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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

Probability, Statistical Mechanics and Quantum Fields

This is a series of seminars taking place (approximately) every week, on Tuesday at 14:00. It covers topics at the interface between probability theory, statistical mechanics and quantum field theory.

Organizers: Ilya Chevyrev, Marcello Porta.

Paweł Duch (EPFL) will give a mini-course on "Singular Stochastic PDEs", in the period March 9-20, 2026. Here you can find the description of the course.

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 four-manifolds

This is an introductory course on 4-manifold topology. Its purpose is to provide general knowledge to students working in related research areas like algebraic and complex geometry.  

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.

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