University of Lethbridge

`Modular curves are the parameter space of elliptic curves with certain torsion structure.' In this talk, I will explain what that sentence means, and how it is related to the usual definition of modular curves in terms of quotients of the complex...

We recall the formalism of arithmetic fundamental groups according to Grothendieck and the circle of ideas surrounding his 1983 letter to Faltings. We discuss also non-abelian reciprocity maps and applications to Diophantine problems.

A Cayley graph Cay(G;S) has the elements of G as its vertices, with g--gs if and only if s is an element of S. There is a natural colouring of the edges of any such graph, by assigning colour s to an edge if it came from the element s of S. For a...

claVision team: Mohammad Akbari, Hossein Naseri and Howard Cheng Abstract: The claVision team from the University of Lethbridge entered the Microsoft Imagine Cup Innovation Competition in 2014 and was chosen as the Canadian winner. In addition, the...

Let $d(n)=\sum_{d|n} 1$. This is known as the divisor function. It counts the number of divisors of an integer. Consider the following shifted convolution sum \begin{equation*} \sum_{an-m=h}d(n) \, d(m) \, f(an, m), \end{equation*} where $f$ is a...

We study the distribution of primes for which the index and order of a modulo p have a fixed common divisor. We will also motivate this problem through previously known results related to Artin's conjecture for primitive roots.

Let $\pi(x;q,a)$ denote the number of primes up to $x$ that are congruent to $a \pmod{q}$. A ``prime number race", for fixed modulus~$q$ and residue classes $a_1,\ldots,a_r$, investigates the system of inequalities $$\pi(x;q,a_1)>\pi(x;q,a_2)> \cdots...

Let $G$ be a compact semisimple Lie group, and let $H$ be a closed subgroup of $G$. Given a linear representation $\tau:H\to\mathrm{End}(V)$ of $H$, one can form an associated vector bundle $\mathcal{V}_\tau \to G/H$ over the homogeneous space $G/H$...

We discuss the solvability of certain linear equations in sparse subsets of the squares. Specifically, we investigate equations of the form \begin{align*} \lambda_1 n_1^2 + \dotsb + \lambda_s n_s^2 = 0, \end{align*} where $s \geq 7$ and the...

We describe heuristics for several well-known density conjectures in prime number theory and elliptic curves theory. In each case we show how one can arrive at explicit expressions for density constants. The conjectures include twin prime conjecture...