
Date 
Speaker 
Title 


Sept 28 
everyone 
Organizational meeting and problem session 

at noon
online

(University of Lethbridge)

We will discuss how often to meet, and choose some of the
speakers for the rest of the semester. There will also be an opportunity to share
(math) problems with the rest of the audience.



Oct 5 
Allysa Lumley 
Primes in short intervals: Heuristics and calculations 

at noon
online

(Centre de Recherches Mathématiques, Montréal)

We formulate, using heuristic reasoning, precise conjectures for the range of the number of primes in intervals of length $y$ around $x$,
where $y\ll (\log x)^2$. In particular, we conjecture that the maximum grows surprisingly slowly as $y$ ranges from $\log x$ to $(\log x)^2$.
We will show that our conjectures are somewhat supported by available data, though not so well that there may not be room for some modification. This is joint work with Andrew Granville.



Oct 19 
Elchin Hasanalizade 
Distribution of the divisor function at consecutive integers 

at noon
online

(University of Lethbridge)

The purpose of this talk is to discuss the famous ErdősMirsky conjecture and problems on consecutive values of multiplicative functions that arise in connection with this conjecture. We'll show how the results of Goldston et al. can be used to sharpen Hildebrand's earlier result on a conjecture of Erdős on limit points of the sequence ${d(n)/d(n+1)}$. If time permits. we'll also discuss some open problems.



Oct 26 
Kubra Benli 
Prime power residues modulo $p$ 

at noon
online

(University of Georgia)

Let $p$ be a prime number. For each positive integer $k\geq 2$, it is widely believed that the smallest prime that is a $k$th power residue modulo $p$ should be $O(p^{\epsilon})$, for any $\epsilon>0$. Elliott proved that such a prime is at most $p^{\frac{k1}{4}+\epsilon}$, for each $\epsilon>0$. In this talk we will discuss the distribution of prime $k$th power residues modulo $p$ in the range $[1, p]$, with a more emphasis on the subrange $[1,p^{\frac{k1}{4}+\epsilon}]$ for $\epsilon>0$.



Nov 2 
Iren Darijani 
Colourings of $G$designs 

at noon
online

(University of Lethbridge)

A $G$decomposition of a graph $H$ consists of a set $V$ of vertices of $H$ together with a set $\mathcal{B}$ of subgraphs (called blocks) of $H$, each isomorphic to $G$, that partition the edge set of $H$. A $G$design of order $n$ is a $G$decomposition of the complete graph $K_n$ on $n$ vertices. A complete graph is a simple graph in which every pair of distinct vertices is connected by a unique edge. A $G$design is said to be $k$colourable if its vertex set can be partitioned into $k$ sets (called colour classes) such that no block is monochromatic. It is $k$chromatic if it is $k$colourable but is not $(k  1)$colourable.
The block intersection graph of a $G$design with block set $\mathcal{B}$ is the graph with $\mathcal{B}$ as its vertex set such that two vertices are adjacent if and only if their associated blocks are not disjoint. The chromatic index of a graph $H$ is the least number of colours that enable each edge of $H$ to be assigned a single colour such that adjacent edges never have the same colour.
In this talk we will first see some results on the chromatic index of block intersection graphs of Steiner triple systems. A Steiner triple system of order $v$ is a $K_3$design of order $n$ with $n = v$. We will then discuss some other results on $k$colourings of $e$star systems for all $k \ge 2$ and $e \ge 3$. An $e$star is a complete bipartite graph $K_{1,e}$. An $e$star system of order $n > 1$, is a $G$design of order $n$ when $G$ is an $e$star. Finally, we will see some other results on $k$colourings of path systems for any $k \ge 2$. A path system is a $G$design when $G$ is a path.



Nov 16 
Yash Totani 
On the number of representations of an integer as a sum of $k$ triangular numbers 

at noon
online

(University of Lethbridge)

The $m$th $r$gonal number is defined by $$p_r(m)=[(r2)m^2(r4)m]/2.$$ The problem of finding the number of representations of an integer as a sum of $k$ $r$gonal numbers is well studied, however a lot of interesting related questions are still unresolved. When $r=4$, this problem converts to the classical problem of finding the number of representations of an integer as a sum of $k$ squares.
In this talk, we discuss the case $r=3$ where we study the number of representations of an integer by $k$ triangular (3gonal) numbers. Let $t_k(n)$ be the number of representations of a positive integer $n$ as a sum of $k$ triangular numbers. One of the oldest results in connection with triangular numbers is due to Gauss who proved that $t_3(n)\geq 1$ for any positive integer $n$. We discuss some new results in the theory by Atanasov et al. where they provide exact formulas for $t_{4k}(n)$, which are proved mysteriously. Here, we discuss a method, inspired by the work of Dr. Zafer Selcuk Aygin, which can be used to give a more concrete proof for the result mentioned above using the modular forms of even integral weight. We also obtain an extension of the above and provide formulas for $t_{4k+2}(n)$ using the theory of odd integral weight modular forms for $\Gamma_0(8)$.



Nov 23 
Dave Morris 
Automorphisms of direct products of some circulant graphs 

at noon
online

(University of Lethbridge)

The direct product of two graphs $X$ and $Y$ is denoted $X \times Y$. This is a natural construction, so any isomorphism from $X$ to $X'$ can be combined with any isomorphism from $Y$ to $Y'$ to obtain an isomorphism from $X \times Y$ to $X' \times Y'$. Therefore, the automorphism group $\mathrm{Aut}(X \times Y)$ contains a copy of $(\mathrm{Aut}\, X) \times (\mathrm{Aut}\, Y)$. It is not known when this inclusion is an equality, even for the special case where $Y = K_2$ is a connected graph with only $2$ vertices.
Recent work of B.Fernandez and A.Hujdurović solves this problem when $X$ is a "circulant" graph with an odd number of vertices (and $Y = K_2$). We will present a short, elementary proof of this theorem.



Nov 30 
Milad Fakhari 
The Correction Factors in Artin's Type Problems 

at noon
online

(University of Lethbridge)

In 1927, Emil Artin formulated a conjecture for the density $\delta(g)$ of primes for which a given integer $g$ is a primitive root, i.e., $\langle g \mathop{\mathrm{mod}} p\rangle=\mathbb{F}_p^{\times}$. Such prime $p$ does not split completely in the splitting field of $x^qg$, indicated by $K_q$, for all primes $q\mid p1$. For $g\neq 1$ and nonsquare, Artin conjectured
$$\delta(g)=\prod_p\left(1\frac{1}{[K_p:\mathbb{Q}]}\right).$$
In 1957, computations, carried by Derrick H. Lehmer and Emma Lehmer, revealed that for some $g$ (for example $g=5$) the conjectured density is quite different from the one computed. Artin provided a correction factor to handle this mismatch. In fact, the correction factor appears because of the "entanglement" between number fields $K_n=\prod_{p\mid n}K_p$ for $n$ squarefree when the squarefree part of $g$ is congruent to $1$ modulo $4$. In 1967, Hooley proved the modified Artin's conjecture under the assumption of the generalized Riemann hypothesis.
In this talk, we will present the character sums method formulated by Lenstra, Moree and Stevenhagen which gives us an effective way to find the correction factor and the Euler product of the density in Artin's conjecture and the cyclicity problem for Serre curves. We will observe that this method does not work in some Artinlike problems such as the Kummerian Titchmarsh divisor problem. At the end, we will indicate that how a modification of the character sums method may lead to the correction factor in such problems.

