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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

Algebraic Geometry (Sheaf cohomology)

The course will be divided in two parts, one on sheaves, derived functors, sheaf cohomology and spectral sequences; and a second, which will essentially be a first introduction to the theory of schemes.

Geometric Control Theory

PART 1: Controllability
-) Control systems: accessibility, controllability
-) Controllability of linear systems
-) Families of vector fields. Lie algebras
-) The Frobenious theorem
-) The orbit Theorem and the Krener theorem
-) Symmetric families: the Chow theorem
-) Compatible vector fields
-) Recurrent drift
-) Systems with unbounded controls

PART 2: Optimal control
-) Existence. The Filippov theorem
-) First order conditions. The Pontryagin Maximum Principle
-) Synthesis theory

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