∙ 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

## Non-linear Gross-Pitaevskii dynamics of a 2D binary condensate: a numerical analysis

.## Spin geometry and Index Theorem

The talks will include the topic on Clifford algebra, Spin group and their representation; K, KO theory; characteristic class, spin structure, spin connection, Dirac operator; (pseudo)differential operator, Multiplicative sequence, Chern character, Atiyah-Singer Index theorem.

## On Geometric Quantum Confinement in Grushin-Like Manifolds

.## Gauge Theory

In this course we will give an introduction to gauge theory on smooth 4-manifolds and to its applications in differential topology and algebraic geometry. The following themes are supposed to be covered: