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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

Mathematical Methods for Interacting Condensed Matter Systems

In this course we will discuss rigorous methods for condensed matter systems. In particular, we will focus on cluster expansion techniques for interacting fermions, and on rigorous renormalization group methods. One main focus of the course will be on the application of these tools to the study of transport in interacting two-dimensional models, and in particular on the emergence of universality.

Higher algebra

We start with an informal introduction to abelian categories, discussing the main examples (sheaves of modules on a ringed space). We then define the associated derived category, and revisit the notion of derived functors using a language different from the one in [Harsthorne, Algebraic Geometry]. In particular, we are able to define left derived functors even when the category doesn't have enough projectives.

C* algebras

Course contents
• Basic functional analysis, Banach spaces, linear operators
• Banach algebras, spectrum, Gelfand transform, (holomorphic) functional calculus
• C ∗ -algebras and their basic properties
• Gelfand-Naimark duality between C ∗ -algebras and locally compact Hausdorff spaces
• Continuous functional calculus, positive elements, approximate units
• Basics of von Neumann algebras
• (pure) states and (irreducible) representations
• Gelfand-Naimark-Segal (GNS) construction

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