Mathematics

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.

The aim of the course is to give an introduction to some topics of current interest in the globalgeometry of compact Kähler manifols.Part I – Basic notions. Hermitian and Kähler manifolds, harmonic theory, Kodaira Vanishing andEmbedding, Kempf-Ness Theorem, notions of canonical Kähler metrics.Some textbooks:

This is an advanced course that will explore connections between recent developments in geom- etry, mathematical physics and homotopy theory. Its format will be that of a working seminar. Mirror symmetry has been at the center stage of geometry for the last thirty years. It is a sophisticated dictionary relating the symplectic geometry of a variety X and the algebraic ge- ometry of its mirror Y , and vice versa.