∙ Integrable systems in relation with differential, algebraic and symplectic geometry, as well as with the theory of random matrices, special functions and nonlinear waves, Frobenius manifolds
• Deformation theory, moduli spaces of sheaves and of curves, in relation with supersymmetric gauge theories, strings, Gromov-Witten invariants, orbifolds and automorphisms

• Quantum groups, noncommutative Riemannian and spin geometry, applications to models in mathematical physics

• Mathematical methods of quantum mechanics

• Mathematical aspects of quantum Field Theory and String
Theory

• Symplectic geometry, sub-riemannian geometry

• Geometry of quantum fields and strings

## Algebraic Methods for Quantum Theories

**Topics:**

- introduction to the theory of C*-algebras and von Neumann algebras

- representations

- automorphisms and dynamical systems

- KMS condition

- contraction semigroups

- algebraic formulation of Quantum Mechanics and Statistical Mechanics

- elements on the algebraic and the constructive Quantum Field Theory

## Riemann surfaces and integrable systems

**Program of the course:**