Title: Cohomology theories in the moduli of ring stacks
Abstract: There's recently been a lot of interest in the "stacky" approach to cohomology of schemes, an idea going back to Simpson's de Rham stack. A salient feature of these theories is the appearance of "ring stacks"; in particular, the stack associated to a ring R obtains a natural map to the stack of R-algebra stacks. I'll give some background on the stacky approach and explain how in relevant cases (syntomic, characteristic 0 filtered de Rham, Betti, and l=p étale) this map is fully faithful. One can therefore view all of these stacks as moduli stacks of R-algebra stacks, putting them on the same footing. This is joint work with Dhilan Lahoti.