2024 Simons Collaboration on Perfection in Algebra, Geometry and Topology Annual Meeting

The annual meeting brought together mathematicians with a range of expertise related to the collaboration. It featured both participants currently affiliated with the collaboration, as well as individuals with related research interests. The meeting had a lively atmosphere with substantial informal interactions.

Attendees included 102 in-person participants and additional 14 remote participants. The speakers included both collaboration members as well as other speakers reporting on research of relevance to the collaboration goals.

The talks of Bhargav Bhatt and Akhil Mathew focused on recent foundational developments in the subject. Bhatt discussed a new approach to the 𝑝‑adic Simpson correspondence, which relates vector bundles on the pro-étale site of a smooth 𝑝‑adic variety to Higgs bundles, using the so-called Simpson gerbe. The introduction of this gerbe allows for a generalization of prior work in the subject and explains various dependencies on choices of prior constructions. Mathew discussed new approaches to Dieudonné theory using the prismatic theory. Mathew explained how this gives an understanding of 𝑝-divisible groups and related objects in terms of prismatic 𝐹-gauges, which are sheaves on a certain stack. He also discussed applications related to flat cohomology.

The other six talks discussed applications of perfection techniques to related parts of mathematics.

The talk of Ben Antieau concerned applications to the calculation of algebraic K‑theory of rings of the form 𝑂_𝐾/𝝕^𝑛, where 𝐾 is a finite extension of 𝐐_𝑝, 𝝕 is a uniformizer and 𝑛 ≥ 1. Antieau explained how he and coauthors have developed algorithms using prismatic cohomology to calculate these 𝐾-groups producing tables enumerating them. Prior to this work only various low degree cases were understood.

Jakub Witaszek discussed applications to commutative algebra and birational geometry in mixed characteristic. In particular, he discussed progress on a key problem about the localization properties of certain test ideals in mixed characteristic, which measure the singularities of a ring. Using the 𝑝‑adic Riemann-Hilbert correspondence of Bhatt and Lurie, he explained that the desired properties of these test ideals hold up to small perturbation.

Applications of perfection techniques to automorphic forms appeared in several of the talks and were the subject of the presentations of Ana Caraiani and Si Ying Lee. Caraini talked about the construction of families of 𝑝‑adic Siegel modular forms and comparisons between two different approaches. Such a comparison would realize an Eichler-Shimura type decomposition for 𝑝‑adic Siegel modular forms. Lee discussed recent developments on the mod-ℓ cohomology of local Shimura varieties establishing a vanishing conjecture for these groups. A key ingredient is the local Langlands correspondence of Fargues-Scholze and new geometric methods for understanding sheaves on the stack of 𝐺-bundles on the Fargues-Fontaine curve.

Juan Esteban Rodríguez Camargo discussed recent developments in the theory of analytic stacks, with an eye towards applications to local Shimura varieties. He discussed how the solid formalism gives rise to analytic versions of the Hodge-Tate and de Rham stacks, as well as giving rise to a theory of locally analytic representations in this context.

The meeting concluded with a talk of Dustin Clausen concerned the connection between perfectoid geometry and chromatic homotopy theory in topology. He discussed the resolution of a conjecture about Galois descent of a certain localized form of algebraic 𝐾‑theory, and how perfection techniques enter into the proof of this topological result.

Benjamin Antieau
Northwestern University

Prismatic Cohomology Relative to Delta-Rings
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Benjamin Antieau will report on joint work with Krause and Nikolaus on prismatic cohomology relative to delta-rings and its role in our machine computations of K-groups.
 

Bhargav Bhatt
IAS & Princeton University

The 𝑝‑adic Simpson Correspondence

Given a smooth rigid variety X over a perfectoid 𝑝‑adic field C, Bhargav Bhatt will explain that the cotangent bundle T*X carries a natural (\’etale) G_m-gerbe P(X) — which we call the Simpson gerbe — whose coherent sheaf theory realizes a 𝑝‑adic Simpson correspondence: Higgs bundles on X twisted by P(X) identify naturally with the generalized local systems of Faltings (nowadays understood as pro-étale vector bundles on X). Any C-local system on X has a canonically attached P(X)-twisted Higgs bundle. From this perspective, several previous results in this topic can be understood as trivializing the gerbe P(X) over suitable loci in T*X. Time permitting, Bhatt will explain an analogous picture for 𝑝‑adic formal schemes; this picture, which partially motivated the discovery of P(X), naturally interpolates between the rigid-analytic story on the generic fibre and the Ogus-Vologodsky correspondence in characteristic p.

This is joint work, very much in progress, with Mingjia Zhang; it is inspired by several recent works in this area (especially that of Ben Heuer).
 

Ana Caraiani
Imperial College London and Bonn

Towards an Eichler-Shimura Decomposition for Ordinary 𝑝‑adic Siegel Modular Forms

There are two different ways to construct families of ordinary 𝑝‑adic Siegel modular forms. One is by 𝑝‑adically interpolating ordinary classes in Betti cohomology, first introduced by Hida and then given a more representation-theoretic interpretation by Emerton. The other is by 𝑝‑adically interpolating ordinary classes in coherent cohomology, once again pioneered by Hida and generalised in recent years by Boxer and Pilloni. Ana Caraiani will explain these two constructions and then discuss joint work with James Newton and Juan Esteban Rodríguez Camargo, very much in progress, that aims to compare them.
 

Dustin Clausen
IHES

Perfection in Chromatic Homotopy Theory

Dustin Clausen will introduce the circle of ideas surrounding the “Chromatic Nullstellensatz” of Burklund-Schlank-Yuan, which opens the door to the use of perfection in chromatic homotopy theory. Then Clausen will describe an application of this, namely joint work with Robert Burklund where we settle the general case of the Ausoni-Rognes Galois descent conjecture in chromatic algebraic 𝐾‑theory, completing the substantial progress made in previous joint work with Mathew-Naumann-Noel.
 

Akhil Mathew
University of Chicago

F-gauges and Barsotti-Tate Groups

Akhil Mathew will give an exposition of prismatic Dieudonné theory, as developed by Anschütz and Le Bras, and some generalizations, in the setting of prismatic F-gauges.

Joint work with Keerthi Madapusi and Zachary Gardner.
 

Juan Esteban Rodríguez Camargo
Columbia University

𝑝‑adic D-modules and Shimura Varieties

In this talk, Juan Esteban Rodríguez Camargo will explain how the recent developments in the theory of analytic geometry and 𝑝‑adic D-modules can be applied to the study of 𝑝‑adic automorphic forms of (local) Shimura varieties.
 

Jakub Witaszek
Princeton University

Singularities in Mixed Characteristic via the Riemann-Hilbert Correspondence

In this talk, Jakub Witaszek will discuss how one may study mixed characteristic singularities using the 𝑝‑adic Riemann-Hilbert correspondence.

This talk is based on joint work with Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi, Karl Schwede, Kevin Tucker and Joe Waldron.
 

Si Ying Lee
Stanford University

Torsion Vanishing for Some Shimura Varieties

Si Ying Lee will discuss joint work with Linus Hamann on generalizing the torsion-vanishing results of Caraiani-Scholze and Koshikawa for the cohomology of Shimura varieties. Si Ying Lee will apply various geometric methods to understand sheaves on Bun_G, the moduli stack of G-bundles on the Fargues-Fontaine curve. The method showcases that the behavior of the torsion cohomology localized at a semi-simple L-parameter is related to the perversity of Hecke eigensheaves with Hecke eigenvalue given by that parameter. If time permits, Si Ying Lee will also discuss generalizations of the torsion-vanishing conjecture.

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