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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

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

This course aims to introduce students to the topic of 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.

Integrable systems and wave motion

Course content

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

Log Calabi-Yau geometry

Calabi-Yau manifolds play a central role in algebraic/differential

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