Workshop on geometry and prismatic cohomology

Stacky approach to Dieudonne theory I

I will begin with a quick introduction to crystalline Dieudonne theory. Then I'll discuss a geometric reformulation of this theory using crystalline cohomology of classifying stacks.

Prismatic crystals and $D$-modules 

In this talk, we will explain how prismatic crystals can be realized as certain $D$-modules via the stacky approach of Drinfeld and Bhatt–Lurie, thereby yielding a classification of the former. 

Purity for torsors under pseudo-finite groups

In the case of an imperfect base field k, the category of finite k-groups may be enlarged to the category of pseudo-finite k-groups. The latter are often positive dimensional and include restrictions of scalars of finite groups, as well as many groups crucial in the study of pseudo-reductive groups and pseudo-abelian varieties. I will show that purity theorems for torsors over geometrically regular k-schemes continue to hold for pseudo-finite groups. Time permitting, I will also discuss such purity torsors under pseudo-proper and pseudo-complete groups. The talk is based on joint work with Alexis Bouthier and Federico Scavia.

Stacky approach to Dieudonne theory II

In this talk, I will explain how an extension of the geometric reformulation of Dieudonne theory from my previous talk combined with the stacky approach to prismatic cohomology (due to Drinfeld and Bhatt--Lurie) leads to a classification of finite flat group schemes over fairly general rings of mixed characteristics. 

Simpson gerbe and p-adic nonabelian Hodge theory (Part 1)

For a smooth rigid space X over a perfectoid field in characteristic zero, we define the Simpson gerbe for X and study its geometry properties. We will then explain how Cartier-duality and Fourier-Mukai theory for it realize the p-adic Simpson correspondence. In case X has a smooth integral model, this is also closely related to a similar picture for the Hodge-Tate stack of the model. The talks are based on joint work in progress with Bhargav Bhatt.

Algebraicity of stacks of sections over the syntomification of a formal scheme

I'll motivate and briefly explain results in recent work with Gardner that construct formal algebraic stacks from certain other stacks living over the syntomification of p-adic formal schemes. This has applications to the classification of p-divisible groups, and of finite flat p-power torsion commutative group schemes and their fppf cohomology (latter joint w. Mondal), stacks of isogenies, Rapoport-Zink spaces and Igusa stacks associated with exceptional groups (joint w. S-Y Lee), and the construction of special cycles on Shimura varieties.

Pursuing logarithms

This talk introduces motivic spectra without A¹-invariance, motivating it via its connections to prismatic cohomology. I will compare this approach to logarithmic homotopy theory of Binda–Park–Østvær, and discuss how A¹-colocalization offers a new perspective to logarithmic cohomology groups. We will conclude with some geometric applications. 

Simpson gerbe and p-adic nonabelian Hodge theory (Part 2)

For a smooth rigid space X over a perfectoid field in characteristic zero, we define the Simpson gerbe for X and study its geometry properties. We will then explain how Cartier-duality and Fourier-Mukai theory for it realize the p-adic Simpson correspondence. In case X has a smooth integral model, this is also closely related to a similar picture for the Hodge-Tate stack of the model. The talks are based on joint work in progress with Bhargav Bhatt.

The sheared Hodge-Tate stack.

Let X be a smooth formal scheme X over O_C.  I will explain

an integral interpolation between the two constructions:

(1) The p-adic Simpson correspondence between the category of

pro-etale vector bundles over the generic fiber of X and the category

of

twisted Higgs bundles over the same space;

(2) The category of modules with integrable connection over the

special fiber of X and the category

twisted Higgs bundles over the same scheme.

Concretely, I will construct  a G_m-gerbe over the cotangent bundle

T^*X to X, whose restriction to the special fiber is represented by

the algebra of differential operators on the special fiber of X and

whose restriction to the generic fiber is the Simpson gerbe.

The talk is based on a joint work  with Bhatt, Kanaev, Mathew, and Zhang.