Algebraic geometers often study nonsmooth varieties by resolving their singularities, and some of the main problems in the area have centered on this technique. Group actions have their own singular sets, namely the set of points at which the isotropy (stabilizer) subgroup is not minimal. For example, the involution of any Euclidean space which maps x to -x has the origin as its only singular point, while the typical linear cyclic action on a complex vector space has a singular set which is a union of eigenspaces.
The topology of nonsingular, that is free, group actions is easier to think about than the general case, since these amount to fiber bundles under reasonable conditions. We can resolve away most of, and sometimes all of, the singular behavior in a smooth group action by the same process of blowing up that the algebraic geometers exploit. The talk will present this form of resolution of singularities, its features and limitations, and an equivariant version of a problem of Nash.