**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)

**Title:**

**Abstract:**

**February 2:**

**Speaker:**

**Title:**

**Abstract:**

** February 9:**

**Speaker:**

**Title:**

**Abstract:**

** 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:**

**Speaker:**

**Title:**

**Abstract:**

** March 9:**

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

**Title:**

**Abstract:**

** 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:**

**Speaker:**

**Title:**

**Abstract:**

** March 23:**

**Speaker:** No seminar: spring break

**Title:**

**Abstract:**

** March 30:**

**Speaker:** Daniel Le

**Title:**

**Abstract:**

** 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

**Title:**

**Abstract:**

** 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)

__ May 4:__ (Note this is after the term ends)

** 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.