Equivariant moduli problems and the Euler class
Invariants in geometry are often defined by integrating differential forms over the moduli space of solutions of a partial differential equation. Usually this moduli space is neither compact nor smooth, where the nonsmoothness comes from nontrivial isotropy of a group action.
In this talk I explain how to construct invariants if the moduli space is compact and the group acts with finite isotropy. This situation arises, e.g., from the Seiberg-Witten equations and the "symplectic vortex equations". I present two approaches. The first one is based on the equivariant Thom class; the second one uses multivalued perturbations and weighted branched manifolds.