Date 
Speaker 
Title 

Aug 27 
Allysa Lumley 
Distribution of Values of $L$functions associated to Hyperelliptic Curves over Finite Fields 
at noon
in C630

York University

In 1992, Hoffstein and Rosen proved a function field analogue to Gau\ss' conjecture (proven by Siegel) regarding the class number, $h_D$, of a discriminant $D$ by averaging over all polynomials with a fixed degree. In this case $h_D=\text{Pic}(\mathcal{O}_D)$, where $\text{Pic}(\mathcal{O}_D)$ is the Picard group of $\mathcal{O}_D$.
Andrade later considered the average value of $h_D$, where $D$ is monic, squarefree and its degree $2g+1$ varies. He achieved these results by calculating the first moment of $L(1,\chi_D)$ in combination with Artin's formula relating $L(1,\chi_D)$ and $h_D$. Later, Jung averaged $L(1,\chi_D)$ over monic, squarefree polynomials with degree $2g+2$ varying. Making use of the second case of Artin's formula he gives results about $h_DR_D$, where $R_D$ is the regulator of $\mathcal{O}_D$.
For this talk we discuss the complex moments of $L(1,\chi_D)$, with $D$ monic, squarefree and degree $n$ varying. Using this information we can describe the distribution of values of $L(1,\chi_D)$ and after specializing to $n=2g+1$ we give results about $h_D$ and specializing to $n=2g+2$ we give results about $h_DR_D$.


Sept 10 
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.


Sept 17 
Darcy Best 
Transversal This, Transversal That 
at noon
in C630

Monash University

We will discuss several results related to transversals in Latin squares (think
"Sudoku") and other Latin squarelike objects. These results centre around showing
the existence of, the number of, or the structure of transversals in different cases.
Several patterns have been unearthed via computation that will also be discussed.
These patterns are often related to the permanent of certain 01 matrices that can
be used to count the number of transversals in Latin squares.


Sept 24 
Nathan Ng 
Mean values of $L$functions 
at noon
in C630

(University of Lethbridge)

In the past twenty years there has been a flurry of activity in
the study of mean values of $L$functions. This was precipitated by
groundbreaking work of Keating and Snaith in which they modelled
these mean values by random matrix theory. In this talk I will survey the
best known asymptotic results for the mean values of $L$functions in certain families
(the Riemann zeta function on the half line and/or quadratic Dirichlet $L$functions
at the central point $s=1/2$).


Oct 1 
Lee Troupe 
Distributions of polynomials of additive functions 
at noon
in C630

(University of Lethbridge)

How is the set of values of an arithmetic function distributed? In a seminal
1940 paper, Erdős and Kac answered this question for a class of additive
functions satisfying certain mild hypotheses, a class which includes the
numberofprimedivisors function. Using ideas from both probability and
number theory, they showed that the values of these additive functions tend
toward a Gaussian normal distribution. In the intervening years, this
"ErdősKac class" of additive functions has been broadened to include
certain compositions of arithmetic functions, as well as arithmetic functions
defined on natural sequences of integers, such as shifted primes and values of
polynomials. In this talk, we will discuss recent joint work with Greg Martin
(UBC) which further expands the ErdősKac class to include arbitrary sums
and products of additive functions (satisfying the same mild hypotheses).


Oct 4 
Khoa Dang Nguyen 
Linear recurrence sequences and some related results in arithmetic dynamics and ergodic theory 
at 12:15pm
in D631

University of Calgary

First we introduce some results about certain simple diophantine equations
involving linear recurrence sequences. Then we present 2 applications. The
first application involves the socalled Orbit Intersection Problem for the
arithmetic dynamics of semiabelian varieties and linear spaces. The second
application involves certain ergodic averages for surjective endomorphisms of
compact abelian groups. Parts of this come from joint work with Dragos Ghioca.


Oct 15 
Muhammad Khan 
Fractional clique $k$covers, vertex colorings and perfect graphs 
at noon
in C630

(University of Lethbridge)

The relationship between independence number, chromatic number, clique number,
clique cover number and their fractional analogues is wellestablished for
perfect graphs. Here, we study the clique $k$cover number $cc_{k}(G)$ and the
fractional clique $k$cover number $cc_{fk}(G)$ of a graph $G$. We relate
$cc_{fk}(G)$ to the fractional chromatic number of the complement
$\overline{G}$, obtaining a Nordhaus–Gaddum type result. We modify the method
of Kahn and Seymour, used in the proof of the fractional
Erdős–Faber–Lovász conjecture, to derive an upper bound on
$cc_{fk}(G)$ in terms of the independence number $\alpha(G')$ of a particular
induced subgraph $G'$ of $G$. When $G$ is perfect, we get the sharper relation
$cc_{fk}(G)= k cc_{1}(G)= k\alpha(G) \le cc_{k}(G)$. This is in line with a
result of Grötschel, Lovász, and Schrijver on clique covers of perfect
graphs. Moreover, we derive an upper bound on the fractional chromatic number
of any graph, which is tight for infinitely many perfect as well as nonperfect
graphs. This is joint work with Daya Gaur (Lethbridge).


