Xin Zhang (UIUC)
Gap Distribution in Circle Packings
Given a configuration of finitely many tangent circles, one can form a packing of infinitely many circles by Möbius inversions. Fixing one circle from such a packing, we study the distribution of tangencies on this circle via the spectral theory of automorphic forms. Specifically, we will use Anton Good's theorem to show that these tangencies are uniformly distributed when naturally ordered by a growing parameter, and the limiting gap distribution exists, which is conformally invariant. This is a joint work with Zeev Rudnick.
Detchat Samart (UIUC)
Feynman integrals and special values of 
-functions
Over the past few decades, Feynman integrals have been studied extensively in order to formulate the standard model of particle physics (i.e., the theory explaining how fundamental particles interact). In certain cases, these integrals can be expressed as periods associated to algebraic varieties and are conjectured to encode some interesting arithmetic information about the varieties. More precisely, numerical evidence suggests that their evaluations are, up to simple factors, special values of 
-functions. In this talk, we will briefly explain what Feynman integrals are and present a recent result on Feynman integral evaluations related to critical values of 
-functions of 
surfaces, which was discovered numerically by D. Broadhurst.
Jack Shotton (U Chicago)
Local deformation rings when 
Given a mod 
representation of the absolute Galois group of 
, consider the universal framed deformation ring 
parametrising its lifts. When 
and 
are distinct I will explain a relation between the mod 
geometry of 
and the mod 
representation theory of 
, that is parallel to the Breuil-Mézard conjecture in the 
case. I will give examples and say something about the proof, which uses automorphy lifting techniques.
Harold Diamond (UIUC)
PNT Equivalences and Nonequivalences for Beurling primes
In classical prime number theory there are several asymptotic formulas that are said to be ``equivalent'' to the Prime Number Theorem. (This notion is colloquial, not mathematical: it means that the formulas can be deduced from each other by relatively simple arguments.) We show conditions under which analogues of these formulas do or do not hold for Beurling generalized numbers.
Ciprian Demeter (IU Bloomington)
The proof of Vinogradov's Mean Value Theorem
I will present some history, implications to number theory and elements of our proof of VMVT. Joint work with Jean Bourgain and Larry Guth.
Wushi Goldring (Washington U St. Louis)
Algebraicity of automorphic representations: The functoriality approach
The Langlands correspondence for number fields predicts a precise distinguished subclass of all automorphic representations that should be in correspondence with pure motives (and so also with their various realizations: certain Galois representations, pure Hodge structures, etc.). In particular, many automorphic representations which are initially defined by analytic and/or representation-theoretic means should have deep algebraic properties, the simplest being that their Hecke eigenvalues should be algebraic. In this talk, I will focus on the possibilities and limitations of using Langlands functoriality to prove such algebraicity results. I'll begin by explaining how automorphic representations naturally break-up into several classes, which are motivated by both geometry and representation theory. I will then discuss some general negative results, some positive examples and some open problems about when it is possible to ``move'' from one of these classes to another one by means of functoriality.
Michael Magee (Yale)
Lattice point count and continued fractions
In this talk I’ll discuss a lattice point count for a thin semigroup inside 
. It is important for applications that one can perform this count uniformly throughout congruence classes and for arbitrary moduli. The approach to counting is dynamical - with input from both the real and finite places. At the real place one brings ideas of Dolgopyat concerning oscillatory functions into play. At finite places, a rapid mixing property is supplied by expansion of Cayley graphs and injected into the thermodynamical formalism. The expansion of the relevant Cayley graphs was first established by Bourgain and Gamburd (for prime places) and extended to arbitrary moduli by Bourgain and Varjú. Until recently it was only known how to apply these expansion results in the thermodynamical setting for squarefree moduli. I’ll discuss a decoupling method (developed in work with Bourgain and Kontorovich) that allows arbitrary moduli to be treated. This talk is based on joint works with Oh and Winter, and with Bourgain and Kontorovich.
Vlad Serban (Northwestern)
On p-adic strengthenings of the Manin-Mumford conjecture
Let 
be an abelian variety or a product of multiplicative groups 
and let 
be an embedded curve. The Manin-Mumford conjecture (a theorem by work of Lang, Raynaud et al.) states that only finitely many torsion points of 
can lie on 
unless 
is in fact a subgroup of 
. I will show how these purely algebraic statements extend to suitable analytic functions on open 
-adic unit poly-disks. These disks occur naturally as weight spaces parametrizing families of 
-adic automorphic forms for 
over a number field 
. When 
, the "Hida families" in question play a crucial role in the study of modular forms. When 
is imaginary quadratic, I will explain how our results show that Bianchi modular forms are sparse in these 
-adic families.
Atul Dixit (IIT Gandhinagar)
Overpartition analogues of partitions associated with the Ramanujan/Watson mock theta function 
Let 
denote the third order mock theta function of Ramanujan and Watson. Recently George E. Andrews, Ae Ja Yee and I showed that 
is the generating function of 
, the number of partitions of a positive integer 
such that all odd parts are less than twice the smallest part. We also studied the associated smallest parts partition function 
and proved some congruences for the same. Very recently, we considered the overpartition analogue of 
, namely, 
. Finding an alternate representation for the generating function of 
turns out to be difficult in this case. We devise a new seven parameter 
-series identity which generalizes a deep identity of Andrews (as well as its generalization by R. P. Agarwal), and then specialize it, along with the use of some identities in basic hypergeometric series, to arrive at an alternate representation in terms of a 
and an infinite series involving the little 
-Jacobi polynomials. We also prove some congruences for 
and for the overpartition analogue of 
. This is joint work with George E. Andrews, Daniel P. Schultz and Ae Ja Yee.
Florin Boca (UIUC)
Correlations of Farey fractions revisited
Existence of the correlation measures for the Farey sequence and a way of expressing them were investigated in a 2005 joint work of Zaharescu and myself. In this talk we we will discuss some potential applications to estimating the moments of eigenvalues of a certain interesting family of (non-random) large matrices. This is work in progress with Maksym Radziwill and Alexandru Zaharescu.
Michael Lipnowski (Duke)
Statistics of abelian varieties over finite fields
Joint work with Jacob Tsimerman. Let B(g,p) denote the number of isomorphism classes of g-dimensional abelian varieties over the finite field of size p. Let A(g,p) denote the number of isomorphism classes of principally polarized g dimensional abelian varieties over the finite field of size p. We derive upper bounds for B(g,p) and lower bounds for A(g,p) for p fixed and g increasing. The extremely large gap between the lower bound for A(g,p) and the upper bound B(g,p) implies some statistically counterintuitive behavior for abelian varieties of large dimension over a fixed finite field.
Caroline Turnage-Butterbaugh (Duke)
Large gaps between zeros of Dedekind zeta-functions of quadratic number fields
Let 
be a quadratic number field with discriminant 
. The Dedekind zeta-funciton attached to 
can be expressed by 
for 
, where 
is the Riemann zeta-function, the character 
is the Kronecker symbol associated to 
, and 
is the corresponding Dirichlet 
-function. Combining Hall’s Method with the twisted second moment of 
, we show that there are infinitely many gaps between consecutive zeros of 
which are greater than 
times the average spacing. This is joint work with Hung Bui and Winston Heap.
Ilya Vinogradov (Princeton)
Effective equidistribution of horocycle lifts
We give a rate in the problem of equidistribution of lifted horocycles in the space of unimodular two-dimensional lattice translates. The ineffective version is due to Elkies and McMullen and relies on Ratner's Theorem. The approach used here builds on recent works of Strombergsson and of Browning and the author.