Bhargav Bhatt - Prismatic cohomology and applications: Overview

February 16, 2022 - This is the first in a series of three Minerva Lectures
Prismatic cohomology is a recently discovered cohomology theory for algebraic varieties over p-adically complete rings. In these lectures, I will give an introduction to this notion with an emphasis on applications. The first lecture will be an overview, explaining the problems this theory was designed to solve as well as its origins (partially in algebraic topology). The second lecture will be dedicated to the notion of a prismatic crystal, which sheds new light on some classical objects in both number theory (such as Galois representations over p-adic fields) and algebraic geometry (such as the Deligne-Illusie theorem for de Rham cohomology). In the final lecture, I will discuss a somewhat surprising recent use of prismatic cohomology in establishing an analog of the Kodaira vanishing theorem for algebraic varieties over p-adically complete rings.
The work covered in these talks is due to several authors, including Clausen, Drinfeld, Lurie, Mathew, Morrow, Scholze and myself.