Joint IAS/PU Arithmetic Geometry

Recent work of Drinfeld, Bhatt, and Lurie provides a “geometrization” of the theory of prismatic cohomology, where, for a p-complete commutative ring R, one produces various algebraic stacks (“prismatizations”) whose coherent cohomology identifies with the prismatic cohomology of R (and variants thereof). It turns out that using the theory of “topological Hochschild homology”, one can also construct the prismatization of (p-complete) commutative ring *spectra*; the latter are objects from homotopy theory which generalize the usual notion of commutative rings. Although our extension of the theory of prismatization uses a lot of technology to set up, it has some concrete applications: I will discuss how it can be used to understand Hodge theory in characteristic p>0 for smooth proper schemes of dimension>p, extending work of Deligne and Illusie. Time permitting, I also hope to discuss the Drinfeld formal group, and how this theory leads to some interesting generalizations of the q-logarithm and q-hypergeometric series. (A large part of) this is joint work with Jeremy Hahn, Arpon Raksit, and Allen Yuan.