** 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.