at noon
in C630

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 $h_DR_D$.

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.

at noon
in C630

We will discuss several results related to transversals in Latin squares (think "Sudoku") and other Latin square-like 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 0-1 matrices that can be used to count the number of transversals in Latin squares.

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 $s=1/2$).

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 number-of-prime-divisors 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 "Erdős-Kac 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 Erdős-Kac class to include arbitrary sums and products of additive functions (satisfying the same mild hypotheses).

First we introduce some results about certain simple diophantine equations involving linear recurrence sequences. Then we present 2 applications. The first application involves the so-called 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.

at noon
in C630

(University of Lethbridge)

The relationship between independence number, chromatic number, clique number, clique cover number and their fractional analogues is well-established 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 non-perfect graphs. This is joint work with Daya Gaur (Lethbridge).

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".

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 $\{24, 32, 36, 40, 42, 45\}$.

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 L-functions 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 L-function approach leads to Dirichlet's theorem for modular forms that gives a count of Fourier coefficients of modular forms over primes in arithmetic progressions.

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 lift-off for bounding this sum is the explicit formula, which pulls the zeros of the Riemann zeta-function 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 $\mu (n)$.

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.

at noon
in C630

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 self-sufficient 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 self-sufficient zero is enough to ensure that the bias is well defined.