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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

Standard and less standard asymptotic methods

In every branch of mathematics, one is sometimes confronted with the

Introduction to Topological Recursion theory and moduli spaces of curves

Topological Recursion (TR) can be thought as a universal procedure, or algorithm, to generate solutions of enumerative geometric problems related directly or indirectly to moduli spaces of curves. 
 - the input is the so-called spectral curve (think e.g. an algebraic curve with two particular meromorphic functions on it), 
 - the output is an infinite list of numbers (think e.g. Gromov-Witten invariants of some kind).

Introduction to Noncommutative Geometry

 These lectures focus on the latest ’layer’ Riemannian and Spin of Noncommutative Geometry (NCG). Its central concept, due to A. Connes, is ’spectral triple’ which consists of an algebra of operators on a Hilbert space and an analogue of the Dirac operator. A prototype is the canonical spectral triple of a Riemannian spin manifold which will be described starting with the basic notions of multi-linear algebra and differential geometry.

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