Number Theory and Representation Theory Archives for Fall 2024 to Spring 2025
The (non-uniform) Hrushovski-Lang-Weil estimates
When: Wed, September 6, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Shuddhodan Kadattur Vasudevan (IHES, France) - https://www.ihes.fr/~kvshud/
Abstract: In 1996 using techniques from model theory and intersection
theory, Hrushovski obtained a generalization of the Lang-Weil estimates.
Subsequently, the estimate has found applications in group theory,
algebraic dynamics, and algebraic geometry. We shall discuss an l-adic
proof of these estimates' non-uniform version and the rationality of the
associated generating function.
An invitation to the theory of diamonds (minicourse)
When: Mon, September 11, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Ian Gleason (University of Bonn) - https://ianandreigf.github.io/Website/
Abstract: The purpose of this talk is to discuss the notion of an Artin v-stack following Fargues--Scholze and keeping in mind the example of G-bundles on the Fargues--Fontaine curve. We will not assume the audience has background in analytic geometry and for this reason the discussion will be informal, but it will build up from the basic notions.
Ideally by the end of the talk the following questions will be answered in superficial terms:
What is an adic space?
What is a perfectoid space?
What is a proetale cover?
What is a v-cover?
What is a diamond?
What is a v-stack?
What is an Artin v-stack?
Why do number theorist care?
NOTE: the talk will continue for a second hour 3:00-4:00 pm, in Kirwan 1313.
Finiteness of the stack of p-adic local shtukas
When: Wed, September 13, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Ian Gleason (University of Bonn) - https://ianandreigf.github.io/Website/
Abstract: Rapoport-Zink spaces and local Shimura varieties have played an important role in the study of the local Langlands correspondence for reductive groups. In the Berkeley notes, Scholze and Weinstein discuss their moduli spaces of p-adic local G-shtukas that are vast generalizations of local Shimura varieties. In this talk we discuss the stack of p-adic local G-shtukas and explain its relation to the moduli problem introduced by Scholze and Weinstein. Our main theorem states that the stack of local G-shtukas (appropriately bounded) is an Artin v-stack.
Unitarity of Arthur packets for real groups
When: Wed, September 27, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Jeff Adams (IDA) -
Abstract: Arthur conjectured the existence of sets of representations of a real reductive group satisfying various properties. A definition of these "Arthur packets" was given in 1992 by Adams, Barbasch and Vogan. They proved all of the conjectural properties with the significant exception of the fact these representations are unitary.
Unitarity of "unipotent" packets is known: by work of Barbasch/Sun/Ma/Zhu for classical groups (a series of arXiv papers), for classical groups, and Adams/Miller/Vogan for exceptional groups (using the Atlas software). The general case reduces to the unipotent case. This requires a generalization of endoscopic lifting, which includes both real and cohomological induction.
Analytification and GAGA over the Fargues-Fontaine curve
When: Mon, October 16, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Kiran Kedlaya (UCSD) - https://kskedlaya.org/
Abstract: The Fargues-Fontaine curve is an exotic but important object associated
to an algebraically closed nonarchimedean field of characteristic p
(e.g., a completed algebraic closure of the field of p-adic numbers). In
fact there are two such objects, an "algebraic" object in the category
of schemes and an "analytic" object in the category of adic spaces, and
there is a form of the GAGA theorem relating coherent sheaves on the
two. After reviewing all of this in detail, we formulate a similar GAGA
theorem for proper schemes over the FF curve. The proof of this requires
some newer ingredients which adapt certain standard properties of
classical affinoid algebras to affinoid algebras over an affinoid
subspace of the FF curve. Some of these ingredients are due to my former
PhD student Peter Wear.
Rigid inner forms and the Bernstein decomposition for L-parameters
When: Wed, November 1, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Peter Dillery (UMD) -
Abstract: Aubert, Moussaoui, and Solleveld have formulated a version of the Bernstein decomposition for enhanced L-parameters. Their results are the first step in a general strategy to reduce local Langlands correspondences to the supercuspidal case. In this talk I’ll explain how to modify the theory, which uses Arthur’s enhancements of L-parameters, to fit into Kaletha’s framework of rigid inner forms. The chief technical ingredient is a more conceptual description of the generalized Springer correspondence for disconnected groups. A focus will be placed on examples and applications of this work, which is joint with David Schwein (Bonn).
Unitary representations of semisimple Lie groups and conical symplectic singularities
When: Wed, November 8, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Lucas Mason-Brown (Oxford) -
Abstract: One of the most fundamental unsolved problems in representation theory is to classify the set of irreducible unitary representations of a semisimple Lie group. In this talk, I will define a class of such representations coming from filtered quantizations of certain graded Poisson varieties. The representations I construct are expected to form the "building blocks" of all unitary representations.
