∙ 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

## Integrable systems from moduli spaces of stable curves

The first part of this course will introduce the moduli spaces of stable curves and the main results about their geometric structure and their Deligne-Mumford compactification. Studying the topology of such spaces leads directly to the question of describing their cohomology and intersection theory. We will introduce some examples of cohomlogy classes on the moduli spaces and eventually the notion of tautological ring as a natural subring of cohomology whose structure is sufficiently well behaved.

## Random polynomial systems, Kahler geometry and the momentum map

**Lecture 1:** On counting solutions of polynomial systems

- Bézout's theorem
- Smale's 17-th problem
- Shortcomings of Bézout's theorem
- Sparse polynomial systems, and the mixed volume

**Lecture 2:** Differential forms

- Multilinear algebra over R
- Complex differential forms
- Kähler geometry
- The coarea formula, using bundles.
- Projective space

**Lecture 3:** Reproducing kernel spaces