Bialynicki-Birula decompositions and the Hilbert scheme of points

In the talk I will briefly describe how a group action can be used to analyse a moduli space (or more generally, a functor) via a generalization of the Bialynicki-Birula decomposition. As a half-of-the-talk-example I will explain
what can be said for the Hilbert scheme of points on A^n (n>2) and in particular how to exhibit its components. In the last part I'll carefully
review open questions: on the one hand, the newly exhibited smooth components are open to direct or experimental investigation and on the other hand, the new methods may help to answer classical open questions about those Hilbert schemes.