ReMp-AHT Seminar

This is the webpage for the weekly seminar on Remedial Mathematics for \(p\)-Adic Hodge Theory, a learning seminar that covers background material needed to better understand and appreciate Bhargav Bhatt's course Topics in Arithmetic Geometry: \(p\)-adic Hodge theory taught at Princeton in Fall 2025.

Logistics

Talk Titles

1. Dhruv: Organizational Meeting and a Review of \(p\)-adic Formal Schemes

   • Logistics: 1100-1200 on Monday 09/08 in Fine Hall Common Room
   • Abstract: We will finalize the meeting time and place, discuss topics to be covered, and distribute the talks. If time permits, we will quickly review the theory of formal schemes from Hartshorne §II.9.

2. Michael: Derived Categories and the Derived \(p\)-Completion

   • Logistics: 1100-1200 on Monday 09/15 in B04
   • Abstract: What is a derived X? We shall find out! We will explain how to take an intersection in a derived sense, and how to form the derived completion of a module, and hopefully help you not fear the adjective derived.
   • Handout: Can be found here.

3. Kenta: What the **** is a stack and why should I care?

   • Logistics: 1100-1200 on Monday 09/22 in B04
   • Abstract: Schemes are the basic object of study in algebraic geometry, which are usually defined as ringed spaces. We introduce an alternative formalism, the so-called functor-of-points perspective on schemes. This motivates the definition of stacks, which are certain functors on rings. The main insight of Grothendieck (please correct my history!) was that one can really think of such functors as geometric objects, and develop the theory of sheaves on them.
   • Handout: Can be found here.

4. Hari: Witt Vectors and \(\delta\)-Rings

   • Logistics: 1100-1200 on Monday 09/29 in B04
   • Abstract: We motivate and define \(p\)-typical Witt vectors classically, as a useful way to study \(p\)-adic fields. We will also introduce what a \(\delta\)-ring is and how they relate to Witt vectors via a more geometric interpretation, time permitting.
   

5. Frankie: Perfectoid Rings

   • Logistics: 1100-1200 on Monday 10/06 in B04
   • Abstract: According to MAT 517, a perfectoid ring is a quotient of a perfect prism by its distinguished ideal. My aim for today is to discuss an alternative (one might even say intrinsic) characterisation of perfectoid rings. Time permitting, the tilting equivalence and Tate perfectoid rings should also make an appearance.
   

6. Quanlin: Adic Spaces I

   • Logistics: 1100-1200 on Monday 10/27 in B04
   • Abstract: We give a rapid overview of the theory of adic spaces most relevant to Bhargav's class, topics including basic definitions, analytic adic spaces, rigid spaces, separated and proper morphisms, the generic fiber construction, the étale site, and overconvergent sheaves.
   

7. Inés: Almost Mathematics

   • Logistics: 1100-1200 on Monday 11/03 in B04
   • Abstract: We will construct the almost category, and give some basic properties. Time permitting we will prove a first case of the almost purity theorem.
   

8. Frankie [TBD]: All Things Perfectoid II

   Abstract: About perfectoid spaces. [TBD]

9. Quanlin [TBD]: Adic Spaces II

   Abstract: [TBD]

10. Meenakshi: Higgs Bundles and the Classical Non-Abelian Hodge Correspondence over \(\mathbb{C}\)

    Abstract: [TBD]

11. [TBD]: Non-Abelian Hodge Theory in Characteristic \(p\) à la Ogus-Vologodsky

    Abstract: [TBD]

Resources

1. Lectures notes on The Hodge-Tate Decomposition via Perfectoid Spaces for lectures given by Bhatt at the Arizona Winter School 2017.
2. CMI Summer School Notes on \(p\)-Adic Hodge Theory by Brinon-Conrad.
3. Notes on prismatic cohomology by Kedlaya.
4. Several Approaches to Non-Archimedean Geometry by Conrad.
5. Theory of \(p\)-adic Galois Representations by Fontaine-Ouyang. This contains a lot of foundational stuff in the first chapter.

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