A panoramic view of prismatic cohomology (part 2 of 2)

This is the second of two talks described below.

Arithmetic geometry over the p-adic numbers has historically had an excess of good cohomology theories (e.g., de Rham, crystalline, Hodge-Tate, étale). One of the main goals of p-adic Hodge theory has been to study the relationship between these various theories with the intention of not only clarifying them individually, but to work towards a 'grand unifying cohomology theory' which generalizes them all. Steady progress towards these goals has been made since the beginning of the subject in the early 1980s. That said, the recent (since ~2018) work of Bhatt, Drinfeld, Lurie, Morrow, and Scholze (among others) on the notion of prismatic cohomology has gotten us considerably closer to realizing this program in its true form. In the first of these two lectures, I will give a very high-level motivation for prismatic cohomology: what type of object is it roughly, and what sort of properties do we wish it to satisfy. In the second of these two lectures I will discuss joint work with Abhinandan (Juisseu) showing that even formal properties of prismatic cohomology discussed in the first lecture are enough to clarify and generalize old questions: the relationship between de Rham and crystalline cohomology (in ramified settings), as in the celebrated paper paper of Berthelot and Ogus.