## Riemann surfaces and Integrable Systems

### Prof. Tamara Grava

Contents:

- Integrable systems:
- random NxN Hermitian matrices,
- Toda lattice,
- large N limit and continum limit of Toda lattice,
- graphical enumeration.

- Riemann surfaces:
- definition of Riemann surfaces, classification, uniformization theorem, Kleinian groups;
- functions and forms on Riemann surfaces. The Riemann-Roch theorem.
- Hurwitz theorem, definitions and open problems.
- theta-functions and Jacobi varieties.
- application to Integrable Systems: nonlinear integrable PDEs (periodic solutions).
- KP equation, NLS equation, KdV equation and spectral theory - small dispersion limit.