SL2 Seminar

We survey work over the last 50 years advancing our understandingof cohomology of groups. We begin with results of Daniel Quillen which have influenced all that follows. We mention stability results of Cline, Parshall, Scott, and van der Kallen relating the rational cohomology of a simple algebraic group $G$ defined over a finite field k and the cohomology of associated finite groups $G(K)$ of $K$-points of $G$ for $K/k$ finite. We proceed to discuss cohomology in the category of (strict polynomial) functors, especially joint work with Andrei Suslin, leading to many stable computations of cohomology. Time permitting, we shall mention various basic properties which remain to be proved. For example, unanswered questions abound about the rational cohomology of the Heisenberg group $U_3$ and its associated finite groups $U_3(k)$.