Here “derived commutative ring” refers to a nonconnective generalization of animated commutative rings,
first defined by Mathew; in other words, it’s a common home for the homotopy theories of simplicial and cosimplicial
commutative rings. I first encountered the notion while studying Hochschild homology and de Rham cohomology.
In this talk, I’ll describe a different presentation of the category. It’s an instance of a fairly simple and very general pattern,
of which there are other interesting instances too. I plan to also discuss one more, a notion of “synthetic E-infinity ring”,
and the relevance of this to the study of topological Hochschild homology and prismatic cohomology (this part will be
based on joint work with Devalapurkar, Hahn, and Senger).