Representation Theory and Number Theory Seminar, 2017-2018

Spring 2018: Friday, 3:30-4:30 PM, LCB 215

January 12:
Title: Organizational meeting at 2:45 in JWB 308

January 19:
Title: No seminar

January 26:
Speaker: No Seminar (Steven Sam Special Colloquium 3-4pm, JWB 335)

February 2:

February 9:

February 16:
Speaker: Shiang Tang
Title: Full-image p-adic Galois Representations and Galois Deformation Theory
Abstract: The classical inverse Galois problem asks whether or not every finite group appears as the Galois group of an extension of the rationals. It is still wide open for finite groups of Lie type, for example, SL_2(F_p). On the other hand, number theorists are interested in reductive algebraic groups over p-adic fields for various reasons. We then ask an analogue of the inverse Galois problem in the p-adic world, for example, whether or not there is a Galois extension of the rationals whose Galois group is an open subgroup of SL_2(Z_p). This kind of question fits into the study of continuous p-adic Galois representations valued in a connected reductive group. Representations into general linear groups, general symplectic groups or general orthogonal groups may be constructed by studying the action of the Galois group over Q on the cohomology of an algebraic variety over Q. Representations into exceptional algebraic groups appear in the work of Nick Katz, Stefan Patrikis and Zhiwei Yun.
In this work, we construct p-adic Galois representations using Galois deformation theory, which first appeared in Barry Mazur's work and became one of the central tools in modern algebraic number theory after Andrew Wiles published his proof of Fermat's Last Theorem. We will explain a strategy (due to Ravi Ramakrishna and others) for “deforming" a continuous Galois representation in characteristic p to characteristic zero, together with its application to the p-adic analogue of the inverse Galois problem.

February 23:
Speaker: Peter Trapa
Title: Reducible characteristic cycles of Harish-Chandra modules and the Kashiwara-Saito singularity.
Abstract: Characteristic cycles of Harish-Chandra modules are a subtle geometric invariant of considerable interest. For example, computing them would give complete information about Arthur packets at the real place. In this talk, we give new examples of reducible characteristic cycles for representations of U(p,q). The singularity controlling the reducibility turns out to be closely related to the so-called Kashiwara-Saito singularity (which turns up in many interesting places).

March 2:

March 9:
Speaker: Special AG seminar, Yuri Tschinkel (2:30-3:30, LCB 215)

March 9:
Speaker: AG/NT seminar, Wieslawa Niziol (3:30-4:30, LCB 215)
Title: Cohomology of p-adic Stein spaces
Abstract: I will discuss a comparison theorem that allows us to recover p-adic (pro-)etale cohomology of p-adic Stein spaces with semistable reduction over local rings of mixed characteristic from complexes of differential forms. To illustrate possible applications, I will show how it allows us to compute cohomology of Drinfeld half-space in any dimension and of its coverings in dimension one. This is a joint work with Pierre Colmez and Gabriel Dospinescu.

March 16:

March 23:
Speaker: No seminar: spring break

March 30:
Speaker: Daniel Le (Toronto)
Title: The weight part of Serre's conjecture
Abstract: In the 70's, Serre conjectured that all two-dimensional odd irreducible continuous mod p Galois representations arise from modular forms. A decade later, he conjectured a recipe for the weight and level of the modular forms in terms of the Galois representations--a recipe which would play a key role in the proof of Fermat's Last Theorem. In Serre's original context, these conjectures are now known. We survey recent conjectures and results about the weight part of Serre's conjecture for more general automorphic forms. The main ingredient is a description of local Galois deformation rings using local models. This is joint work with Le Hung, Levin, and Morra.

