Number Theory and Representation Theory Archives for Fall 2025 to Spring 2026
A prismatic approach to special values of zeta functions
When: Wed, September 4, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Logan Hyslop (UCLA) - https://loganhyslop.github.io/
Abstract: Work of Schneider, Neukirch, and Milne provides formulas for integer values of zeta functions associated to smooth proper schemes over a finite field in terms of Euler characteristics on p-adic cohomology theories. Later, Geisser extended this to general finite type schemes over a finite field under the assumption of strong resolution of singularities. We will discuss a simplification of Milne’s contribution using the Nygaard filtration, as well as progress towards removing the assumption of resolution of singularities from Geisser’s proof.
Non-basic rigid packets for discrete L-parameters
When: Mon, September 9, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Peter Dillery (UMD) -
Abstract: We formulate a new version of the local Langlands correspondence for discrete L-parameters which involves (Weyl orbits of) packets of representations of all twisted Levi subgroups of a connected reductive group G through which the parameter factors and prove that this version of the correspondence follows if one assumes the pre-existing local Langlands conjectures. Twisted Levi subgroups are crucial objects in the study of supercuspidal representations; this work is a step towards deepening the relationship between the representation theory of p-adic groups and the Langlands correspondence. This is joint work with David Schwein (Bonn).
Quasisemisimple actions on reductive groups and buildings (joint with Adler and Lansky)
When: Wed, September 11, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Loren Spice (TCU) -
Abstract: In the early 21st century, Prasad and Yu proved that, for a field of characteristic exponent p ≥ 1, the identity component of the group of fixed points of a finite group $\Gamma$ of prime-to-p order acting on a reductive group $\tilde G$ is itself a reductive group; and that, for a local field of residual characteristic p, the building of the resulting fixed-point group is the fixed-point set for the action of $\Gamma$ on the building of $\tilde G$.
Applications by Adler and Lansky to lifting require analogous statements for quasisemisimple actions, which are those for which there are a Borel subgroup $\tilde B$ of $\tilde G$, and a maximal torus in $\tilde B$, that are preserved by the action of $\Gamma$. The fixed-point group remains reductive in this case, at least after smoothing. Unfortunately, the directly analogous statement about buildings is too strong; but, fortunately, the directly analogous statement about buildings is too strong, so that we can explore the interesting ways that the building of the fixed-point group can fail to fill out the full fixed-point set in the building of $\tilde G$, and conditions that recover the exact analogue of the Prasad–Yu statement.
On the Hasse-Weil zeta functions of Kottwitz simple Shimura varieties
When: Mon, September 16, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Tom Haines (UMCP) -
Abstract: Kottwitz introduced certain compact ''fake'' unitary group Shimura varieties and determined their local Hasse-Weil zeta functions at primes of good reduction. For primes where the level is arbitrarily deep, the local Hasse-Weil zeta functions were further studied in the case of signature $(1,n-1)$ (by Xu Shen), and in the case of arbitrary signature but under the assumption that the group at $p$ is a product of Weil restrictions of general linear groups, by Scholze and Shin. In this talk, I will explain joint work with Jingren Chi in which we generalize the work of Scholze-Shin by allowing the group at $p$ to be any inner form of a product of Weil restrictions of general linear groups. New phenomena arise when the group at $p$ is not quasi-split, for example a crucial vanishing property of twisted orbital integrals of the test functions at $p$.
The stable wavefront set of theta representations
When: Wed, September 25, 2024 - 2:00pmWhere: Kirwan Hall 0305 (note special room)
Speaker: Edmund Karasiewicz (NUS) -
Abstract: The Fourier coefficients of theta functions have featured prominently in numerous number theory applications and constructions in the Langlands program. For example, they play an important role in the recent work of Friedberg-Ginzburg generalizing the theta correspondence to higher covering groups. For their construction one wants to know the wavefront set of the theta representations, i.e. the largest nilpotent orbit with nonvanishing Fourier coefficient.
To investigate these Fourier coefficients, it can be valuable to study the analogous local question. In this talk we consider local depth 0 theta representations and describe how to compute their stable wavefront set. This is joint work with Emile Okada and Runze Wang.
Ultra-Galois theory, and an analogue of the Kronecker-Weber theorem for rational function fields over ultra-finite fields
When: Mon, September 30, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Dong Quan Ngoc Nguyen (UMD) - https://www-math.umd.edu/people/postdocs-and-visitors/item/1771-dongquan.html
Abstract: In this talk, I will talk about my recent work that establishes a correspondence between Galois extensions of rational function fields over arbitrary fields $\mathbb{F}_s$ and Galois extensions of the rational function field over the ultraproduct of the fields $\mathbb{F}_s$. As an application, I will discuss an analogue of the Kronecker-Weber theorem for rational function fields over ultraproducts of finite fields. I will also describe an analogue of cyclotomic fields for these rational function fields that generalizes the works of Carlitz from the 1930s, and Hayes in the 1970s. If time permits, I will talk about how to use the correspondence established in my work to the inverse Galois problem for rational function fields over finite fields.
