In this talk I'll report some initial progress on a new project. The cohomology groups of symmetric groups have long been known, but for example both the ring structure and the pairing with the homology of symmetric groups (the Dyer-Lashof algebra) are open questions in general. I have started to study these questions by looking at "linear subvarieties" of spaces of unordered configurations. I will start with basic background, including the standard invariant theory approach to group cohomology. Next I'll review the cohomology of ordered configuration spaces, which is the model I'd like to follow for the significantly harder unordered setting. Then I'll give some explicit calculations in the cohomology of S_{4}. Finally, I'll indicate my main reason for interest in the project, which is to better understand invariants of homotopy groups coming from loopspace theory. |