Connected components in the moduli space of L-parameters
When: Mon, November 20, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Sean Cotner (University of Michigan) -
Abstract: There have been several recent approaches to defining a moduli space of L-parameters over Z[1/p], in order to obtain refined versions of the local Langlands conjecture "at all primes away from p at once". I will discuss the approach of Dat-Helm-Kurinczuk-Moss, including some basic results and examples. There is a conjectural description of the connected components of this moduli space, and I will describe recent work on this conjecture.
The affine Grassmannian as a presheaf quotient
When: Mon, November 27, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Kęstutis Česnavičius (Institut de Mathématique d'Orsay, Université Paris-Saclay) -
Abstract: The affine Grassmannian of a reductive group $G$ is usually defined as the étale sheafification of the quotient of the loop group $LG$ by the positive loop subgroup. I will discuss various triviality results for $G$-torsors which imply that this sheafification is often not necessary.
Unramified Grothendieck--Serre for isotropic groups
When: Wed, November 29, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Kęstutis Česnavičius (Institut de Mathématique d'Orsay, Université Paris-Saclay) -
Abstract: The Grothendieck–Serre conjecture predicts that every generically trivial torsor under
a reductive group $G$ over a regular semilocal ring $R$ is trivial. We establish this for unramified $R$ granted that $G$ is totally isotropic, that is, has a “maximally transversal” parabolic $R$-subgroup. We also use purity for the Brauer group to reduce the conjecture for unramified R to simply connected $G$—a much less direct such reduction of Panin had been a step in solving the equal characteristic case of Grothendieck–Serre. The talk is based on joint work with Roman Fedorov.
A geometric interpretation of H^3
When: Mon, December 4, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Libby Taylor (UMCP) -
Abstract: When $X$ is a variety, it is well understood how to interpret $H^0(X,\mathcal O^*_X)$ and $H^1(X,\mathcal O^*_X)$, which are, respectively, the group of invertible functions and the group of invertible sheaves. The group $H^2(X,\mathcal O^*_X)$ is similarly well-understood via either Azumaya algebras or $\mathbb G_m$ gerbes on $X$. A similar description for $H^3(X,\mathcal O^*_X)$, however, has been more elusive. In this work, we will explain how to provide such a geometric interpretation using what we call 2-Azumaya algebras on $X$. This is joint work with Danny Krashen.
Recent developments in p-adic geometry and the local Langlands correspondence
When: Wed, December 6, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Ian Gleason (University of Bonn) -
Abstract: The local Langlands correspondence, proposed by Langlands as the non-abelian analogue of local class field theory for non-Archimedean local fields, has been established for various reductive groups over the years. However, a construction in full generality was lacking until the recent proposal by Fargues and Scholze, who introduced semi-simplified Langlands parameters for every reductive group. Their construction employs Scholze's theory of diamonds, a modern approach to p-adic analytic geometry. This talk begins by revisiting historical developments in number theory that led to the formulation of the local Langlands correspondence. Subsequently, we explore the relevance of p-adic analytic geometry to the study of the local Langlands program and the impact these innovative methods have on our comprehension of the local Langlands correspondence.
Quantum ergodicity on the Bruhat-Tits building for PGL(3, F) in the Benjamini-Schramm limit
When: Mon, December 11, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Carsten Peterson (Paderborn University) -
Abstract: Originally, quantum ergodicity concerned equidistribution properties of Laplacian eigenfunctions with large eigenvalue on manifolds for which the geodesic flow is ergodic. More recently, several authors have investigated quantum ergodicity for sequences of spaces which "converge" to their common universal cover and when one restricts to eigenfunctions with eigenvalues in a fixed range. Previous authors have considered this type of quantum ergodicity in the settings of regular graphs, rank one symmetric spaces, and some higher rank symmetric spaces. We prove analogous results in the case when the underlying common universal cover is the Bruhat-Tits building associated to PGL(3, F) where F is a non-archimedean local field. This may be seen as both a higher rank analogue of the regular graphs setting as well as a non-archimedean analogue of the symmetric space setting.
Factorization structures in the geometric Langlands correspondence
When: Wed, December 13, 2023 - 4:00pmWhere: Kirwan Hall 3206
Speaker: Justin Campbell (University of Chicago) - https://sites.google.com/view/justincampbell/home
Abstract: The geometric Langlands program was originally inspired by number theory, but has come to incorporate ideas from homotopy theory and mathematical physics. Factorization structures are closely related to both the theory of E_n-algebras and the notion of locality in conformal field theory. In recent joint work with Sam Raskin, we established some basic equivalences in the local geometric Langlands theory using factorization methods, including a version of the derived Satake equivalence. I will explain some of the main ideas behind this work and discuss forthcoming applications.