Representations of p-adic groups: Reducing problems to the depth-zero case (Note unusual day and location)
When: Fri, November 8, 2024 - 2:00pmWhere: Kirwan Hall 1308
Speaker: Jeff Adler (American University) -
Abstract: Suppose that G is a connected reductive group over a nonarchmidean local field F. The Bernstein decomposition expresses the category of smooth representations of G(F) has a (usually infinite) product of subcategories. It has long been known that each of these subcategories is equivalent to the category of modules over some algebra. We will show that, up to isomorphism, only finitely many algebras arise, and will describe their structure. As a corollary, each such subcategory is equivalent to a category of “depth-zero” representations of a smaller group.
Elliptic Witt vectors
When: Mon, November 11, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: James Borger (Australian National University/Columbia University) - https://maths-people.anu.edu.au/~borger/
Abstract: The usual Witt vector construction can be viewed as coming from the isogenies of the multiplicative group G_m. This suggests that there might exist analogous constructions which, in the same way, come from isogenies of other commutative algebraic groups. In this talk, I'll explain how this works for elliptic curves. I'll also explain what the corresponding elliptic analogues of lambda-rings and delta-rings are. I'll finish with a conjectural elliptic, or GL_2, analogue of Chebotarev plus Kronecker-Weber.This was inspired by (and no doubt overlaps with) the work of Charles Rezk. It is joint with Lance Gurney.
Asymptotic Fundamental Lemma I
When: Mon, November 18, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Griffin Wang (IAS) - https://math.griffin.wang/
Abstract: In my recent work on a geometric proof of the endoscopic fundamental lemma forspherical Hecke algebras, there are many new features not present in its Lie
algebra analogue originally proved by B.C.~Ng\^o.
One of such new features is a
combinatorial analogue of the fundamental lemma, called the asymptotic
fundamental lemma (AsFL). Not only does it imply the
original fundamental lemma when combined with results about
multiplicative Hitchin fibrations, it also connects the mysterious restriction
functor for Kashiwara crystals with the transfer factor, hinting at
its potential role in functoriality in general.
In the first talk of this two-talk series, I will explain what fundamental lemma
is and the motivation behind. I will then discuss how to translate it into
geometry, and how it leads to the asymptotic fundamental lemma.
Asymptotic Fundamental Lemma II
When: Wed, November 20, 2024 - 2:00pmWhere: Kirwan Hall 1311
Speaker: Griffin Wang (IAS) - https://math.griffin.wang/
Abstract: This is the second talk in a two-talk series. Continuing from theprevious one, I will present some examples of asymptotic fundamental lemma and
explain how Kashiwara crystals and Langlands--Shelstad transfer factor fit in
the picture.
Poincaré Duality in p-adic Analytic Geometry
When: Tue, November 26, 2024 - 3:30pmWhere: Kirwan Hall 3206
Speaker: Bogdan Zavyalov (Princeton University) - https://bogdanzavyalov.com/
Abstract: Rigid-analytic spaces are geometric objects described by convergent power series over the field of p-adic numbers Q_p. Just as complex-analytic spaces provide a robust framework for analytic geometry over C, rigid-analytic spaces offer a natural setting for analytic geometry over Q_p. In this talk, I will give a gentle introduction to the theory of rigid-analytic spaces and then discuss a version of Poincaré Duality for these spaces, as conjectured by Peter Scholze in 2012.
Riemann--Hilbert correspondence: from complex analysis to p-adic geometry
When: Wed, December 4, 2024 - 3:30pmWhere: Kirwan Hall 3206
Speaker: Haoyang Guo (University of Chicago) - https://sites.google.com/umich.edu/hyg
Abstract: One of Deligne's foundational results in complex geometry is the Riemann--Hilbert correspondence, which relates the monodromy representations of complex algebraic varieties to algebraic differential equations, and provides an affirmative answer to Hilbert's 21st problem. Recent developments in arithmetic geometry have revealed analogous correspondences in the p-adic setting. In this talk, we will review the history and report on the recent progress in p-adic geometry.
Eisenstein cocycles for imaginary quadratic fields
When: Mon, December 9, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Romyar Sharifi (UCLA) - https://www.math.ucla.edu/~sharifi/
Abstract: I will discuss a construction of maps from the homology of Bianchi spaces for an imaginary quadratic field F to second K-groups of ray class fields of F. These maps are “Eisenstein” in the sense that they factor through the quotient by the action of an Eisenstein ideal way from the level. These long-expected maps are direct analogues of known explicit maps in the setting of modular curves and cyclotomic fields. We use a refinement of a method Venkatesh and I developed for constructing Eisenstein cocycles, which I’ll explain. This is joint work with E. Lecouturier, S. Shih, and J. Wang.
