Bredon cohomology of equivariant configuration spaces

Consider the configuration space of n distinct points in Euclidean space.  The singular cohomology is computed classically, with the answer having interesting connections to combinatorics and geometry.  If G is a finite group, one can attempt the analogous computation for RO(G)-graded equivariant (Bredon) cohomology of configuration spaces of points in a G-representation.  The classical approach runs into trouble and one has to instead find a different method.  This talk will report on recent work with Christy Hazel on one such method, which has led to some success but not a complete resolution of the problem.  We will also discuss some open questions about the cohomology of a point that are related to these issues.