Time: Monday 2:30- 3:50 pm,  Zoom link (please contact the organizers for the link)

Room:  SCHM 103 (Purdue)

Schedule and Plan:  

Here is the Introduction and  Plan 

1. Jan 12:  Overview: Explain the main goal of this semester. Speaker: Shubhodip Mondal (Purdue)

            Title: Overview of prismatic F-gauges.

           Abstract: This talk will be an introduction and a brief overview of prismatic F-Gauges and their connection to p-adic cohomology theories.

2. January 22 (Note the changed time: 4:00-5:20 pm)   Prismatic cohomology. Speaker: George Nicolas Diaz-Wahl (Purdue) 

Title: Overview of Prismatic Cohomology

Abstract: We will define the notion of prisms and prismatic cohomology, emphasizing the examples of crystalline, perfect, and Breuil-Kisin prisms. We will also review the Nygaard filtration on prismatic cohomology in the quasi-syntomic setting

3. Jan 26. Review Stacks and anything needed for Chapter 2. Speaker Mansimar Singh (Purdue)

Title: Stacks

Abstract : The goal of this talk is to introduce stacks with a focus on quotient stacks, especially the classifying stack BG. We begin with prestacks, or categories fibered in groupoids, and work through concrete examples. To understand quotient prestacks, we will introduce G-torsors and discuss their local triviality. We then explain how to pass from prestacks to stacks via descent theory, recalling Grothendieck topologies and focusing on the fpqc topology as the correct notion for descent. Throughout, the emphasis will be on motivation and examples, leading to a conceptual understanding of classifying stack BG.

4. Feb 2. Review the main results from Chapter 2. Speaker: Zichuan Wang (IU)

Title: Stacky approach to filtered de Rham cohomology

Abstract: Following chapter 2, we review quasicoherent sheaves on quotient stacks and representations of formally completed vector bundles, with examples for A^1/G_m and the de Rham stack A^{1,dR}. We then review the realization of (filtered) de Rham cohomology via stacks obtained by "transmutation". Over a perfect field k of char. p, we discuss combining the Hodge and conjugate filtered stacks by gluing their Witt vector models, which serves as a “mod-p prelude” to the Nygaard/Syntomic constructions in later chapters.

5.  Feb 9. §3.1: The prismatization over k. Speaker: Matthias Strauch (IU)

Title: The prismatization over k

Abstract: We will cover some of the material in sections 3.1 and 3.2 in Bhatt's paper on Prismatic F-gauges. We start with the prismatization construction in 3.1.1. Then we discuss the definition of the Nygaard filtration following Theorem 3.2.1 (without proof). Finally, our goal is to explain the actual construction of the Nygaard filtration using the de Rham Witt complex following Def. 8.1 in the paper "Topological Hochschild Homology and integral p-adic Hodge Theory" by Bhatt, Morrow, and Scholze (https://arxiv.org/pdf/1802.03261).

6. Feb 16. §3.3 The (Nygaard) filtered prismatization. Speaker: Zhilin Luo (Purdue)

7. Feb 23. §3.4 Gauges over k. Speaker: Ruipeng Zou (Purdue)

8. Mar 2. §4.1, §4.2. Speaker

References and useful links (also see the website of learning seminar during Fall 2025 )

Prismatic F-gauge (Bhatt's notes for a class taught at Princeton, which will be our main reference)
Prisms and Prismatic Cohomology (Bhatt and Scholze's paper on prismatic cohomology)
Prismatization (Drinfeld's article on the stacks \Sigma, \Sigma', \Sigma'', also called ``prismatization", ``Nygaard filtered prismatization" and ``syntomification" of Z_p)
Absolute prismatic cohomology (Bhatt and Lurie's article on ``prismatization", which is called Cartier-Witt stack in their paper)
Dieudonn\'e theory via cohomology of classifying stacks and prismatic F-gauges (Mondal's article containing an approach to prismatic F-gauges via quasisyntomic descent)
An algebraicity conjecture of Drinfeld and the moduli of p-divisible groups (article of Gardner-Madapusi containing an exposition of the stacks we need equipped with suitable derived structures)
Prismatic F-gauges and a result of T. Liu (containing a summary of the stacks we need)