MENU

You are here

Home

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

Riemann surfaces and integrable systems

Content:

Part 1

1.1Short review of the classical theory of finite-dimensional integrable systems
1.2 Be-Hamiltonian Geometry and Lax pair
1.3 The Toda system
1.4 The Korteweg de Vries equation: direct and inverse scattering on the line with decreasing initial data
1.5 Long time asymptotic for the solution of the KdV equation with decreasing initial data and Deift-Zhou steepest descent method
1.6 The  Cauchy problem for the KdV equation with periodic initial data and action-angle variables.

Algebraic Surfaces


The lectures will follow the book : "Complex Algebraic Surfaces" by Arnauld Beauville

 

Non-commutative Geometry

Program:
0. Introduction.
1. Exterior and Clifford algebras, Spin groups, spinors.
2. Spin structures.
3. Dirac operator.
4. Some analytic properties. Spectral triple.
5. Other characteristic features: dimension (finite summability),
   regularity (smoothness), finiteness & projectivity, reality,
   first order, orientation, Poincare duality.
6. Statement of the ‘reconstruction theorem’ of A. Connes.
7. NCG Examples: noncommutative tori, spheres,
   finite spectral triple for the Standard Model.

Pages

Sign in