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: 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• Handout: Can be found here.
• 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.
• Logistics: 1100-1200 on Monday 11/10 in B04
• Abstract: Perfectoid spaces are adic spaces locally modeled on perfectoid rings, as introduced by Frankie. Any adic space has a pro-étale cover which is perfectoid, so can be recovered using descent data. Moreover, tilting provides a functor from perfectoid spaces to perfectoid spaces in characteristic \(p\), which “preserves all topological information.” Thus, tilting allows us to translate difficult questions in mixed characteristic to questions in characteristic \(p\). For a perfectoid space in mixed characteristic, tilting amounts to “forgetting the structure morphism to \(\operatorname{Spa}(\mathbb{Z}_p)\),” so one is tempted to say perfectoid spaces in mixed characteristic are equivalent to “perfectoid spaces in characteristic \(p\) with a morphism to \(\operatorname{Spa}(\mathbb{Z}_p)\).” We will introduce diamonds, which is a category in which all the air quotes can be made precise.
• Logistics: 1100-1200 on Monday 11/24 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. (Continued from last time.)
• Handout: Can be found here.
• Logistics: 1200-1300 on Wednesday 12/03 in B04
• Abstract: We've learned a lot of fancy machinery like perfectoid spaces and almost mathematics, but in this talk we'll be stepping back to learn about the classical non-abelian Hodge theory over (gasp) $\mathbb{C}$! (But don't worry, Michael will do a characteristic $p$ version next week!) After reviewing some important concepts in Hodge theory, this talk will introduce $\lambda$-connections and Higgs bundles, and sketch the ideas behind Simpson's correspondence relating representations of the fundamental group with certain Higgs bundles. From there we'll learn about Hitchin fibrations, and, time permitting, possibly introduce period maps.
• Logistics: 1100-1200 on Monday 12/08 in B04
• Abstract: Over \(\mathbb{C}\), the Riemann-Hilbert correspondence tells us that an ODE is equivalent to its local system of solutions. Generalizing this, last time Meenakshi taught us about the non-abelian Hodge correspondence; this is a certain isomorphism between the moduli stack of local systems on a curve \(X\) and the moduli stack of ODEs on that same curve \(X\).
In characteristic \(p\), a similar story emerges. We will start with Cartier descent, which is a positive characteristic variant of the Riemann-Hilbert correspondence, and then discuss how to extend it to a non-abelian Hodge correspondence.