Graded affine Hecke algebras, residues and unitary representations
When: Wed, December 11, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Eric Opdam (University of Amsterdam) - https://staff.fnwi.uva.nl/e.m.opdam/
Abstract: I will discuss analytic aspects of the representation theory of graded
affine Hecke algebras, which originally arose in the context of certain
completely integrable models. This has applications to the automorphic
spectrum of reductive groups, and to the unitarisability of certain
representations of real reductive groups.
An Equivariant Main Conjecture in Iwasawa Theory and Applications
When: Fri, December 13, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Cristian Popescu (UCSD and IAS Princeton) - https://mathweb.ucsd.edu/~cpopescu/
Abstract: I will discuss the statement and proof of an equivariant main conjecture in Iwasawa theory, building upon recent work of Dasgupta-Kakde on the Galois module structure of the Selmer groups defined by Burns-Kurihara-Sano. I will make connections with my earlier joint results with Greither in geometric Iwasawa theory and give applications to the Galois module structure of the even Quillen K-groups of rings of algebraic integers. This is based on recent joint work with Rusiru Gambheera.
Maximal cliques of sets of strongly orthogonal roots
When: Wed, February 5, 2025 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Qëndrim Gashi (UMCP) -
Abstract: Strongly orthogonal roots first appeared in the works of Harish-Chandra and Kostant. We will discuss a surprising relationship between existence problems in finite geometries, existence of certain maximal cliques of sets of strongly orthogonal roots, and classical results from extremal combinatorics.
Algebraic cycles and the Hitchin fibration
When: Thu, March 27, 2025 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Davesh Maulik (MIT) - https://math.mit.edu/~maulik/
Abstract: In the first lecture, given a proper map $f: X \rightarrow Y$, I introduced the perverse filtration on the cohomology of $X$, which measures the singularities of the fibers of $f$. In this talk, when $X$ is an abelian fibration over $Y$, I will explain a technique for studying this filtration via the Fourier-Mukai transform on DCoh(X), the derived category of coherent sheaves of X. This approach is a natural extension of ideas of Beauville and Deninger-Murre for studying Chow groups of abelian schemes. As an application, we get a proof of the P=W conjecture, introduced in the last lecture, but also other conjectures lifting these filtrations to Chow groups. Joint work with Junliang Shen and Qizheng Yin.
D-equivalence conjecture for hyperkahler varieties of K3^[n] type.
When: Fri, March 28, 2025 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Davesh Maulik (MIT) - https://math.mit.edu/~maulik/
Abstract: The D-equivalence conjecture of Bondal and Orlov predicts that birational Calabi-Yau varieties have equivalent derived categories of coherent sheaves. I will explain how to prove this conjecture for hyperkahler varieties of K3^[n] type (i.e. those that are deformation equivalent to Hilbert schemes of K3 surfaces). This is joint work with Junliang Shen, Qizheng Yin, and Ruxuan Zhang.
Investigating singularities of moduli spaces with analytic number theory
When: Mon, March 31, 2025 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Matthew Hase-Liu (Columbia University) - https://www.math.columbia.edu/~mhaseliu/index.html
Abstract: I will explain how one can use ideas from analytic number theory and Manin's conjecture to understand the geometry of moduli spaces of curves on smooth low degree hypersurfaces. In particular, when e is large compared to g, the moduli space of degree e maps from smooth genus g curves to an arbitrary smooth hypersurface of low degree has at worst terminal singularities. Using a spreading-out argument together with a result of Mustata, we reduce the problem to counting points over finite fields on the jet schemes of these moduli spaces. We solve this counting problem by developing a suitable geometric interpretation of the circle method. This is joint work with Jakob Glas.
Discriminants and motivic integration
When: Wed, April 23, 2025 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Oscar Kivinen (Aalto University) - https://math.aalto.fi/~kivineo3/
Abstract: Several different invariants can be attached to an isolated hypersurface singularity using motivic integration. In the Euler characteristic limit, these invariants are all related in a straightforward way, but the relationships between the motivic versions are more difficult to understand. We will discuss the case of plane curves in detail, including connections to knot Floer homology, Hilbert schemes, and the Igusa zeta function. Based on joint work with Oblomkov and Wyss.
Gm-equivariant degenerations of del Pezzo surfaces
When: Mon, April 28, 2025 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Junyao Peng (Princeton University) -
Abstract: We study special Gm-equivariant degenerations of a smooth del Pezzo surface X induced by
valuations that are log canonical places of (X,C) for a nodal anti-canonical curve C. We show that the space of special valuations in the dual complex of (X,C) is connected and admits a locally finite partition into sub-intervals, each associated to a Gm-equivariant degeneration of X. This result is an example of higher rank degenerations of log Fano varieties studied by Liu-Xu-Zhuang, and verifies a global analog of a conjecture on Kollár valuations raised by Liu-Xu. For del Pezzo surfaces with quotient singularities, we obtain a weaker statement about the space of special valuations associated to a normal crossing complement.
