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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

Computations in Algebraic Geometry

The course aims to teach the classical techniques for parameterising certain locally closed subschemes of the Hilbert and Quot schemes of points, namely Hilbert–Samuel strata. There will be a particular focus on using the software Macaulay2 in algebraic geometry. Five topics will be covered in five lectures during the course. An additional lecture will be devoted to explicit computations using the Macaulay2 software. Lecture notes written in collaboration with Dott.

A course on non-negative polynomials

We will explore the theory of nonnegative polynomials from both classical and modern perspectives, highlighting connections with convex and algebraic geometry, semidefinite programming, and current research directions.

Enumerative Geometry

This course will provide an introduction to modern enumerative geometry, with special focus on localisation and motivic techniques. We will cover the classical Atiyah-Bott localisation formula, its enhancement to the virtual setup, and use these techniques to compute Donaldson-Thomas invariants of 3-folds. Then we will move to motivic invariants of moduli spaces of sheaves and give examples of their calculation for curves and surfaces, mixing localisation techniques with the Bialynicki-Birula decomposition.

Introduction to Topological Field Theories

The course provides a brief introduction to Topological Field Theories as infinite dimensional generalisation of localisation formulae in equivariant cohomology. It starts with an introduction to Duistermaat-Heckman theorem, equivariant cohomology and Atiyah-Bott formula and their extension on supermanifolds. It then continues with supersymmetric quantum mechanics and its relation with Morse theory, gradient flow lines and Morse-Smale-Witten complex.

Differential Geometry

Description:
In the first part of the course we will cover the basic theory of compact complex manifolds, in particular those admitting a strongly compatible Riemannian metric (i.e. a Kähler metric), especially in the case of vanishing Ricci curvature (i.e. Calabi-Yau manifolds). In the second part we will study a remarkable class of submanifolds of Calabi-Yau manifolds, known as special Lagrangian submanifolds, following ideas of Thomas, Yau, Joyce and Li.

Mathematics of Many-Body Quantum Systems

The course will discuss rigorous methods for the study of quantum statistical mechanics models of importance in condensed matter physics. The focus will be on analytic techniques that allow to describe in a quantitative way the large scale behavior of many-body systems in the thermodynamic limit.Program:

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