Mathematics

Generalities. The goal of this Ph.D. course is to present some aspects of classical results in Riemannian geometry relying on the representation theory of some orthogonal group. Most of these results build on work of Weyl [12] on the representation theory and the invariant theory of classical Lie groups. Since the general theory is too wide for such a course, some emphasis will be given to concrete examples, particularly in low dimension. Explicit calculations will be carried out where necessary to illustrate relevant techniques.

• Topics:

1. Representing homology classes by submanifolds and the genus function.
2. Cobordisms and basic topological invariants.

This course presents techniques used to rigorously approach the analysis of statistical mechanical systems of interacting fermions on a lattice.

Topics:

1. Universality and Critical phenomena
2. Gaussian integration, Feynman graphs and Linked Cluster Theorem
3. Grassmann variables and Grassmann Gaussian integration
4. Perturbation theory for Fermions using Grassmann variables
5. Brydges-Battle-Federbush formula with applications.

• Description:

 We will review several constructions of smooth four-dimensional manifolds that have proven useful to unveil inequivalent smooth structures.  

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.