April 6:
Speaker: Eric Marberg (HKUST)
Title: From Klyachko models to representations of extended Hecke algebras
Abstract: A model for a finite group is a set of characters, each obtained by inducing a linear character of a subgroup, whose sum contains each of the group's irreducible characters with multiplicity one. In the 1980s, Klyachko found natural models for the finite general linear groups. These models deform, in at least a formal sense, to models for the symmetric groups on passing from finite fields to the mythical field with one element. In a preprint from last year, Lusztig studies a representation of an "extended Hecke algebra" on the set of involutions in an extended Weyl group. This talk will trace a sequence of extensions and generalizations leading from Klyachko' classical models to Lusztig's cutting edge construction. Along the way, I will survey some interesting connections between the representations interpolating between Klyachko and Lusztig and things like the unipotent characters of reductive groups, Schubert calculus, and the combinatorics of K-orbit closures.

April 13:
Speaker: Stefan Patrikis

April 20:
Speaker: Goran Muic, University of Zagreb
Title: Some results on the Schwartz space of G/Gamma
Abstract: Let G be a connected semisimple Lie group with finite center. Let Gamma be a discrete subgroup of G. We study closed admissible irreducible subrepresentations of the space of distributions S(G/Gamma)' defined by Casselman, and their relations to automorphic forms on G/Gamma when Gamma is a congruence subgroup. If time permits we will discuss analogous results for adelic groups.

April 27, 11:00am-12:00pm, JWB 333: (Note unusual time and place)
Speaker: Lars Thorge Jensen, Max Planck Institute (Bonn)
Title: The ABC of p-Cells
Abstract: The Hecke category is a categorification of the Hecke algebra that plays an important role in (geometric) representation theory. Using this categorification, I will introduce a positive characteristic analogue of the famous Kazhdan-Lusztig basis of the Hecke algebra, called the p-canonical or p-Kazhdan-Lusztig basis. If time permits, I will mention connections between the p-Kazhdan-Lusztig basis and the representation theory of reductive algebraic groups. Motivated by the very rich theory of Kazhdan-Lusztig cells, I study cells with respect to the p-Kazhdan-Lusztig basis. Throughout the talk, I will use SL_2 as a running example. In the end, I will give a complete description of p-Cells in finite type A and mention some interesting results in finite types B and C.

May 4, 2:30-3:30pm, LCB 225: (Note unusual time and place)
Speaker: Nicolle E.S. González, University of Southern California
Title: A Categorification of the Boson-Fermion Correspondence
Abstract: The Boson-Fermion correspondence is a fundamental relationship in mathematical physics relating the Fock spaces of two elementary particles, bosons and fermions. Given that bosonic and fermionic Fock spaces are irreducible representations of the Heisenberg and Clifford algebras, the correspondence yields a way of relating these two actions via the theory of vertex operators. In this talk we will explain this correspondence from an algebraic and combinatorial perspective and discuss a categorification of this relationship.

Fall 2017: Friday, 3:30-4:30 PM, LCB 215

August 25:
Title: Organizational meeting and student modularity seminar (Moss)

September 1:
Title: Student modularity seminar (Moss)

September 8:
Title: Student modularity seminar (Patrikis)

September 15:
Title: Student modularity seminar (Patrikis)

September 22:
Speaker: Anna Romanov, The University of Utah
Title: A Kazhdan-Lusztig algorithm for Whittaker modules
Abstract: The category of Whittaker modules for a complex semisimple Lie algebra generalizes the category of highest weight modules and displays similar structural properties. In particular, Whittaker modules have finite length composition series and all irreducible Whittaker modules appear as quotients of certain standard Whittaker modules which are generalizations of Verma modules. Using the localization theory of Beilinson-Bernstein, one obtains a beautiful geometric description of Whittaker modules as twisted sheaves of D-modules on the associated flag variety. I use this geometric setting to develop an analogue of Kazhdan-Lusztig algorithm for computing the multiplicities of irreducible Whittaker modules in the composition series of standard Whittaker modules.

