** January 6:**

**Speaker: ** Peter Wear, UC San Diego

**Title: **
Perfectoid covers of abelian varieties and the weight-monodromy conjecture

**Abstract: **
The theory of perfectoid spaces was initially developed by Scholze to
prove new cases of the weight-monodromy conjecture. He constructed perfectoid covers
of toric varieties that allowed him to translate results from characteristic p to
characteristic 0. We will give an overview of Scholze’s method, then explain how to
use an analogous construction for abelian varieties to prove the weight-monodromy
conjecture for complete intersections in abelian varieties.

** January 27:**

**Speaker: ** Adam Brown, IST Austria

**Title: **
Contravariant forms on Whittaker modules

**Abstract: **
Contravariant forms are symmetric bilinear forms on modules over a semisimple
complex Lie algebra which satisfy an invariance property with respect to the action
of the Lie algebra. Certain well-studied modules, such as highest weight modules,
admit a unique contravariant form up to scaling. In this talk I will outline joint
work with Anna Romanov, classifying contravariant forms on irreducible Whittaker
modules. Specifically, we will discuss the non-uniqueness of contravariant forms on
irreducible Whittaker modules and the resulting implications for potential algebraic
notions of duality in the category of Whittaker modules.

** February 3:**

**Speaker: ** Mathilde Gerbelli-Gauthier, University of Chicago

**Title: **
Cohomology of Arithmetic Groups and Endoscopy

**Abstract: **
How fast do Betti numbers grow in a congruence tower of compact arithmetic
manifolds? The dimension of the middle degree of cohomology is proportional to the
volume of the manifold, but away from the middle the growth is known to be
sub-linear in the volume. I will explain how automorphic representations and the
phenomenon of endoscopy provide a framework to understand and quantify this slow
growth. Specifically, I will discuss how to obtain explicit bounds in the case of
unitary groups using Arthur’s stable trace formula. This is joint work in progress
with Simon Marshall.

** February 10:**

**Speaker: ** Patrick Daniels, University of Maryland

**Title: **
A Tannakian framework for G-displays and Rapoport-Zink spaces

**Abstract: **
Formal p-divisible groups over a p-adic ring are equivalent to linear
algebraic objects called displays. In this talk, we present a Tannakian framework
for group-theoretic analogs of displays, which correspond to formal p-divisible
groups with additional structures. We use these G-displays to define a Rapoport-Zink
functor which generalizes the purely group-theoretic one of Bueltel and Pappas, and
we show that this functor recovers the classical one of Rapoport and Zink in the
unramified EL-type situation. Representability of this functor in general would
provide integral models for local Shimura varieties.

__ February 24 3-4pm LCB 323:__ (Note unusual room)

__ Friday February 28 3:30-4:30 LCB 215:__ (Note unusual day, time, and room)

__ April 6, 3:30-4:30pm:__ (Remote Seminar)

** August 29:**

**Speaker: **

**Title: ** Brief Organizational Meeting

**Abstract: **

** August 26:**

**Speaker: ** Stephen Miller, Rutgers

**Title: ** An update on the sphere packing problem

**Abstract: ** I will discuss recent work on the optimal arrangement of points
in euclidean space. In addition to the solution to the sphere packing
problem in dimensions 8 and 24 from 2016, the "Universal Optimality"
conjecture has now been established in these dimensions as well. This shows
that E8 and the Leech lattice minimize energy for any completely monotonic
function of distance-squared. Previously there were no proofs in any
dimension > 1 that a particular configuration minimizes energy for any
nonzero example of such a potential. Beyond giving a new proof of the
sphere packing results, Universal Optimality also gives information about
long-range interactions. The techniques reduce the problem to a question in
single-variable calculus, which is ultimately solved using modular forms
and methods introduced by Viazovska. This will be a colloquium-style talk.
(Joint work with Henry Cohn, Abhinav Kumar, Danylo Radchenko, and Maryna
Viazovska)

** September 9:**

**Speaker: ** Gil Moss, Utah

**Title: **
The Whittaker model of Serre's universal unramified module

**Abstract: **
Let F be a nonarchimedean local field with residue field of order q, and
let l be a prime different from p. The mod-l representation theory of the
F-points of reductive groups can exhibit very different behavior than the
complex theory. We will discuss the interplay between Whittaker models and
systems of spherical Hecke eigenvalues for representations of GL_n(F). The
results can be applied toward an open conjecture in the theory of mod-l
automorphic forms.

