∙ 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 problem of evaluating an infinite sum numerically and trying to guess its exact value, or of recognizing the precise asymptotic law of formation of a sequence of numbers {A_n} of which one knows, for instance, the first couple of hundred values. The course will tell a number of ways to study both problems, some relatively standard (like the Euler-Maclaurin formula and its variants) and some much less so, with lots of examples. Here are three typical examples: 1.

## 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.

## Noncommutative Geometry: Banach and C∗-algebras + Selected topics

The first part of the course will supply some basic materials in functional analysis that are frequently used in noncommutative geometry. More details can be found in the attached syllabus.

The lectures will be streamed via Teams with the same link ( Tuesday and Thursday 11-13 weekly, the last one is the week of Oct 25):