A theorem by J.D.S. Jones from 1987 identifies the cohomology of the free loop space of a simply connected space with the Hochschild homology of the singular cochain algebra of this space. There are very strong relations between the Floer homology of cotangent bundles in symplectic geometry and the homology of free loop spaces of closed manifolds.
J'exposerai des travaux en collaboration avec Julien Marché, dans lesquels
nous décrivons l'action du mapping class group sur les composantes connexes
de l'espace des représenations du groupe de surface de genre 2 dans PSL(2,R).
Recall that the geometric dimension $gd(G)$ of a group $G$ is the smallest dimension of a space on which $G$ acts in such a way that fixed point sets of finite subgroups are contractible. For many prominent classes of groups (e.g. for amenable groups, lattices in classical Lie groups, mapping class groups, groups of outer automorphisms of free groups...) one has equality between the geometric dimensions and the virtual cohomological dimension. On the other hand, there are some examples showing that these two notions of dimension might well differ.