Date 
Speaker 
Title 

Sep 11 
everyone 
Open problem session 
at noon
in C630


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.


Sep 18 
PengJie Wong 
Nearly supersolvable groups and Artin's conjecture 
at noon
in C630

(University of Lethbridge)

Let $K/k$ be a Galois extension of number fields with Galois group $G$, and let $\rho$ be a nontrivial irreducible representation of $G$ of dimension $n$. The Artin holomorphy conjecture asserts that the Artin $L$function attached to $\rho$ extends to an entire function.
It is wellknown that when $n=1$, this conjecture follows from Artin reciprocity. Also, by the works of Langlands and many others, we know that this conjecture is valid for $n=2$ under certain conditions. However, in general, the Artin holomorphy conjecture is wildly open.
In this talk, we will discuss how elementary group theory plays a role in studying the Artin holomorphy conjecture and introduce the notion of "nearly supersolvable groups". If time allows, we will explain how such groups lead to a proof of the Artin holomorphy conjecture for Galois extensions of degree less than 60.


Sep 25 
Muhammad Khan 
The contact graphs of totally separable packings 
at noon
in C630

(University of Lethbridge)

Contact graphs have emerged as an important tool in the study of translative packings
of convex bodies and have found numerous applications in materials science. The
contact number of a packing of translates of a convex body is the number of edges
in the contact graph of the packing, while the Hadwiger number of a convex body is
the maximum vertex degree over all such contact graphs. In this talk, we investigate
the Hadwiger and contact numbers of totally separable packings of convex bodies, known
as the separable Hadwiger number and the separable contact number, respectively. We
show that the separable Hadwiger number of any smooth strictly convex body in dimensions
$d = 2, 3, 4$ is $2d$ and the maximum separable contact number of any packing of $n$
translates of a smooth strictly convex domain is $\lfloor 2n  2\sqrt{n} \rfloor$. Our
proofs employ a characterization of total separability in terms of hemispherical caps
on the boundary of a smooth convex body, Auerbach bases of finite dimensional real
normed spaces, angle measures in real normed planes, minimal perimeter polyominoes
and an approximation of smooth $o$symmetric strictly convex domains by, what we call,
Auerbach domains. This is joint work with K. Bezdek (Calgary) and M. Oliwa
(Calgary).


Oct 2 
Andrew Fiori 
The average number of quadratic Frobenius pseudoprimes 
at noon
in C630

(University of Lethbridge)

Primality testing has a number of important applications. In particular in
cryptographic applications the complexity of existing deterministic algorithms
causes increasing latency as the size of numbers we must test grow and the
number of tests we must run before finding a prime grows aswell. These
observations lead one to consider potentially nondeterministic algorithms
which are faster, and consequently leads one to consider the false positives
these algorithms yield, which we call pseudoprimes.
In this talk I will discuss my recent work with Andrew Shallue where we study Quadratic
Frobenius Pseudoprimes. I shall describe our results on an asymptotic lower bounds on
the number of false positives. These results represent a generalization of those
ErdosPomerance concerning similar problems for (Fermat) pseudoprimes.


Oct 16 
Lee Troupe 
Normally Distributed Arithmetic Functions 
at noon
in C630

(University of British Columbia)

In the late 1930s, Paul Erdős attended a seminar at Cornell University
given by Mark Kac, who suspected that divisibility by primes satisfies a
certain "statistical independence" condition. If this were true, the
central limit theorem could be used to show that the number of distinct prime
factors of $n$, as $n$ varies over the natural numbers, is normally
distributed, with mean $\log\log n$ and standard deviation $\sqrt{\log\log n}$.
Erdős used sieve methods to confirm Kac's intuition, and the resulting
ErdősKac theorem is a foundational result in the field of probabilistic
number theory. Many different proofs of and variations on the ErdősKac
theorem have been given in the intervening decades. This talk will highlight
some of these results and the techniques used to obtain them, including recent
work of the speaker and Greg Martin (UBC).


Oct 23 
Sam Broadbent, Habiba Kadiri, and Kirsten Wilk 
Sharper bounds for Chebyshev functions $\boldsymbol{\theta(x)}$ and $\boldsymbol{\psi(x)}$ 
at noon
in C630

(University of Lethbridge)

In this talk we report on some research projects from summer 2017 supported by NSERCUSRA.
In the first part of the project, we surveyed all existing explicit results from the past 60 years on prime counting functions,
with a special focus on $\theta(x)$ (counting $\log p$ for each prime $p \le x$).
In the second part, we provided new bounds for the Chebyshev function $\psi(x)$ based on a recent zero density result for the zeros of the Riemann zeta function (due to KadiriLumleyNg).
Finally, we have established the current best results for the prime counting function $\theta(x)$ for various ranges of $x$.
(Joint work with Noah Christensen, Allysa Lumley, and Nathan Ng)


Oct 30 
Akshaa Vatwani 
Variants of equidistribution in arithmetic progression 
at noon
in C630

(University of Waterloo)

It is well known that the prime numbers are equidistributed in arithmetic
progression. Such a phenomenon is also observed more generally for a class of
multiplicative functions. We derive some variants of such results and give an
application to tuples of squarefree integers in arithmetic progression.
We also discuss an interesting application that relates to the Chowla conjecture on
correlations of the Möbius function, and show its relevance to the twin prime conjecture.


