2025 Simons Collaboration on Perfection in Algebra, Geometry and Topology Annual Meeting

The second annual meeting of the collaboration brought together researchers formally affiliated with the collaboration as well as mathematicians with related research interests. Reflecting the breadth of the collaboration, the meeting covered a range of topics in algebraic geometry, number theory, and topology. It was a lively event with many informal interactions, in addition to the scheduled presentations.

Attendees included 89 in-person participants and an additional 15 remote participants. The speakers included both collaboration members as well as other individuals covering relevant research, including areas with potential future applications of perfection techniques. The meeting opened with a talk by Toby Gee on prismatic 𝐹-gauges and their applications to Galois representations in number theory. Classical work in the area has focused on representations with ℚ_𝑝-coefficients where the notions of crystalline, semistable, and de Rham representations were introduced by Fontaine. With ℤ_𝑝-coefficients much less is known.

Gee presented new results, joint with Bhatt and Kisin, proving a structure theorem for the Breuil–Kisin modules attached to residual representations of crystalline representations, and explained their new approach using prismatic 𝐹-gauges.

This talk was followed by presentations discussing perfection techniques from a number of different perspectives.

Karl Schwede discussed new approaches to singularity theory in mixed characteristic. Modeled on the theory of 𝐹-pure singularities for varieties over positive characteristic fields, he explained a new notion perfectoid pure singularities for varieties over ℤ_𝑝, and discussed new results and expectations relating perfectoid pure singularities to log canonical singularities in the sense of the minimal model program.

Tomer Schlank discussed an application of perfection techniques in chromatic homotopy theory. He explained how the computation of a basic invariant, Hopkins’ Picard group, for 𝐾(𝑛)-local spectra can be reduced to a cohomological calculation in 𝑝-adic geometry that Schlank and coauthors completed. Key inputs came from the work of Colmez and Niziol and their joint work with Dospinescu.

The meeting featured two talks about results in more classical number theory obtained using perfectoid methods. The Langlands correspondence relates Galois representations to automorphic forms and is expected to be realized through the cohomology of sheaves on Shimura varieties. Mingjia Zhang discussed her ongoing work with Caraiani and Hamann on the intersection cohomology of certain Igusa stacks and minimal compactifications of such stacks. She also explained how this can be related to the cohomology of Shimura varieties of abelian type and the Langlands correspondence. The cohomology of Shimura varieties also played a key role in the result presented by Naomi Sweeting. The Bloch–Kato conjecture discussed here relates the order of vanishing of certain L-functions to other invariants called Selmer ranks. Sweeting explained her new results affirming this conjecture.

for Galois representations arising from automorphic forms on the group 𝐺𝑆𝑝(4), and the key inputs from perfectoid methods for the construction of cohomology classes used in the arguments.

The talk of Tasho Kaletha was also about the local Langlands correspondence, but from a more classical group-theoretic viewpoint. He surveyed recent results on the formulas for certain key functions arising in the local Langlands correspondence and the connection with the stack of 𝐺-bundles on the Fargues–Fontaine curve, thereby relating it to perfectoid geometry.

In a different direction Alberto Vezzani discussed 𝑝-adic cohomology from the perspective of motives and 𝐀¹-homotopy theory in the style of Voevodsky and Ayoub. He outlined his key foundational work on a category of 𝑝-adic analytic motives and explained how this can be used to obtain a number of results about 𝑝-adic cohomology theories. This includes defining relative rigid cohomology and giving a new construction of the Hyodo–Kato operator in 𝑝-adic Hodge theory.

The meeting concluded with a talk by Jacob Lurie who explained a vast new vision for prismatic homotopy theory. Current prismatic theory is a cohomological machinery and Lurie explained a framework for analogous prismatization machinery in the context of 𝐾-theory, topological Hochschild homology, and other homotopical invariants.

Toby Gee
Imperial College London

Applications of F-gauges

Toby Gee will present a survey talk about Bhatt–Lurie and Drinfeld’s theory of F-gauges and their applications in the work of various authors.
Tasho Kaletha
University of Bonn

Harish–Chandra Characters and the Langlands Conjectures

A fundamental theorem of Harish–Chandra assigns to an irreducible complex-valued representation of a reductive real or p-adic group a conjugation-invariant function, called its character, which uniquely determines the representation. For p-adic groups, the character has a reflection in the sheaf theory on the space of G-bundles on the Fargues–Fontaine curve.

