∙ 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

## Riemann surfaces and integrable systems

- Riemann surfaces: definitions and examples
- Holomorphic and meromorphic functions on Riemann surface
- Compact Riemann surface: genus, monodromy, homology
- Differentials on Riemann surface and integration
- Riemann bilinear relation
- Jacobi variety and Abel theorem
- Divisors and Riemann-Roch theorem

- Jacobi inversion problem and theta functions
- Integrable systems: the Toda Lattice with periodic boundary conditions
- Integrable systems with random initial data and connection with the theory of random ma

## Point processes and random measures

The theory of point process is soon turning 500, yet very many questions remain unanswered.

The first part of the course will give a gentle introduction to point processes, with a special emphasis on determinantal point processes.

The second part will be devoted to corresponding random measures such as the Gaussian multiplicative chaos. We will start from the basics, no prerequisites assumed and will go on to topics of current research.

## Volterra series, shuffles and permutations

Volterra series are universal asymptotic expansions of solutions of well-posed evolution equations. These series have rich algebraic and combinatorial structures which we will focus on. The mentioned structures are all related to the combinatorial analysis of words, trees and permutations.

## Introduction to the constructive renormalization group for Fermionic theories

In this course I will present a set of techniques which make it

possible to rigorously implement the renormalization group for several

euclidean Fermionic field theories and related models in equilibrium

statistical physics (such as planar Ising and dimer models). This will

mainly be illustrated with the example of the planar Ising universality

class.

## Moment maps in differential geometry

:This course aims to show that many PDEs studied in differential geometry can be considered as moment map equations for some infinite-dimensional group action. This point of view gives a unifying perspective for the study of various problems and can provide important insights by establishing a parallel with known phenomena in finite dimensions. We will be most interested in explaining a formal connection between the existence of solutions to PDEs of geometric interest and algebraic stability conditions, along the lines of the Kempf-Ness Theorem.

## Discrete Integrable Systems

## Introduction to Smooth Ergodic Theory

Smooth Ergodic Theory is the study of dynamical systems on smooth manifolds from a probabilistic and statistical perspective.

## Advanced Mathematical Physics A

See homepage of Trieste University

## Integrable systems and wave motion

Course content

- From the incompressible Euler equation to the Korteweg - de Vries (KdV) equation
- KdV and Schrödinger: The Inverse Scattering Method.
- The Hamiltonian and bi-Hamiltonian settings for equations of KdV type.
- Group actions and Hamiltonian reductions. Systems of Calogero-Moser and Toda type.
- Stratified flows. The Green-Naghdi and Miyata-Camassa-Choi equations.
- Hamiltonian aspects of sharply stratified flows and boundary effects."