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Geometry and Mathematical Physics

∙ 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

Dirac operators on manifolds with boundary

This is an introductory course on the analysis of Dirac operators on manifolds with boundary. Our goal is to prepare the background to deal with a number of applications of elliptic boundary value problems for Dirac operators which are frequently encountered in Geometry and in Mathematical Physics.

Geometry and quantization of moduli spaces of Higgs bundles

 The ubiquity of moduli spaces of semi-stable higgs bundles on a smooth projective curve both in mathematics and physics is rather impressive. These moduli spaces have proven to be grounds of extremely fruitful interaction between the two disciplines. As an example, the techniques developed by physicists to quantize a symplectic manifold and to quantize a com- pletely integrable Hamiltonian system when applied to these moduli spaces yield remarkable mathematical results. E.

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