Junxian Li (UIUC)
A lower bound for the least prime in an arithmetic progression
Fix 
a positive integer, and let 
be coprime to 
. Let 
denote the smallest prime equivalent to 
, and set 
to be the maximum of all the 
. We seek lower bounds for 
. In particular, we show that for almost every 
one has 
, answering a question of Ford, Green, Konyangin, Maynard, and Tao. We rely on their recent work on large gaps between primes. Our main new idea is to use sieve weights to capture not only primes, but also small multiples of primes. We also give a heuristic which suggests that 
. This is joint work with Kyle Pratt and George Shakan.
Vesna Stojanoska (UIUC)
Galois actions on the homology of Fermat curves, and applications
In the late 80ties, Anderson gave a method for describing the Galois action on the singular homology of Fermat curves. In this talk, I will concentrate on Fermat curves of prime exponent p, and the singular homology will have mod p coefficients. I will describe how to explicitly determine the Galois action using Anderson’s work, and then proceed to compute some Galois cohomology groups, which naturally appear when studying certain obstructions to the existence of rational points. This is all joint work in progress with R. Davis, R. Pries, and K. Wickelgren.
Byungchan Kim (Seoul Tech)
OverGuassian Polynomials
We introduce a two variable generalization of Gaussian polynomial. After introducing some basic properties analogous to classical binomial coefficients, I will discuss its roles in q-series and integer partitions. This is joint work with Jehanne Dousse.
Tong Liu (Purdue)
Noncongruence modular form and Scholl representation
In this talk, I will report the progress of the research on noncongruence modular form via the attached Galois representation (Scholl representation). As Scholl representations are motivic, they are expected to correspond to automorphic representations according to the Langlands philosophy. I will show this is the case for certain situations (potentially GL(2)-type) and explain how (potential) automorphy of Scholl representation relates to some standard conjectures of noncongruence modular form. This is joint work of Winnie Li and Ling Long.
Bruce Berndt (UIUC)
Ramanujan's formula for 
Let 
denote the Riemann zeta function. If 
is a positive integer, a famous formula of Euler provides an elegant evaluation of 
. However, little is known about 
. In Ramanujan's earlier notebooks, we find a formula for 
which is a natural analogue of Euler's formula. We provide its history, indicate why it is "interesting," and show its connections with other mathematical objects such as the Dedekind eta function, Eisenstein series, and period polynomials.
Harold Diamond (UIUC)
A proof of 
for Beurling generalized numbers
In classical prime numbertheorythereareseveral asymptotic formulas said to be ``equivalent'' to the Prime Number Theorem. One of these assertions is that M(x), the summatory function of the Moebius function, is o(x). Implications between these formulas are different for Beurling generalized numbers (g-numbers). We deduce the g-number version of M(x) = o(x) using the PNT and a crude O-bound on the distribution of g-integers.
Joseph Vandehey (Ohio State)
Analyzing rationals by simpler rationals
The (decimal) expansions of rational numbers continue to be a rather mysterious object. For instance, we expect most rational numbers to have a periodic expansion that involves a 7, but how soon should we expect that 7 to appear? In this talk, we will discuss a new method of relating the expansion of a given rational number to rationals with smaller denominators, by means of a new differencing method for exponential sums that is highly effective for exponential sums with an exponential argument.
Xin Zhang (UIUC)
Finding integers from orbits of thin subgroups of SL(2, Z)
Let 
be a finitely generated, non-elementary Fuchsian group of the second kind, and 
be two primitive vectors in 
. We consider the set 
, where 
is the standard inner product in 
. Using Hardy-Littlewood circle method and some infinite co-volume lattice point counting techniques developed by Bourgain, Kontorovich and Sarnak, together with Gamburd's 5/6 spectral gap, we show that if 
has parabolic elements, and the critical exponent 
of 
exceeds 0.995371, then a density-one subset of all admissible integers (i.e. integers passing all local obstructions) are actually in 
, with a power savings on the size of the exceptional set (i.e. the set of admissible integers failing to appear in 
). This supplements a result of Bourgain-Kontorovich, which proves a density-one statement for the case when 
is free, finitely generated, has no parabolics and has critical exponent 
.
Ae Ja Yee (Penn State)
Singular overpartitions
Singular overpartitions, which were defined by George Andrews, are overpartitions whose Frobenius symbols have at most one overlined entry in each row. In his paper, Andrews obtained interesting results on singular overpartitions; in particular, one result relates a certain type of singular overpartitions with a subclass of overpartitions. In this talk, I will introduce partitions with dotted parity blocks and give a combinatorial proof of Andrews' result. I will also discuss some refinements on Andrews' result.
Rahul Krishna (Northwestern)
A new approach to Waldspurger's formula
I will present a new trace formula approach to Waldspurger's formula for toric periods of automorphic forms on 
. The method is motivated by interpreting Waldspurger's result as a period relation on 
, which leads to a strange comparison of relative trace formulas. I will explain the local results needed to carry out this comparison, mention some technical hurdles, and discuss some optimistic dreams for extending these results to high rank orthogonal groups.
William Banks (Missouri)
Consecutive primes and Beatty sequences
Beatty sequences are generalized arithmetic progressions which have been studied intensively in recent years. Thanks to the work of Vinogradov, it is known that every Beatty sequence S contains "appropriately many" prime numbers. For a given pair of Beatty sequences S and T, it is natural to wonder whether there are "appropriately many" primes in S for which the next larger prime lies in T. In this talk, I will show that this is indeed the case if one assumes a certain strong form of the Hardy-Littlewood conjectures. This is recent joint work with Victor Guo.
Xianchang Meng (UIUC)
Chebyshev's bias for products of 
primes
For any 
, we study the distribution of the difference between the number of integers 
with 
or 
in two different arithmetic progressions, where 
is the number of distinct prime factors of 
and 
is the number of prime factors of 
counted with multiplicity . Under some reasonable assumptions, we show that, if 
is odd, the integers with 
have preference for quadratic non-residue classes; and if 
is even, such integers have preference for quadratic residue classes. This result confirms a conjecture of Hudson. However, the integers with 
always have preference for quadratic residue classes. Moreover, as 
increases, the biases become smaller and smaller for both of the two cases.