In 2017, in https://arxiv.org/abs/1703.03545, Totaro initiated the study of de Rham cohomology of classifying stacks of reductive groups relating it to some purely representation-theoretic data via Hodge-to de Rham spectral sequence. He was able to explicitly identify de Rham cohomology with the singular cohomology in most examples and conjectured that at least an inequality of dimensions should hold in general. I will talk about joint work https://arxiv.org/abs/2105.05319 with A.Prikhodko where among other things we proved this conjecture using prismatic cohomology. I will discuss some particular examples as well as the general strategy of the proof. If time permits I will also briefly talk about the results of our more recent paper https://arxiv.org/abs/2211.17227 where a version of rational Hodge theory was established for all Artin stacks with a smooth d-Hodge-proper integral model. This implies some new results on crystallinity of etale cohomology in the schematic setting as well.
