Date 
Speaker 
Title 

Jan 14 
everyone 
Open problem session 
at noon
in W866

(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.


Jan 21 
Amir Akbary 
Ambiguous Solutions of a Pell Equation 
at noon
in D631

(University of Lethbridge)

It is known that if the negative Pell equation $X^2DY^2=1$
is solvable (in integers), and if $(x, y)$ is its solution with smallest positive $x$ and $y$, then all of its solutions $(x_n, y_n)$ are given by the formula
$$x_n+y_n \sqrt{D}=\pm (x+y\sqrt{D})^{2n+1}$$
for $n\in \mathbb{Z}$. Furthermore, a theorem of Walker from 1967 states that
if the equation $aX^2bY^2=\pm 1$ is solvable, and if $(x, y)$ is its solution with smallest positive $x$ and $y$, then all of its solutions $(x_n, y_n)$ are given by
$$x_n\sqrt{a}+y_n \sqrt{b}=\pm (x\sqrt{a}+y\sqrt{b})^{2n+1}$$
for $n\in \mathbb{Z}$. We describe a unifying theorem that includes both of these results as special cases. The key observation is a structural theorem for the nontrivial ambiguous classes of the solutions of Pell equations $X^2DY^2=\pm N$.
This talk is based on the work of Forrest Francis in an NSERC USRA project in summer 2015.


Feb 4 
Hadi Kharaghani 
Two new classes of Hadamard matrices 
at noon
in D631

(University of Lethbridge)

A Hadamard matrix $H$ of order $4n^2$ is said to be skewregular if
it is of skewtype and the absolute values of the row sums are all $2n$.
It is conjectured that for each odd integer $\boldsymbol{n}$ there is a skewregular
matrix of order $\boldsymbol{4n^2}$.
A Hadamard matrix $H$ of order $m$ is said to be balancedly splittable
if there is an $\ell\times m$ submatrix $H_1$ of $H$ such that inner products for
any two distinct column vectors of $H_1$ take at most two values.
It is conjectured that only (Sylvester) Hadamard matrices of order $\boldsymbol{4^n}$
are balancedly splittable.
The existence and applications of these two very interesting classes of matrices
to Hadamard diagonalizable strongly regular graphs, maximal equiangular lines set,
doubly regular tournament, and unbiased Hadamard matrices will be discussed in detail.


Feb 11 
Nathan Ng 
Discrete moments of the Riemann zeta function 
at noon
in D631

(University of Lethbridge)

In this talk I will consider the discrete moments
\[
J_{k}(T) = \sum_{0 < \gamma < T} \zeta'(\rho)^{2k},
\]
where $\zeta(s)$ is the Riemann zeta function, $\rho=\beta+i \gamma$ is a nontrivial zero of $\zeta(s)$, and $T >0$.
In the 1980's Steve Gonek and Dennis Hejhal (independently) studied these moments and
proposed a conjecture for the size of $J_k(T)$. I will give a survey of the known results
towards the GonekHejhal conjecture on $J_k(T)$. If time permits, I will present several new results.


Feb 25 
Jack Klys

CohenLenstra heuristics and counting number fields 
at noon
in D631

(University of Calgary)

We will discuss the CohenLenstra heuristics, both in the classical and
nonabelian setting. In particular we will make the connection between these
heuristics and the problem of counting extensions of quadratic fields with
fixed Galois groups, and when knowledge of the latter implies such heuristics.
Finally we will discuss our recent work on the nonabelian case and counting
unramified 2group extensions of quadratic fields.


Mar 4 
Joy Morris 
Counting Digraphical Regular Representations (DRRs) 
at noon
in D631

(University of Lethbridge)

A Digraphical Regular Representation (DRR) for a group $G$ is a directed
graph whose full automorphism group is the regular representation of $G$. In
1981, Babai showed that with five small exceptions, there is at least one DRR
for every finite group.
A Cayley digraph on a group $G$ is a group that contains the regular representation
of $G$ in its automorphism group. So when a Cayley digraph is a DRR, its automorphism group
is as small as possible given that constraint.
The question of whether or not requiring a certain level of symmetry (in this case, a
Cayley digraph) makes having additional symmetry more likely is a natural one, particularly
in light of results by Erdős and others proving that almost every graph is asymmetric.
In some 1981 and 1982 papers, Babai and Godsil conjectured that as $r \to \infty$, over all
groups $G$ of size $r$, the proportion of Cayley digraphs that are DRRs tends to $1$. They
proved this to be true for the restricted family of nilpotent groups of odd order.
I will talk about recent joint work with Pablo Spiga in which we prove this conjecture.