Oct 22 
Andrew Fiori 
The Least Prime in the Chebotarev Density Theorem 
at noon
in C630

(University of Lethbridge)

The classic theorem of Chebotarev tells us that for any Galois extension $L/K$ of
degree $n$ the proportion of primes of $K$, whose Frobenius conjugacy class is a
given conjugacy class of the Galois group is proportional to the size of that
conjugacy class. If one interprets this as a statement that the Frobenius of a
randomly chosen prime is uniformly distributed, then a natural consequence is
that if we begin selecting primes at random, by the time we select roughly
$n \log(n)$ primes, we should expect to encounter every conjugacy class at least
once. Given that selecting the first $m$ primes is hardly random, and there are
infinitely many fields it is hardly surprising that this expectation will often
not be met by simply looking at the first $m$ unramified degree one primes.
None the less, there are many known and conjectured upper bounds, relative to
the absolute discriminant of $L$, on the smallest prime for the Chebotarev theorem.
In this talk we will discuss several aspects of this problem, including, as
time allows, some recent work on computationally verifying some of these
conjectures for all fields with small discriminants and on the discovery, by
way of this computational verification, of an infinitely family of fields for
which the smallest prime in the Chebotarev theorem is "large".


Oct 29 
Dave Morris 
Cayley graphs of order $kp$ are hamiltonian for $k < 48$ 
at noon
in C630

(University of Lethbridge)

For every generating set $S$ of any finite group $G$, there is a corresponding
Cayley graph $\mathrm{Cay}(G;S)$. It was conjectured in the early 1970's that
$\mathrm{Cay}(G;S)$ always has a hamiltonian cycle, but there has been very little
progress on this problem. Joint work with Kirsten Wilk has established the
conjecture in the special case where the order of $G$ is $kp$, with $k < 48$
and $p$ prime. This was not previously known for values of $k$ in the set
$\{24, 32, 36, 40, 42, 45\}$.


Nov 5 
PengJie Wong 
Dirichlet's Theorem for Modular Forms 
at noon
in C630

(University of Lethbridge)

Dirichlet's theorem on arithmetic progressions states that for any $(a,q)=1$,
there are infinitely many primes congruent to $a$ modulo $q$. Such a theorem
together with Euler's earlier work on the infinitude of primes represents the
beginning of the study of Lfunctions and their connection with the distribution
of primes.
In this talk, we will discuss some ingredients of the proof for the theorem.
Also, we will explain how such an Lfunction approach leads to Dirichlet's
theorem for modular forms that gives a count of Fourier coefficients of modular
forms over primes in arithmetic progressions.


Nov 19 
Kirsty Chalker 
Perron's formula and explicit bounds on sums 
at noon
in C630

(University of Lethbridge)

Previously, in this seminar series, we have heard about explicit bounds on
\[
\psi (x) := \displaystyle{\sum_{n \leq x} \Lambda (n)},
\]
which refers to the von Mangoldt function $\Lambda (n).$ The point of liftoff for bounding
this sum is the explicit formula, which pulls the zeros of the Riemann zetafunction into
the equation. However, there are other sums for which using an explicit formula is currently
unconditionally impossible. In this talk, I will outline the work of my current thesis, in
which I prove bounds for a somewhat general function
$\displaystyle{\sum_{n \leq x} \frac{a_n}{n^s}}$ with $a_n, s \in \mathbb{C},$ and apply
these bounds to the sums
\[
M(x) := \sum_{n \leq x} \mu (n)~~\text{and}~~m(x) = \sum_{n \leq x} \frac{\mu (n)}{n},
\]
which refer to the Möbius function $\mu (n)$.


Nov 26 
Farzad Maghsoudi 
Finding Hamiltonian cycles in Cayley graphs of order $6pq$ 
at noon
in C630

(University of Lethbridge)

Suppose $G$ is a finite group of order $6pq$ such that $p$ and $q$ are distinct
prime numbers. It is conjectured that, if $S$ is any generating set of $G$, then
there is a Hamiltonian cycle in $Cay(G;S)$. The talk will discuss a special case
of this problem which is solved.


Dec 3 
Lucile Devin 
Continuity of the limiting logarithmic distribution in Chebyshev's bias 
at noon
in C630

University of Ottawa

Following the framework of Rubinstein and Sarnak for Chebyshev's bias, one
obtains a limiting logarithmic distribution $\mu$. Then assuming that the
zeros of the $L$functions are linearly independent over $\mathbf{Q}$, one can
show that the distribution $\mu$ is smooth.
Inspired by the notion of selfsufficient zeros introduced by Martin and Ng, we
use a much weaker hypothesis of linear independence to show that the distribution
$\mu$ is continuous. In particular the existence of one selfsufficient zero is
enough to ensure that the bias is well defined.

