2026 Simons Collaboration on Perfection in Algebra, Geometry and Topology Annual Meeting

The 2026 annual meeting of the collaboration brought together 107 in-person participants and 27 online participants for two days of talks and discussion. The event featured a mixture of talks by collaboration PIs (Česnavičius, Emerton, Nizioł, Olsson) and other speakers (Chan, Hahn, Petrov, and Tsimerman) on a variety of topics related to the collaboration’s research focus. The meeting was lively with many informal discussions.

The meeting opened with Collaboration Director Martin Olsson providing an overview of the motivation and goals of the collaboration, progress achieved during the first 2.5 years of the collaboration, and current and future research. He highlighted the importance of the collaboration in fostering interactions among researchers in the field, and the research impacts of these collaborations.

This was followed by a talk by Alexander Petrov who discussed joint work of his with Heuer and Vologodsky on an analogue of the Riemann-Hilbert correspondence for t-connections on adic spaces. Their theory gives a lifting of the p-adic Simpson correspondence of Faltings and Heuer to the period ring B⁺_dR.

The following talk by Charlotte Chan concerned representation theory of p-adic groups and realizing their representations through geometry. Professor Chan began by motivating the general theory with more classical results on characterizing irreducible representations by the traces on certain elements, after which she discussed very recent results due to her and coauthors on Deligne–Lusztig theory. Connections with local Langlands was also briefly discussed.

Jeremy Hahn discussed his joint work with Ishan Levy and Andrew Senger. One of the important motivating properties of crystalline cohomology of a p-adic scheme is that it only depends, in a functorial manner, on the reduction modulo p. Professor Hahn discussed variants of this result for other theories, such as prismatic, syntomic, and ´etale cohomology, that imply that these theories depend functorially on the reduction modulo pⁿ for a fixed n.

The last talk of the first day was by Matthew Emerton who gave a survey of recent developments in the p-adic Langlands program. He motivated the general modern theory with a discussion of GL₂ and the classical theory of modular form and towers of modular curves. He then connected this with recent work on more general locally symmetric spaces and their cohomology and the modularity of Galois representations.

One of the great results of the late 20th century is the proof by Faltings of the Shafarevich conjecture, which states that there are only finitely many isomorphism classes of abelian varieties over a number field with good reduction outside a fixed finite set of places. The second day began with Jacob Tsimerman discussing his joint work with Bakker and Shankar generalizing this result to the setting of Shimura varieties.

Wiesława Nizioł gave a survey of developments in analytic geometry, with an eye towards applications to p-adic Langlands. Professor Nizioł discussed the p-adic proétale cohomology of analytic varieties illustrating the general theory with key examples.

The meeting concluded with a talk by Kęstutis Česnavičius about his joint work with Bouthier and Scavia. He discussed recent progress on classical questions of Grothendieck-Serre about trivializing torsors in the Zariski topology. Basic examples include the classical theory of Brauer groups as well as results about twists of varieties. The proofs require extensive work on pseudo-finite, pseudo-complete, and pseudo-proper groups and rests on broadly-applicable purity results for torsors under such groups.

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

Kęstutis Česnavičius
CNRS, Sorbonne Université

Generically Trivial Torsors under Algebraic Groups

For a smooth variety X over a field k and a smooth k-group scheme G, Grothendieck and Serre predicted that every generically trivial G-torsor over X trivializes Zariski locally on X. In this talk, Kęstutis Česnavičius will explain a resolution of the Grothendieck–Serre problem, the main new case being when k is imperfect, in which pseudo-reductive and quasi-reductive groups play a central role. The argument is built on new purity and extension theorems for torsors valid for pseudo-finite, pseudo-proper, and pseudo-complete groups, and it also rests on several other new results on algebraic groups over imperfect fields. The talk is based on joint work with Alexis Bouthier and Federico Scavia.
 

