Topics and Lecturers

Quantum Field Theory in Curved Space-times
Abhay Ashtekar (Pennsylvania)

In this mini-course I will first introduce the mathematical theory of linear quantum fields on curved space-times and then briefly discuss applications to cosmology and black hole evaporation. The first part of the course will be pretty much self contained, assuming only the knowledge of quantum field theory in Minkowski space-time, while the second part will be more in the style of seminars.


Introduction to the Superstring.
Nathan Berkovits (São Paulo)

In these lectures, I will introduce the basic ingredients of the superstring and describe the advantages and disadvantages of the different formalisms. I will assume that students are familiar with general relativity and quantum field theory, but will not assume familiarity with bosonic string theory. The lectures will be based on the paper:
Covariant quantization of the superparticle using pure spinors

Introduction to elliptic fibrations with a view toward string theory.
Mboye Esole (Harvard)

I will give an introduction to the geometric and arithmetic properties of elliptic fibrations. Several techniques of algebraic geometry related to their topological invariants and the resolutions of their singularities will also be introduced. I will explain how recent developments in string theory and super-conformal field theories connect elliptic fibrations to representation theory and combinatorics.


Deformation quantization and group actions
Simone Gutt (Brussels)

We shall introduce deformation quantizations, formal and convergent, and study their link with group actions. We shall introduce a nice subclass of homogeneous symplectic manifolds.


Principal fiber bundles in noncommutative geometry
Christian Kassel (Strasbourg)

Principal fiber bundles play a fundamental role in geometry and topology. In this course we will investigate the concept of noncommutative principal fiber bundle, which is the analogue of a classical principal fiber bundle in the framework of noncommutative geometry. Many examples of such objects appear in the world of quantum groups and in what physicists call quantum group gauge theory.
After a brief introduction to Hopf algebras, I will define noncommutative principal fiber bundles and give examples. We will next investigate various themes such as (i) functoriality, (ii) the noncommutative analogues of trivial principal fiber bundles, (iii) universality questions.