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

4-manifolds

The main goal of the course is to provide a concise introduction to 4-manifold topology. A list of topics to be covered is as follows.

Topics in Advanced Mathematical Physics

The course will discuss the mathematics of many-body quantum mechanics, with a focus on the rigorous derivation of effective theories for complex quantum systems. Topics to be covered include:

-) Introduction to quantum mechanics. The hydrogenic atom. Uncertainty principles, stability of matter of the first kind.

-) Bosons and fermions, density matrices. Introduction to large Coulomb systems, as models for atoms and molecules.

-) Lieb-Thirring inequalities, semiclassical approximations.

Noncommutative Geometry

The course focuses 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.

Introduction to Topological Field Theories

The course provides a brief introduction to Topological Field Theories as infinite dimensional generalisation of classical localisation formulae in equivariant cohomology. It starts with an introduction to these latter subjects (Duistermaat-Heckman theorem, equivariant cohomology and Atiyah-Bott formula) and their extension on supermanifolds. It then continues supersymmetric quantum mechanics and its relation with Morse theory, gradient flow lines and Morse-Smale-Witten complex.

Pages

Sign in