∙ 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

## Morse Theory and Related Topics

The course includes:

- Classical Morse theory for smooth functions and regular variational problems.
- Morse theory for nonsmooth and, in particular, piecewise smooth functions.
- Symplectic interpretation of the Lagrange multipliers rule. Maslov index and its relation to the Morse index.
- Floer homology.

The only prerequisite beyond basic mathematical background is the elementary homology theory. No knowledge of symplectic or Riemannian geometry is expected.

I do not plan to follow any book. Standard sources are: