L2

In this lecture, I will review Atiyah's definition of a topological quantum field theory. I'll then sketch the definition of a more elaborate structure, called an "extended topological quantum field theory", and describe a conjecture of Baez and Dolan which gives a classification of these extended theories.

Coefficients of the characteristic polynomial are generators of the ring of polynomial functions on the space of matrices which are invariant under the conjugation. This was generalized by Chevalley to general reductive groups. By looking closely on the centralisers, one is lead to a very natural 2-category attached to Chevalley characteristic morphism. This abstract, but yet elementary, construction helps one to understand the symmetries of the fibres of the Hitchin fibration, as well as those of affine Springer fibers.

We will also explain how these groups of symmetries are related to the notion of endoscopic groups, which was introduced by Langlands in his stabilisation of the trace formula. We will also briefly explain how the symmetry groups help one to acquire a rather good understanding of the cohomology of the Hitchin fibration and eventually the proof of the fundamental lemma in Langlands' program.

I shall show how cellularity may be used to obtain presentations of the
endomorphism algebras in question, both in the classical and quantum cases.

In classical economic theory, one discounts future gains or losses at a constant rate: one pound in t years is worth exp(-rt) pounds today. There are now very good reasons to consider non-constant discount rates. This gives rise to a problem of time-inconsistency: a policy which is optimal today will no longer be optimal tomorrow. The concept of optimality then no longer is useful. We introduce instead a concept of equilibrium solution, and characterize it by a non-local variant of the Hamilton-Jacobi equation. We then solve the classical Ramsey model of endogenous growth in this framework, using the central manifold theorem

We classify certain linear representations of finite groups with a large orbit. This is motivated by a question on the number of conjugacy classes of a finite group.