∙ 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

## 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

## Introduction to sub-Riemannian geometry

- Isoperimetric problem and Heisenberg group.
- Sub-Riemannian length and metric.
- Rashevskii-Chow theorem.
- Existence of length-minimizers.
- Normal and abnormal geodesics.
- Hamiltonian setting; Hamiltonian characterization of geodesics.
- The endpoint map and the exponential map; conjugate and cut points.
- Nonholonomic tangent space.
- Popp volume and Hausdorff measure.
- Sub-Laplacian and sub-Riemannian heat equation.
- Lie groups and left-invariant sub-Riemannian structures.