Categorical spectra and higher Galois theory
When: Thu, December 14, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Naruki Masuda (JHU) - https://nmasuda2.github.io/
Abstract: Categorical spectra are categorifications of spectra (in the sense of homotopy theory) defined by a sequence (X_n) of pointed higher categories with identifications of the X_n and the endomorphisms of the basepoint of X_(n+1). I will give examples and explain how they carry interesting non-classical information. In particular, I will define the deeper Brauer groups and speculate on how we might proceed to understand their structure using higher Galois theory.
K-Theory of Adic Spaces
When: Wed, January 24, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Grigory Andreychev (IAS Princeton) - https://www.ias.edu/scholars/grigory-andreychev
Abstract: We will explain a new approach to the K-theory of analytic adic spaces via condensed mathematics. Its main advantage is that it allows us to define K-theory in that context as the (continuous) K-theory of a certain dualizable stable $\infty$-category associated to an analytic adic space (as opposed to previous constructions that produced the K-theory spectrum somewhat indirectly). In particular, it makes the study of its general properties much more feasible; for instance, it is possible to prove a very general descent statement without any restrictive hypotheses on the adic space. Moreover, one can prove an analog of the GRR theorem in this context.
Unitary representations of real groups and localization theory for Hodge modules
When: Wed, January 31, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Dougal Davis (University of Melbourne) -
Abstract: Wilfried Schmid and Kari Vilonen have conjectured that the unitarity of an irreducible representation of a real reductive group can be read off from a canonical filtration, the Hodge filtration. This filtration arises naturally from Beilinson-Bernstein localization and the deep theory of mixed Hodge modules on the complex flag variety. In this talk, I will explain a proof of this conjecture, obtained in recent joint work with Vilonen (arXiv:2309.13215). Our proof completes a sketch of Adams, Trapa and Vogan (based on the Atlas algorithm) by establishing two missing ingredients: a version of localization for Hodge modules and a wall-crossing theory for their Hodge filtrations. Time permitting, I may also indicate some applications to the orbit method, to appear in joint work with Lucas Mason-Brown.
The mixed characteristic geometric Casselman--Shalika formula
When: Mon, February 5, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Ashwin Iyengar (JHU) - https://ashwiniyengar.github.io/
Abstract: I will discuss joint work in progress with Milton Lin, where we attempt to geometrize the Casselman--Shalika formula over p-adic local fields. The Casselman--Shalika formula gives an explicit description of the action of the spherical Hecke algebra of a split p-adic reductive group G on the space of unramified Whittaker functions. For equal characteristic local fields, this formula was geometrized by Ngô--Polo and Frenkel--Gaitsgory--Vilonen independently in the early 2000s, by replacing a statement about functions by a statement about the cohomology of certain sheaves on the affine Grassmannian for the group G. We use Zhu's construction of the Witt vector affine Grassmannian to come up with an analogous proof in mixed characteristic.
Higher Theta Series, I
When: Tue, February 27, 2024 - 3:30pmWhere: Kirwan Hall 3206
Speaker: Zhiwei Yun (MIT) -
Abstract: See https://www-math.umd.edu/research/distinguished-lecture-series/lectures-algebra-number.html
Higher Theta Series, III
When: Thu, February 29, 2024 - 3:30pmWhere: Kirwan Hall 3206
Speaker: Zhiwei Yun (MIT) -
Abstract: See https://www-math.umd.edu/research/distinguished-lecture-series/lectures-algebra-number.html
Relative Langlands duality, past and future.
When: Mon, March 4, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Yiannis Sakellaridis (Johns Hopkins University) - https://math.jhu.edu/~sakellar/
Abstract: Since Riemann's 1859 report on the zeta function, it is known that certain automorphic L-functions can be represented as ("period") integrals, which often proves analytic properties such as the functional equation. The method was advanced by Jacquet, Piatetski-Shapiro, Rallis, and many others since the 1970s, giving rise to the "relative" Langlands program. It turns out that the relationship between periods and L-functions reflects a duality between certain Hamiltonian varieties for a reductive group and its Langlands dual group. I will set up this duality in a limited setting (joint work with David Ben-Zvi and Akshay Venkatesh), and speculate on how it might be expanded in the future.
On the strongly regular locus of the inertia stack of Bun_G
When: Wed, March 6, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Daniel Gulotta (University of Utah) -
Abstract: Langlands functoriality predicts that if two reductive groups over a p-adic field are inner forms of each other, then the representation theory of the groups should be related.
One can compare representations of the groups in the following ways:
- One can compare the trace distributions of the representations.
- One can find a "local shtuka space" that has an action of both groups, and look at how the groups act on its cohomology.
Fargues-Scholze have constructed local shtuka spaces in great generality and proved that their cohomology is not too large. One would like to relate the trace distributions of the two group actions on the cohomology. Hansen-Kaletha-Weinstein proved an identity relating the elliptic parts of the trace distributions. I will explain some progress in extending the identity beyond the elliptic parts.
