Cohomology of configuration spaces with coefficients and arithmetic statistics: a progress report

In a paper a few years ago with Akshay Venkatesh and Craig Westerland, we explained how to use results on stable cohomology of Hurwitz spaces to derive results about the variation of class groups in families of quadratic extensions of rational function fields over finite fields (i.e. we said something about the Cohen-Lenstra problem.)  I will report on more recent progress in this direction, including some work by me and some work by others, and in particular try to make a case based on work of Will Sawin and some not-yet-baked work by me, TriThang Tran, and Westerland, that people in this area need to stop worrying and love perverse sheaves.