Charlotte Chan
University of Michigan

Positive-Depth Deligne–Lusztig Varieties and Character Sheaves

Representation theory and the geometry of flag varieties are deeply intertwined. For finite groups of Lie type, Deligne and Lusztig’s breakthrough work in 1976 defined Frobenius-twisted versions of flag varieties whose cohomology realizes all representations of these groups. Lusztig’s theory of character sheaves further revolutionized the subject in the 1980s, yielding a perverse-sheaf-theoretic basis of the vector space of class functions. In the last quarter-century, generalizations of Deligne-Lusztig varieties and character sheaves have allowed us to study representations of p-adic groups using explicit geometric methods. In this talk, Charlotte Chan will describe recent advances in this subject and their relationship to the Langlands program.
 

Matthew Emerton
University of Chicago

Developments in the p-Adic Langlands Program

Matthew Emerton will give an overview of some recent developments in the p-adic Langlands program. Along the way, Emerton hopes to provide motivation for the program, both from various more “classical” problems in number theory, such as questions of local-global compatibility in the cohomology of locally symmetric spaces, and problems of modularity or automorphy of Galois representations, and from the geometric perspective of Fargues–Scholze.
 

Jeremy Hahn
Massachusetts Institute of Technology

Crystallinity and Redshift

The “crystalline miracle” states that the de Rham cohomology of a smooth p-adic formal scheme X is functorially determined by its mod p reduction \(X_{p=0}\): namely, it is isomorphic to the crystalline cohomology of \(X_{p=0}\). For many other invariants, such as prismatic cohomology, syntomic cohomology, or algebraic K-theory, we study the extent to which the invariant is determined by \(X_{p^n=0}\) for any fixed n. This is joint work with Ishan Levy and Andrew Senger.
 

Wiesława Nizioł
CNRS, Sorbonne Université

Cohomology of the Drinfeld Tower for \(GL_2(Q_p)\)

Wiesława Nizioł will present what we know about p-adic pro-étale and étale cohomology of the Drinfeld tower in dimension 1 and its relationship to p-adic local Langlands correspondence for \(GL_2(Q_p)\). Based on a joint work with Pierre Colmez and Gabriel Dospinescu.
 

Martin Olsson
University of California, Berkeley

Simons Collaboration on Perfection: Progress and Next Steps
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Martin Olsson will review some of the main goals and activities of the collaboration, and highlight achievements of the first 2.5 years. Olsson will then introduce key mathematical ideas through particular results in the field, which will also set the stage for the lectures that follow, and discuss future goals.
 

Alexander Petrov
Massachusetts Institute of Technology

On Riemann-Hilbert Correspondence for \(\mathbb{B}_dR^+\)-Local Systems

Given a smooth proper adic space \(X\) over \(B_dR^+\), one expects a relation between vector bundles with a t-connection on \(X\) and local systems over the period sheaf \(\mathbb{B}_dR^+\) on the pro-étale site of rigid-analytic variety \(X\times_{B_dR^+}C\). In this talk, Alexander Petrov will discuss the extent to which such a relation is known. In particular, the choice of an exponential on \(C\) provides an equivalence between rank 1 objects in these categories, lifting Faltings–Heuer’s Simpson correspondence (for line bundles). This talk is based on joint works with Ben Heuer and Vadim Vologodsky.
 

Jacob Tsimerman
University of Toronto

Geometric Shafarevich Conjecture for Exceptional Shimura Varieties

The Shafarevich conjecture is concerned with finiteness results for families of \(g\)-dimensional principally polarized abelian varieties over a base \(B\). Famously, Faltings settled the arithmetic case of \(B=O_{K,S}\). In the case where \(B\) is a curve over a finite field, finiteness can never be true as one may always compose with Frobenius. In this setting, we show that one may recover the theorem by considering families up to p-power isogenies.

We formulate an analogous statement for Exceptional Shimura varieties \(S\), and describe a proof in the generically ordinary setting. The formulation itself requires the development of integral canonical models of such varieties, which we also discuss.

This is joint work with Ben Bakker and Ananth Shankar.

Watch a playlist of all presentations from this meeting here.