Mar 11 
PengJie Wong 
On Siegel Zeros 
at noon
in D631

(University of Lethbridge)

Siegel zeros (or LandauSiegel zeros) are potential counterexamples to the
generalised Riemann hypothesis (for $L$functions). Such zeros, if exist, have
to be "very close" to 1 over the complex plane.
In this talk, we will discuss some results concerning Siegel zeros and their
applications. If time permits, we will talk about the corresponding
"SiegelWalfisz theorems" for certain $L$functions.


Mar 18 
Lee Troupe 
Sums of divisors 
at noon
in D631

(University of Lethbridge)

What happens when you add up the divisors of an integer? This seemingly
innocuous question has motivated mathematicians across the ages, from antiquity
to the present day. In this talk, we'll survey some conjectures and results on
the functions $s(n)$, the sum of the proper divisors of an integer $n$, and
$\sigma(n) = s(n) + n$, the sum of all divisors of $n$. The ancient Greeks
derived religious significance from certain values of these functions; we'll
do no such thing in this talk. However, by asking simple questions whose answers
turn out to be very complicated – if they're known at all – we will see that
an air of mystery continues to surround these two fascinating functions. The
first half of this talk, at the very least, will be extremely accessible to
everyone; tell your students!


Mar 25 
Qing Zhang 
On the holomophy of adjoint Lfunction for GL(3) 
at noon
in D631

(University of Calgary)

Lfunctions associated with automorphic forms are vast generalizations of
Riemann zeta functions and Dirichlet Lfunctions. Although the theory of
Lfunctions play a fundamental role in number theory, it is still largely
conjectural. If $\pi$ is an irreducible cuspidal automorphic representation
of $GL_n$ over a number field $F$ and $\tilde \pi$ is its dual representation,
it is conjectured that the Dedekind zeta functions $\zeta_F(s)$ (which is the
Riemann zeta function when $F=\mathbb{Q}$) "divides" the RankinSelberg
Lfunction $L(s,\pi\times \tilde \pi)$, i.e., the quotient
$L(s,\pi\times \tilde \pi)/\zeta_F(s)$ (which is called the adjoint Lfunction
of $\pi$) should be entire. For $n=2$, this conjecture was proved by
GelbartJacquet. In this talk, I will give a sketchy survey of constructions
of some Lfunctions, including the RankinSelberg Lfunction
$L(s,\pi_1\times \pi_2)$, and report our recent work on the above conjecture
when $n=3$. This is a joint work with Joseph Hundley.


Apr 1 
Quanli Shen 
The ternary Goldbach problem with primes in positive density sets 
at noon
in D631

(University of Lethbridge)

The ternary Goldbach problem stated that every odd integer greater than $5$
can be written as sums of three primes. This was proved by Vinogradov for all
sufficiently large odd integers and completely proved by Helfgott and Platt.
On the other hand, by using Green's transference principle method one can extend
Vinogradov's result to a density version. In this talk, I will talk about
recent results in this direction.


May 30 
Majid Shahabi 
The appearance of padic numbers 
at 11am
in C630


In this talk, we present an introduction to padic numbers and their appearance
in number theory. In particular, we give a proof of Ostrowski's theorem.


June 24 
Cameron Franc 
Classification of some threedimensional vertex operator algebras 
at noon
in C630

(University of Saskatchewan)

Vertex operator algebras (VOAs) as discussed in this talk
are graded complex vector spaces with a collection of many bilinear
operations satisfying some intricate identities. They first arose in
mathematics and physics via the study of string theory and conformal
field theories, in the study of the Monster group and moonshine, and
in the study of the representation theory of affine Lie algebras. From
their inception it has been known that VOAs are deeply connected with
the theory of modular forms.
In this talk we will explain how results on vector valued modular
forms can be used to classify VOAs satisfying certain finiteness
conditions. Our classification for VOAs with exactly three simple
representations rests on using arithmetic properties of a family of
modular forms expressed in terms of generalized hypergeometric series.
The classification of the integral specializations of this family
relies on properties of the monodromy, and on results on distributions
of primes in arithmetic progressions. No prior familiarity with VOAs
will be assumed, and we will focus primarily on the number theoretic
aspects of the problem.

