University of Lethbridge

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...

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. Let $M_E(N)$ be the function that counts the number of primes $p$ of good reduction such that $\#E_p(\mathbb{F}_p) = N$ where $N$ is a fixed integer and $E_p(\mathbb{F}_p)$ denotes the group of...

Let $R$ be an integral domain and let $A$ be a finitely generated free abelian group. An elliptic net is a map $W \colon A\longrightarrow R$ with $W(0)=0$, and such that for all $p,q,r,s \in A$, \begin{multline*} W(p+q+s) \, W(p-q) \, W(r+s) \, W(r)\...

In 1979 G. Belyi proved that given any smooth curve $C$ over any number field, there is map $\beta:C \rightarrow \mathbb{P}^1$ such that $\beta$ is unramified outside of three points. This is particularly striking since Belyi's theorem is not true...

We will discuss a few basic properties of "arithmetic groups," which are certain groups of n-by-n matrices with integer entries. By definition, the subject combines algebra (group theory and matrices) with number theory (the integers), but it also...

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.

Let $L(s)$ be the $L$-function associated to a modular form of weight $k$ for the full modular group. Using spectral theory, Anton Good proved that $L(k/2+it) \ll t^{1/6} (\log t)^{5/6}$. Matti Jutila discovered a simpler proof which makes use of the...

Historically functions have been treated as a kind of meta-object. This all changed with the work of Frege, Russell and Church. Furthermore, the challenges of the paradoxes led to the formalisation of type theory by Russell. Since, functions and...

Mathematical texts can be computerised in many ways. At one end there is document imaging, at the other there are proof assistants (Mizar, Isabelle, Coq, etc.).In between, there are typesetting (e.g., LaTeX and MathML) and semantically oriented (e.g...