Date 
Speaker 
Title 

Sept 9 
everyone 
Open problem session 
at noon
in C630

(University of Lethbridge)

Please bring your favourite
(math) problems. Anyone with a problem to share will be given
about 5 minutes to present it. We will also choose most of the
speakers for the rest of the semester.


Sept 16 
Gabriel Verret 
An update on the Polycirculant Conjecture 
at noon
in C630

(University of Auckland, New Zealand)

One version of the Polycirculant Conjecture is that every finite vertextransitive
digraph admits a nontrivial semiregular automorphism. I will give an overview of
the status of this conjecture, as well as describe some recent progress with
Michael Giudici, Ademir Hujdurovic and Istvan Kovacs.


Sept 23 
Selcuk Aygin 
Extensions of RamanujanMordell Formula with Coefficients $1$ and $p$ 
at noon
in C630

(University of Lethbridge)

We use properties of modular forms to extend the RamanujanMordell formula.
Our result yields to formulas for representation numbers by the quadratic form
$\sum_{i=1}^{2a}x_i^2+\sum_{i=1}^{2b}py_i^2$, for all nonnegative integers
$a,b$ and for all odd prime $p$. We obtain this result by computing the Fourier
series expansions of modular forms at all cusps of $\Gamma_0(4p)$.


Oct 7 
Amir Akbary 
The prime number theorem for automorphic $L$functions

at noon
in C630

(University of Lethbridge)

We describe a theorem due to Jianya Liu and Yangbo Ye (Pure and Applied
Mathematics Quarterly, Volume 3, Number 2, 481497, 2007) concerning the prime
number theorem for automorphic $L$functions. We state the theorem and review
the strategy of the proof in comparison with the classical prime number theorem.
An important ingredient is a new version of Perron's formula that represents a
sum of complex numbers as a contour integral plus some specific error terms.


Oct 21 
Khoa Dang Nguyen 
An analogue of Ruzsa's conjecture for polynomials over finite fields 
at noon
in C630

(University of Calgary)

In 1971, Ruzsa conjectured that if $f:\ \mathbb{N}\rightarrow\mathbb{Z}$ with
$f(n+k)\equiv f(n)$ mod $k$ for every $n,k\in\mathbb{N}$ and $f(n)=O(\theta^n)$ with
$\theta < e$ then $f$ is a polynomial. In this paper, we investigate the analogous problem
for the ring of polynomials over a finite field. This is joint work with Jason Bell.


Oct 28 
Quanli Shen 
The fourth moment of quadratic Dirichlet $L$functions 
at noon
in C630

(University of Lethbridge)

In this talk, I will talk about the fourth moment of quadratic Dirichlet
$L$functions. Under the generalized Riemann hypothesis, we showed an asymptotic
formula for the fourth moment. Unconditionally, we established a precise lower
bound.


Nov 4 
PengJie Wong 
Primes in Short Intervals 
at noon
in C630

(University of Lethbridge)

Bertrand's postulate states that there is always a prime in the interval $[x,2x]$
for any $x\ge 1$. Applying the prime number theorem, one may further show that
there is approximately $\int_{x}^{2x}\frac{dt}{\log t}$ primes in $[x,2x]$ for
sufficiently large $x$. There is a more difficult question concerning the
distribution of primes $p$ in short intervals when $[x,2x]$ is replaced by
$[x, x + h]$ for some $h\le x$ and $p$ is required to be congruent to $a$
modulo $q$ for some $(a,q)=1$. In this talk, we will discuss how short
$[x, x + h]$ can be. If time allows, we will sketch a proof of the
BombieriVinogradov theorem in short intervals, which answers such a question.


Nov 18 
Allysa Lumley 
Distribution of Values of $L$functions over Function Fields 
at noon
in C630

(Centre de Recherches Mathématiques, Montréal)

Let $q\equiv 1 \pmod{4}$ be a prime power and $\mathbb{F}_q$ be the finite
field with $q$ elements. Let $1/2<\sigma<1$ be fixed. We consider $D$ a monic
squarefree polynomial in $\mathbb{F}_q[T]$ and $\chi_D$ the Kronecker symbol
associated with $D$. In this talk, we will discuss the distribution of large
values of $\log L(\sigma,\chi_D)$ with $D$ varying over monic squarefree
polynomials with degree $n\to\infty$. We will highlight the expected
similarities to the situation over quadratic extensions of $\mathbb{Q}$ as
well as the surprising differences.


Nov 25 
PoHan Hsu 
Large deviation principle for the divisor function 
at noon
in C630

(Louisiana State University)

Let $\omega(n)$ denote the number of distinct prime divisors of $n$. Let $W(m)$
be a random integer chosen uniformly from $\{n:n\le m, n\in \Bbb{N}\}$. Let
$X(m)$ be $\omega(W(m))$. The celebrated ErdősKac theorem asserts that
\begin{align*}
\frac{X(m)\log\log m}{\sqrt{\log\log m}}\rightarrow N(0, 1),
\end{align*}
where $N(0, 1)$ is the standard normal distribution.
In 2016, Mehrdad and Zhu studied the large and moderate deviations for the
ErdősKac theorem. In this talk, we will give a brief introduction to the
theory. Then we will discuss how to establish the large deviation principle for
$X(m)/\log\log m$. If time allows, we will discuss some generalisations.
This is a joint work with Dr PengJie Wong.


Dec 2 
Andrew Fiori 
Simplicity of ABVpackets for Arthur Type Parameters in GLn 
at noon
in C630

(University of Lethbridge)

In this talk we will discuss a combinatorial approach to studying the geometry
of the moduli space of Langlands parameters for GLn. We will first discuss
several connections between the study of partitions and the study of nilpotent
conjugacy classes. We then generalize some of these ideas to describe a
relationship between multisegments and orbits in the moduli space of Langlands
parameters. Finally we shall explain how these ideas lead to a proof that simple
parameters have simple packets.