Nov 6 
Károly Bezdek 
Bounds for totally separable translative packings in the plane 
at noon
in C630

(University of Calgary)

A packing of translates of a convex domain in the Euclidean plane is said to
be totally separable if any two packing elements can be separated by a line
disjoint from the interior of every packing element. This notion was
introduced by G. Fejes Toth and L. Fejes Toth (1973) and has attracted
significant attention. In this lecture I will discuss the separable analogue
of the classical inequality of N. Oler (from geometry of numbers) for totally
separable translative packings of convex domains and then derive from it some
new results. This includes finding the largest density of totally separable
translative packings of an arbitrary convex domain and finding the smallest
area convex hull of totally separable packings (resp., totally separable soft
packings) generated by given number of translates of a convex domain (resp.,
soft convex domain). Last but not least, we determine the largest covering
ratio (that is, the largest fraction of the plane covered) of totally
separable soft circle packings with given soft parameter. This is a joint work
with Zsolt Langi (Univ. of Technology, Budapest, Hungary).


Nov 20 
Forrest Francis 
Euler's Function on Products of Primes in Progressions 
at noon
in C630

(University of Lethbridge)

Let $\phi(n)$ be Euler's totient function and let $q$ and $a$ be fixed coprime
natural numbers. Denote by $S_{q,a}$ the set of natural numbers whose prime
divisors are all congruent to $a$ modulo $q$. We can establish
\[\limsup_{n \in S_{q,a}} \frac{n}{\phi(n) (\log(\phi(q)\log{n}))^{1/\phi(q)}}
= \frac{1}{C(q,a)},\]
where $C(q,a)$ is a constant associated with a theorem of Mertens. We may then wish to
know whether there are infinitely many $n$ in $S_{q,a}$ for which
\[
\frac{n}{\phi(n)(\log\phi(q)\log{n})^{1/\phi(q)}} > \frac{1}{C(q,a)}
\qquad (*)
\]
is true. In the case $q=a=1$, Nicolas (1983) established that if the Riemann hypothesis is
true, then ($*$) holds for all primorials (products of the form $\prod_{p \leq x} p$), but
if the Riemann hypothesis is false then there are infinitely many primorials for which
($*$) is true and infinitely many primorials for which ($*$) is false.
In this talk we will show that, for some $q>1$, the work of Nicolas can be generalized by
replacing the Riemann hypothesis with analogous conjectures for Dirichlet $L$functions
and replacing the primorials with products of the form
\[\prod_{\substack{p \leq x \\ p \,\equiv \,a \,(\mathrm{mod}\,q)}} p.\]


Nov 27 
Sara Sasani 
A Strongly Regular Decomposition of the Complete
Graph and its Association Scheme 
at noon
in C630

(University of Lethbridge)

A Strongly Regular Graph, SRG$({\color{blue}\nu,\color{blue}k,\color{blue}\lambda,\color{blue}\mu})$, is a $\color{blue}k$regular graph with
$\color{blue}\nu$ vertices such that every two adjacent vertices have $\color{blue}\lambda$ common neighbors, and
every two nonadjacent vertices have $\color{blue}\mu$ common neighbors. For each positive integer $m$, a construction for
$\color{red}{2^m}$ disjoint SRG$({\color{blue}{2^{2m}(2^m+2)}},\color{blue}{2^{2m}+2^m}, \color{blue}{2^m},\color{blue}{2^m})$ will be shown to form a decomposition of the complete
graph with $\color{blue}{2^{2m}(2^m+2)}$ vertices, if the cliques of size $\color{red}{2^{2m}}$ is considered as a strongly regular graph with parameter
$(\color{blue}{2^{2m}(2^m+2)},\color{blue}{2^{2m}1}, \color{blue}{2^{2m}2},\color{blue}{0})$.
By decomposing the cliques and the strongly regular graphs further, we show the existence of a symmetric association scheme with
$\color{red}{2^{m+2}2}$ classes and explain, by an example, how to find the first and second eigenmatrices of the scheme.


Dec 4

Clifton Cunningham 
On the modularity conjecture for abelian varieties over $\pmb{\mathbb{Q}}$ 
at noon
in C630

(University of Calgary)

The modularity theorem tells us that for every elliptic curve $E$ over $\mathbb{Q}$, there is a modular form $f_E$ such that the Lfunction $L(s,f_E)$ for $f_E$ coincides with the Lfunction $L(s,\rho_E)$ for the Galois representation on the Tate module of $E$. In fact, $f_E$ is a cusp form and its level is determined by the conductor of $\rho_E$. Since the modular form $f_E$ determines an automorphic representation $\pi_E$ of $\operatorname{GL}_2(\mathbb{A}_\mathbb{Q})$ with the same Lfunction as $f_E$, we have
\[
L(s,\rho_E) = L(s,\pi_E).
\]
The modularity conjecture for abelian varieties is the obvious generalization of this theorem from the case of onedimensional abelian varieties: For every abelian variety $A$ over $\mathbb{Q}$ there is an automorphic representation $\pi_A$ of a group $G(\mathbb{A}_\mathbb{Q})$ such that
\[
L(s,\rho_A) = L(s,\pi_A),
\]
where $\rho_A$ is the Galois representation on the Tate module of $A$.
In this talk I will describe recent joint work with Lassina Dembélé giving new instances of the modularity conjecture for abelian varieties over $\mathbb{Q}$. Where do we hunt for $\pi_A$? What is the group $G$ over $\mathbb{Q}$? What is the level of $\pi_A$? Can we find a generalized modular form $f_A$ from which $\pi_A$ can be built? I will explain how we use work of Benedict Gross, Freydoon Shahidi and others to answer these questions. I will also explain how thesis work by Majid Shahabi illuminates the level of $\pi_A$.
This is joint work with Lassina Dembélé.