Local models and nearby cycles for $\Gamma_1(p)$-level
When: Wed, April 30, 2025 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Tom Haines (UMCP) -
Abstract: The theory of local models has been a very successful tool for the study of Shimura varieties with parahoric level structure, and the theory is now very developed in that setting. For level structure which is deeper than Iwahori level, many complications arise, and the subject is in its infancy. I will first review the basic theory of local models for Iwahori level, concentrating on the general linear and general symplectic group cases. The main goal will be to explain what can be said about local models when the level structure is $\Gamma_1(p)$, which is slightly deeper than Iwahori level. For PEL Shimura varieties of Siegel type, I will define the local models using a linear algebra incarnation of Oort-Tate generators of finite flat group schemes of order $p$, and then I will explain how one uses a variant of Beilinson-Drinfeld Grassmannians and Gaitsgory's central functor adapted to pro-p Iwahori level, to study the nearby cycles on the special fibers. This is based on joint work in progress with Qihang Li and Benoit Stroh.
A tale of two congruences
When: Mon, May 5, 2025 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Dinesh Thakur (University of Rochester) - https://people.math.rochester.edu/faculty/dthakur2/
Abstract: We will explain how basic congruences of Fermat and Wilson, when transported to the polynomials case, get entangled with each other and with arithmetic derivatives, zeta values. This leads to solution of analogs of some classical open problems in number theory, and it also leads to many interesting open questions. The talk will be accessible to undergraduates
who have taken the first course in abstract algebra.
Hilbert scheme of points on threefolds
When: Mon, May 12, 2025 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Ritvik Ramkumar (Cornell University) - https://sites.google.com/view/ritvikramkumar/home?authuser=0
Abstract: The Hilbert scheme of d points on a smooth variety X, denoted by Hilb^d(X), is an important moduli space with connections to various fields, including combinatorics, enumerative geometry, and complexity theory, to name a few. In this talk, I will focus on the case where X is a threefold, as there are several open questions regarding its singularities. I will describe the structure of the smooth points of this Hilbert scheme and, time permitting, discuss the structure of the mildly singular points. This is all joint (ongoing) work with Joachim Jelisiejew and Alessio Sammartano.
TBA (Engel)
When: Wed, May 14, 2025 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Philip Engel (UIC) - https://philip-engel.github.io/
Explicit candidates for the arithmetic transfer conjecture in the ramified setting.
When: Fri, May 16, 2025 - 4:00pmWhere: Kirwan Hall 3206
Speaker: Thomas Rud (MIT) - https://math.mit.edu/~rud/
Abstract: One aspect of the Langlands programme is to isolate cuspidal representations "distinguished" by some subgroup. Such representations are conjecturally obtained through some transfer from some form of the subgroup. This distinction is defined in terms of nonvanishing of "period integrals".
The Gan-Gross-Prasad conjecture relates special values of certain L-functions to period integrals on classical groups. Jacquet and Rallis proposed an approach through the relative trace formula in the form of a fundamental lemma and a transfer conjecture, which were largely proven in the work of Yun and Zhang respectively.
We will introduce the problem of distinction and periods and the arithmetic Gan-Gross-Prasad setting, which aims to generalize Gross-Zagier formula, relating Neron-Tate heights of Heegner points on modular curves, to special values of derivatives of certain L-functions. Smithling-Rapoport-Zhang have proven arithmetic transfer in one specific example of ramified unitary groups, but did not construct said transfer explicitly. In this talk, I will be introducing a new conjecture aiming to realize the arithmetic transfer explicitly in all ramified cases. This is joint work with Wei Zhang.
Weil-étale cohomology and the ETNC for constructible sheaves
When: Thu, June 5, 2025 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Adrien Morin (University of Copenhagen, Denmark) - https://adrien.morin.perso.math.cnrs.fr/
Abstract: Let X be a variety over a finite field. Given an order R in a semisimple algebra A over the rationals and a constructible étale sheaf F of R-modules over X, one can consider a natural noncommutative L-function associated with F. We will formulate and prove a special value conjecture at negative integers for this L-function, expressed in terms of the Weil-étale cohomology of Lichtenbaum. This result is a geometric analogue of, and implies, the equivariant Tamagawa number conjecture for an Artin motive and its negative twists over a global function field. It also generalizes the results of Lichtenbaum and Geisser on special values at negative integers for zeta functions of varieties, and the work of Burns-Kakde in the case of the equivariant L-functions coming from a finite G-cover of varieties.