For discrete series representations, explicit formulas for these functions were obtained for real groups by Harish–Chandra and play an important role in the theory, but for p-adic groups such formulas have become available in sufficient generality only in the last few years. Tasho Kaletha will survey these formulas, the striking parallel between the real and p-adic cases, and how this enables both a unique characterization, as well as a construction, of the refined local Langlands correspondence under assumptions on the base field.
Jacob Lurie
Institute for Advanced Study

Prismatic Stable Homotopy Theory
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One of the most powerful approaches to the study of algebraic K-theory is the use of trace methods: that is, approximations of K-theory by more computable invariants (such as Hochschild homology). In this talk, Jacob Lurie will describe a (conjectural) extension of this methodology to other cohomological invariants in algebraic geometry.
Tomer Schlank
University of Chicago

On Hopkins’ Picard Group

This talk focuses on the study of the Picard group of K(n)-local spectra, a structure that plays a key role in chromatic homotopy theory. We shall survey how these groups arise in stable homotopy theory and then discuss recent progress on their computation. Specifically for every height n, we give a complete computation of these groups of all large enough primes. Our approach uses recent advances in p-adic geometry to reformulate the problem in terms of Drinfeld’s symmetric space, enabling us to apply results from a paper by Colmez–Dospinescu–Nizioł to resolve it. This is a joint work with Tobias Barthel, Nathaniel Stapleton, and Jared Weinstein.
Karl Schwede
University of Utah

Perfectoid Pure Singularities

In characteristic p, a ring is F-pure if the Frobenius map is pure (for example, split). In this talk we will discuss what it means if a ring has a pure (i.e., split) map to a perfectoid ring (such rings we call perfectoid pure). Inspired by the fact that F-pure singularities are the (conjectured) analog of log canonical singularities, we show that perfectoid pure singularities are log canonical in some cases. We will also explore examples and other properties of perfectoid pure singularities.
Naomi Sweeting
Princeton University

On the Bloch–Kato Conjecture for Some Four-Dimensional Symplectic Galois Representations

The Bloch–Kato Conjecture predicts a relation between Selmer ranks and orders of vanishing of L functions for certain Galois representations. In this talk, Naomi Sweeting will describe new results towards this conjecture in ranks 0 and 1 for the self-dual Galois representations that come from Siegel modular forms on GSp(4) with parallel weight (3,3). The key step is a construction of auxiliary ramified Galois cohomology classes, which then give bounds on Selmer groups; the ramified classes come from level-raising congruences and the geometry of special cycles on Siegel threefolds.

In the talk, Sweeting will explain the key role of torsion-vanishing results for cohomology of Shimura varieties, due to Caraiani–Scholze, Koshikawa, and Lee–Hamann.
Alberto Vezzani
Università degli Studi di Milano

P-Adic Cohomologies Using Homotopy Theory

By drawing parallels to classical work by Monsky–Washnitzer, Elkik, Arabia, and others, we motivate the study of (non-Archimedean) motivic homotopy theory by showing that it can be used to define/re-define rational p-adic cohomology theories and prove new results about them. For example, we show how to define relative rigid cohomology and deduce finiteness properties for it (joint work with V. Ertl), as well as how to define the Hyodo–Kato and the limit Hodge cohomology using motivic nearby cycles, thus getting an associated “motivic” Clemens–Schmid chain complex (joint work with F. Binda and M. Gallauer).
Mingjia Zhang
Princeton University

Intersection Cohomology of Shimura Varieties

The cohomology of Shimura varieties is an important object in number theory, since it provides a geometric setup for the Langlands correspondence over number fields. For noncompact Shimura varieties, compared to the usual or compact support cohomology, the intersection cohomology has a cleaner relation to automorphic forms through its relation to L2 cohomology. We discuss results on the intersection cohomology of Shimura varieties in the context of the Fargues–Scholze local Langlands correspondence. This is based on joint work in progress with Ana Caraiani and Linus Hamann.