** 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: **

**Abstract: **

** November 18:**

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

**Title: **

**Abstract: **

** November 25:**

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

**Title: **Decompositions of Euler-PoincarĂ© presentations and resolutions

**Abstract: ** Work of Bezrukavnikov-Kazhdan-Varshavsky uses an
equivariant system of trivial idempotents of Moy-Prasad groups
to obtain an Euler-PoincarĂ© 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-PoincarĂ© presentation
and the resolution.