On a conjecture of Pappas and Rapoport
When: Mon, March 11, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Patrick Daniels (Skidmore College) -
Abstract: Pappas and Rapoport have recently conjectured the existence of an integral model which admits a universal p-adic shtuka for any Shimura variety with parahoric level structure at p. They prove that there can exist at most one such integral model, and that such a model exists for many Hodge-type Shimura varieties. In this talk I will discuss recent progress on this conjecture in the remaining Hodge-type cases (joint with van Hoften, Kim, and Zhang) and in the abelian type case (joint with Youcis). In particular, the conjecture is now known for all Hodge-type Shimura varieties, and for all abelian-type Shimura varieties when p > 3.
K-theory of dualizable oo-categories, after Efimov
When: Wed, April 10, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Maxime Ramzi (University of Copenhagen, Denmark) - https://sites.google.com/view/maxime-ramzi-en/home?authuser=0
Abstract: In this talk, I will go over the various enlargements of the scope of higher algebraic K-theory, from rings to stable oo-categories, and motivate the introduction of this latest extension to dualizable oo-categories; as well as describe the basics of the theory of dualizable oo-categories, to indicate how to work with them.
Sen Operators and Lie Algebras arising from Galois Representations over p-adic Varieties
When: Mon, April 22, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Tongmu He (IAS Princeton) - https://sites.google.com/view/tongmu
Abstract: Any finite-dimensional p-adic representation of the absolute Galois group of a p-adic local field with imperfect residue field is characterized by its arithmetic and geometric Sen operators defined by Sen and Brinon. We generalize their construction to the fundamental group of a p-adic affine variety with a semi-stable chart, and give a canonical formation of Sen's theory independently of the choice of the chart, which is even new in the case of local fields. Our construction relies on a descent theorem in the p-adic Simpson correspondence developed by Tsuji. When the representation comes from a Q_p-representation of a p-adic analytic group quotient of the fundamental group, we describe the Lie algebra action of its inertia subgroups in terms of the Sen operators, which is a generalization of a result of Sen and Ohkubo.
Dieudonné theory via prismatic F-gauges.
When: Mon, April 29, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Shubhodip Mondal (UBC, Canada) - https://personal.math.ubc.ca/~smondal/
Abstract: In this talk, I will first describe how classical Dieudonne module of finite flat group schemes and p-divisible groups can be recovered from crystalline cohomology of classifying stacks. Then, I will explain how in mixed characteristics, using classifying stacks, one can define Dieudonné module of a finite locally free group scheme as a prismatic F-gauge (prismatic F-gauges have been recently introduced by Drinfeld and Bhatt--Lurie), which gives a fully faithful functor from finite locally free group schemes over a quasi-syntomic algebra to the category of prismatic F-gauges. This can be seen as a generalisation of the work of Anschütz--Le Bras on "prismatic Dieudonne theory" to torsion situations.
Tame Supercuspidals At Very Small Primes
When: Mon, May 6, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: David Schwein (Bonn) -
Abstract: Supercuspidal representations are the elementary particles in the representation theory of reductive p-adic groups. Constructing such representations explicitly, via (compact) induction, is a longstanding open problem, solved when p is large. When p is small, the remaining supercuspidals are expected to have an arithmetic source: wildly ramified field extensions. In this talk I’ll discuss ongoing work joint with Jessica Fintzen that identifies a second, Lie-theoretic, source of new (tame!) supercuspidals: special features of root systems at very small primes.
Semistable p-adic Galois representation in families
When: Wed, May 8, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Zijian Yao (University of Chicago) -
Abstract: Semistable representations were introduced by Fontaine to capture p-adic étale cohomology of varieties with semistable reduction. Structures of these representations play an important role in the Langlands program. In this talk, I will explain some results on how to study semistable representations in families. If time allows, I will explain a relative version of the p-adic monodromy theorem and some other relevant questions (the first part is joint work in progress with Diao, Du and Moon).
Generalizations of Hamiltonian varieties in relative geometric Langlands
When: Fri, May 17, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Sanath Devalapurkar (Harvard University) - https://sanathdevalapurkar.github.io/
Abstract: A recent article/book by Ben-Zvi, Sakellaridis, and Venkatesh proposes a framework for duality between automorphic periods and L-functions via Hamiltonian spaces. In this talk, I will review some of this picture, and describe some of my recent work on using homotopy-theoretic tools to understand this duality. These tools suggest a generalization of (relative) Langlands duality to more "exotic" coefficients; some concrete implications, for instance, include placing "quasi-Hamiltonian spaces" (and a certain generalization thereof) into this framework. Time permitting, I hope to say a bit about Steenrod and Adams operations in this context.