'd-Sokolov type integrable hierarchies of PDEs, and differential operators on the circle. The examples of the KdV and Boussinesq hierarchies.
- Spin chains: Heisenberg, Gaudin and Toda systems. Universal enveloping algebras. Classical and quantum r-matrix theory. The algebraic Bethe Ansatz.