Virtual integration in moduli problems: We will discuss a method of carrying out integration on moduli spaces defined from Fredholm systems. The main point is that we do away with the regularity assumptions that is generally used in such a setup to obtain smooth manifolds (or orbifolds). We will also discuss some applications for such an integration. Most of the constructions will be elementary and will be motivated from finite dimensional examples. This is based on works of Chen-Li-Tian and works in progress with Chen and Hyvrier.

In this talk, I will explain how we can compute the rational homotopy type of symplectomorphism groups of some four-manifolds. After explaining the general framework, I will focus on some rational symplectic four-manifolds for which the analysis can be carried through effectively using pseudo-holomorphic curves techniques and the theory of deformations of complex structures. For those manifolds, we will see that Symp is equivalent to an iterated homotopy pushout of maximal compact subgroups.