** September 16:**

**Speaker: ** Ronno Das, University of Chicago

**Title: **
Points and lines on cubic surfaces

**Abstract: **
The Cayley-Salmon theorem states that every smooth cubic surface in CP^3
has exactly 27 lines. Their proof is that marking a line on each cubic surface
produces a 27-sheeted cover of the moduli space M of smooth cubic surfaces.
Similarly, marking a point produces a 'universal family' of cubic surfaces over M.
One difficulty in understanding these spaces is that they are complements of
incredibly singular hypersurfaces. In this talk I will explain how to compute the
rational cohomology of these spaces. I'll also explain how these purely topological
theorems have (via the machinery of the Weil Conjectures) purely arithmetic
consequences: the average smooth cubic surface over a finite field F_q contains 1
line and q^2 + q + 1 points.

** September 23:**

**Speaker: ** Stefan Patrikis, Utah

**Title: **
Lifting irreducible Galois representations

**Abstract: **
This will be a largely expository talk on "odd" Galois
representations. I will begin with motivation from the theory of
automorphic forms and representation theory of semisimple Lie groups, and
then, with an emphasis on examples, I will discuss some recent results
with Fakhruddin and Khare on lifting odd irreducible mod p Galois
representations to geometric p-adic representations.

** September 30:**

**Speaker: ** Sean Howe, Utah

**Title: **
A(nother) conjecture about zeta functions, or, "it's zeta functions all the
way down."

**Abstract: **
We conjecture a unification of arithmetic and motivic/topological
statistics over finite fields through a natural analytic topology on the ring of
zeta functions. A key step will be to explain exactly what it means to evaluate the
zeta function of a zeta function at a zeta function. This is joint work with
Margaret Bilu.

** October 21:**

**Speaker: ** Gordan Savin, Utah

**Title: **
Exceptional Siegel-Weil formula

**Abstract: **
Joint work with W.T. Gan

** November 4:**

**Speaker: ** Petar Bakic, Utah

**Title: **The local theta correspondence

**Abstract: **
The global theta correspondence is a standard tool in the study
of automorphic forms. To understand it, one would like to have a
description of its local variant, i.e. the local theta (aka Howe)
correspondence. In the first part of the talk, I will go over the basic
setup and the most important general results concerning theta
correspondence. After that, I will discuss the recent work with M. Hanzer
in which we give an explicit description of the local theta correspondence
for dual pairs of Type I.

** November 18:**

**Speaker: ** Andrea Dotto, University of Chicago

**Title: **Functoriality for Serre weights

**Abstract: **
By work of Gee--Geraghty and myself, one can transfer Serre weights from
the maximal compact subgroup of an inner form D* of GL(n) to a maximal compact
subgroup of GL(n). Because of the congruence properties of the Jacquet--Langlands
correspondence this transfer is compatible with the Breuil--Mezard formalism, which
allows one to extend the Serre weight conjectures to D* (at least for a tame and
generic residual representation). This talk aims to explain all of the above and to
discuss a possible generalization to inner forms of unramified groups.

** November 22, 3:30-4:30pm, LCB 225:**

**Speaker: ** Zheng Liu, UC Santa Barbara

**Title: **Doubling archimedean zeta integrals for symplectic and unitary groups

**Abstract: **
In order to verify the compatibility between the conjecture
of Coates--Perrin-Riou and the interpolation results of the p-adic
L-functions constructed by using the doubling method, a doubling
archimedean zeta integral needs to be calculated for holomorphic
discrete series. When the holomorphic discrete series is of scalar
weight, it has been done by Bocherer--Schmidt and Shimura. I will
explain a way to compute this archimedean zeta integral for general
vector weights by using the theory of theta correspondence.

** November 25:**

**Speaker: ** Allen Moy, Hong Kong University of Science and Technology

**Title: **Decompositions of Euler-Poincaré presentations and resolutions

**Abstract: ** Work of Bezrukavnikov-Kazhdan-Varshavsky uses an
equivariant system of trivial idempotents of Moy-Prasad groups
to obtain an Euler-Poincaré presentation of the r-depth Bernstein
projector. Bestvina-Savin, generalizing earlier work of
Schneider-Sthuler, showed this system of Moy-Prasad groups,
allows a resolution of a smooth representation generated by
its depth r-vectors. We report on work in progress with
Gordan Savin establishing a direct sum decomposition of the
equivariant system and therefore of the Euler-Poincaré presentation
and the resolution.