Abstracts
Jim Brown
Congruences for paramodular Saito-Kurokawa lifts and applications
Let φ ∈ Sk (Γ0 (M )) be a newform whose functional equation has sign −1. It is well known
there is a lift of φ to a Siegel modular form fφ ∈ Sk (Γ[M ]) where Γ[M ] ⊂ Sp4 (Q) is theparamodular group. In this talk we specialize the congruence result described in Huixi Li’s
talk to the case the Siegel modular form is fφ . We show there is a congruence between fφand a cuspidal Siegel eigenform with irreducible Galois representation. This congruence
provides evidence for the Bloch-Kato conjecture for φ not covered by previous work. Thisis joint work with Huixi Li.

Stevo Bozinovski
Intrinsic Theory of Divergent Series
Contemporary the dominant theory of divergent series is the summation method theory,in which a divergent series is assigned a finite sum although divergent series are named
divergent because they do nit have a finite sum. A theory presented in this talk appreciates
the fact that divergent series have no sum, and looks for other relations between divergentseries. Some new results related to Riemann zeta function are revealed. This is a joint
work with Adrijan Bozinovski.

Michael Bush
Non-abelian generalizations of the Cohen-Lenstra heuristics
In the last few years, my collaborators and I have proposed a non-abelian version of theCohen-Lenstra heuristics for both real and imaginary quadratic fields in which one replaces
the p-part of the class group (p odd prime) with the Galois group of the maximal unramified
p-extension of the base field. One big difference when making this switch is that the latterobjects can be infinite while the former are always finite. Our conjectures originally onlydealt with the frequency of occurrence of finite p-groups and certain special finite quotientsof the infinite pro-p groups that arise. After giving an overview, I’ll discuss some recent
developments including an extension covering infinite groups. This is joint work with Nigel
Boston and Farshid Hajir.

1Sungkon Chang
Average Number of Zeckendorf Integers
By Zeckendorf’s theorem each positive integer is uniquely written as a sum of distinct nonadjacent terms of the Fibonacci sequence. In this talk we consider sequences {Gn } thatare given by the Fibonacci recurrence but has different initial values, and investigate thenumber of positive integers up to X that are written as a sum of distinct non-adjacentterms of Gn . We also introduce generalizations of this result for so called the N th order
Fibonacci sequence, and for general linear recurrences with positive coefficients.

Kwangho Choiy
Restriction of tempered L-packets for p-adic groups
From the behavior of irreducible smooth representations of a p-adic group restricted to its
closed subgroup with the same derived group, we derive arithmetic information which canbe applied to establish the local Langlands correspondence of the subgroup.
In this talk, we shall focus on the multiplicity of tempered representations in the restriction
and relate it to its counterpart of corresponding irreducible representations in the Langlands
dual side. This relation is utilized to study the internal structure of tempered L-packets.

Will Craig
On Repeated Prime Divisors of Trinomial Discriminants
The discriminant of a trinomial of the form xn ± xm ± 1 takes the form ±nn ± mm (n −m)n−m whenever n and m are relatively prime. Several papers have been written studying
when these values are squarefree, and it has been conjectured that they are squarefree
infinitely often, and in particular, the special case of xn − x − 1, whose discriminant is
nn + (−1)n (n − 1)n−1 , is squarefree infinitely often. The aim of this talk is to generalize
the methods used previously to attack the squarefree to investigate when these numbers
are nth -powerfree, and in the end to prove a weaker form of these conjectures; namely,that the special case nn + (−1)n (n − 1)n−1 is cubefree infinitely often and that whenever m
is fixed, all discriminants of this form are fourth-powerfree infinitely often. Furthermore,
additional observations about these values is discussed, and methods by which the case ofsquarefreeness might be attacked are outlined.

2Nathan Fontes
Explicit Computations of Higher Weight Modular Forms
Modular forms are holomorphic functions on the complex upper half plane with specialconditions imposed upon them. Many people learn the theory of modular forms without
ever seeing how to compute their spaces, so we will examine computational methods onthese structures. This process involves looking at modular symbols and Manin symbols, aswell as the corresponding Hecke operators on them. I will discuss computations on weight
k4 modular forms and their differences from computing weight 2 modular forms.