September 29:
Speaker: Gil Moss, The University of Utah
Title: Toward a local Langlands correspondence in families
Abstract: In 2012 it was conjectured by Emerton and Helm that the local Langlands correspondence for GL(n) of a p-adic field (suitably normalized) should interpolate in \ell-adic families, where \ell is a prime different from p. Recently, Helm reformulated this conjecture in terms of the existence of a map from the integral Bernstein center to a Galois deformation ring. This connects both congruences and moduli spaces for objects on either side of the correspondence. In this talk we will present recent work (joint with David Helm) showing the existence of such a map and describing its image, and we will discuss a speculative generalization to split classical groups.

October 6:
Speaker: Erick Knight, University of Toronto
Title: Patching and the p-adic Jacquet-Langlands correspondence
Abstract: In this talk, I will explain how to use the Taylor-Wiles-Kisin patching method to study the p-adic Jacquet-Langlands correspondence. I will show that the two constructions of the p-adic Jacquet-Langlands correspondence due to myself and Scholze agree, and also determine the locally algebraic vectors inside the representations of the quaternion algebra. This is joint work with Przemyslaw Chojecki.

October 20:
Title: Student modularity seminar (Klevdal)

October 27:
Title: Student modularity seminar (Childers)

November 3:
Speaker: Sean McAfee, The University of Utah
Title: Twisted and Untwisted Cells for Real Reductive Lie Groups
Abstract: In classical Kazhdan-Lusztig theory, the action of the Hecke algebra H(W) on itself allows us to partition the Weyl group W into "cells" which carry information about representations of W and highest weight modules indexed by elements of W. Given an involution of W, we can define a notion of "twisted cells" of W as well. In this talk, I will describe the special relation between twisted and untwisted cells in the classical case, and show how recent work allows us to define twisted and untwisted cells for the set of Langlands parameters for a real reductive linear Lie group.

November 10:
Speaker: Allen Moy, HKUST
Title: An Euler-Poincaré formula for a depth zero Bernstein projector
Abstract: Work of Bezrukavnikov-Kazhdan-Varshavsky uses an equivariant system of trivial idempotents of Moy-Prasad groups to obtain an Euler-Poincaré formula for the r-depth Bernstein projector. We establish an Euler-Poincaré formula for the projector to an individual depth zero Bernstein component in terms of an equivariant system of Peter-Weyl idempotents of parahoric subgroups P associated to a block of the reductive quotient P. This work is joint with Dan Barbasch and Dan Ciubotaru.

November 17:
Speaker: Sean Taylor, LSU
Title: A Mixed Version of the Derived Category of Constructible Sheaves on Toric Varieties
Abstract: In "Koszul duality patterns in representation theory," Beilinson, Ginzburg, and Soergel provided the notion of a mixed abelian category. In this seminal paper, they used this theory to prove a parabolic-singular duality for BGG Category \mathcal{O}. For varieties over fields of characteristic p > 0, Deligne famously introduced the derived category of mixed constructible complexes. However, Deligne's category is not mixed in the above sense. The problem of producing a mixed structure was dealt with by Beilinson, Ginzburg, and Soergel in the abelian case and by Achar and Riche in the derived case. In this talk, we will explain recent results that accomplish producing a mixed structure on the derived category of constructible complexes over any toric variety. In the process, we prove that this also comes equipped with a mixed version of the category of perverse sheaves. It is then shown that some of functors commute with forgetting the mixed structure.

December 1:
Speaker: Preston Wake, UCLA
Title: The rank of Mazur's Eisenstein ideal
Abstract: In his landmark 1976 paper "Modular curves and the Eisenstein ideal", Mazur studied congruences modulo p between cusp forms and an Eisenstein series of weight 2 and prime level N. We use deformation theory of pseudorepresentations to study the corresponding Hecke algebra. We will discuss how this method can be used to refine Mazur's results, quantifying the number of Eisenstein congruences. Time permitting, we'll also discuss some partial results in the composite-level case. This is joint work with Carl Wang-Erickson.