2025 Simons Collaboration on Perfection in Algebra, Geometry and Topology Annual Meeting
Meeting Goals:
The Simons Collaboration on Perfection in Algebra, Geometry and Topology Annual Meeting will feature lectures by experts in the collaboration’s areas of focus highlighting exciting recent developments and providing an overview of the most pressing problems in the area and directions for the future. Topics to be covered include many aspects of mixed characteristic geometry including connections with algebraic K-theory, commutative algebra, and number theory. Speakers include both regular participants in collaboration activities as well as researchers with relevant expertise and results, and the meeting attendees will include a number of established mathematicians as well as a large number of mathematicians entering the field more recently. In addition to featuring talks by leading researchers, the meeting will provide many opportunities for informal interactions.
Speakers
Toby Gee, Imperial College London
Tasho Kaletha, University of Michigan
Jacob Lurie, Institute for Advanced Study
Tomer Schlank, Hebrew University of Jerusalem
Karl Schwede, University of Utah
Naomi Sweeting, Princeton University
Alberto Vezzani, Università degli Studi di Milano
Mingjia Zhang, Princeton University
Previous Meeting:
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Thursday, March 13, 2025
9:30 AM Toby Gee | Applications of F-gauges 11:00 AM Mingjia Zhang | Intersection Cohomology of Shimura Varieties 1:00 PM Alberto Vezzani | P-Adic Cohomologies Using Homotopy Theory 2:30 PM Tomer Schlank | On Hopkins' Picard Group 4:00 PM Karl Schwede | Perfectoid Pure Singularities Friday, March 14, 2025
9:30 AM Tasho Kaletha | Harish-Chandra Characters and the Langlands Conjectures 11:00 AM Naomi Sweeting | On the Bloch-Kato Conjecture for Some Four-Dimensional Symplectic Galois Representations 1:00 PM Jacob Lurie | Prismatic Stable Homotopy Theory -
Toby Gee
Imperial College LondonApplications 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 BonnHarish–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 StudyPrismatic Stable Homotopy Theory
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 ChicagoOn 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 UtahPerfectoid 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 UniversityOn 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 MilanoP-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 UniversityIntersection 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.