Stephen M Gagola III
Multiplicative properties of partitions
Here we solve a problem that was proposed by Ken Ono and Christine Bessenrodt. Onoand Bessenrodt proved that the number of partitions of n, say p(n), satisfies p(a)p(b) ¿p(a + b) for a, b ¿ 1 and a + b ¿ 9 by using a result of Lehmer and asked whether acombinatorial proof exists. Here we show that such a proof exists and also show that theproof carries over to the analogous inequality for k-regular partitions with k 2. For 2 k
6, these inequalities were first proven to hold for k-regular partitions by Olivia Beckwith
and Christine Bessenrodt using similar methods to the p(n) case.

Hugh Geller
Ramanujan Type Congruences for Klingen-Eisenstein Series over Γn0 (N )In 2014, Kikuta and Takemori published results on degree n Siegel modular forms statingthat for all but finitely many primes p in any ring of algebraic integers, mod pm cusp forms
are congruent to true cusp forms of the same weight. In this talk, we discuss work inprogress to generalize their result to Siegel modular forms over the congruence subgroupΓn0 (N ).

Jon Grantham
Yet Another Conjecture of Goldbach: Preliminary Results
Let A be the set of numbers a for which a2 + 1 is prime. Goldbach conjectured that everya ∈ A, (a > 1) can be written in the form a = b + c, for b, c ∈ A. We present verificationof the conjecture as well as conditional results related to it.

Marie Jameson
Congruences for generalized Frobenius partitions
In 1984, George Andrews introduced k-colored generalized Frobenius partitions, whosecounting function cφk (n) is a generalization of the partition function. In this talk, we will
discuss congruences for generalized Frobenius partitions.
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Huixi Li
Congruence primes of Hilbert Siegel eigenforms
Congruences between modular forms play an important role in number theory. For example,
it is an important ingredient in the proof of the Herbrand-Ribet theorem and the Iwasawa
main conjecture for GL2 . In this presentation I will provide a sufficient condition for aprime ` to be a congruence prime for a Hilbert Siegel eigenform f for a large class of totallyreal fields F via a divisibility of a special value of the standard L-function associated tof . In the special case that F = Q and f is an Ikeda lift, we recover an earlier result ofBrown-Keaton as a special case of our main theorem. This is joint work with Jim Brown.

Aleksander MorganBounded generation of SL2 over S-integers of algebraic number fields withinfinitely many units.
Let O be the ring of S-integers in a number field k. We prove that if the group of units O×
is infinite, then every matrix in G = SL2 (O) is a product of at most 9 elementary matrices.
As a consequence, we obtain that G is boundedly generated as an abstract group.

Jeremy Rouse
Integers represented by positive-definite quadratic forms and Petersson innerproducts
Let Q be a positive-definite quaternary quadratic form with integer coefficients. Westudy the problem of giving bounds on the largest positive integer n that is locally represented by Q but not represented. Assuming that n is relatively prime to D(Q), thedeterminant of the Gram matrix of Q, we show that n is represented provided thatn  max{N (Q)3/2+ D(Q)5/4+ , N (Q)2+ D(Q)1+ }. These results are proven by bounding the Petersson norm of the cuspidal part of the theta series, which is accomplished using
an explicit formula for the Weil representation due to Scheithauer.

James Rudzinski
Monotone Organization of Symmetric Chains
A symmetric chain decomposition (SCD) of a Boolean lattice is a partition into symmetricchains. In this talk we present and analyze an algorithm which organizes the chains ofa SCD into a tree. This allows each element of the decomposition to be constructed in
an increasing or ”monotone” fashion. One could use this to efficiently check for certain
monotone properties such as connectivity in subgraphs of a graph.
4Tom WrightReframing the $620 problem in terms of other, equally unsolvable problems
In this talk, we clean up a heuristic by Pomerance to show exactly how the infamous$620 problem would follow from some reasonable conjectures about primes in arithmetic
progressions and smoothness of various p − 1’s and p + 1’s.

Dan Yasaki
On the cohomology of congruence subgroups of GL3 over the Eisenstein
integers
Cohomology provides a framework for doing explicit computations of certain types of automorphic forms. In joint work with Mark McConnell and Paul Gunnells, we develop
the geometric techniques needed to carry out this computation for GL3 over the imaginary
quadratic field of discriminant -3. In this talk, I will discuss some of the techniques involved
and present some of the interesting phenomena we observed.

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