University of Lethbridge

Motohashi (following Selberg) developped in 1978 a method to prove the Hoheisel and Linnik theorem without any deep mention of the zeros. We will present this approach. The first lecture will be a sketch of the argument and will introduce the tools...

We will present a surprising exact and truncated Perron summation formula as well as several other formulaes. We will proceed by comparing the relative strenghts of the smoothing method versus Perron summation.

While the two fields named in the title seem unrelated, there is a strong link between them. Indeed, random matrix theory makes predictions in both fields, and some of these predictions can be verified rigorously on the number theory side. This...

We will describe the celebrated ABC conjecture, due to Oesterle and Masser, formulated in 1985. We will explain one of the motivations behind this conjecture and discuss a refinement of this conjecture put forward by Baker in 1996.

In 1983, Maru\v si\v c posed the problem of determining which positive integers $n$ have the property that every vertex-transitive graph of order $n$ is isomorphic to a Cayley graph of some group. Such an integer $n$ is called a Cayley number. Much...

Let $a,q$ be relatively prime integers. Then consider $$ \psi (x; q, a) = \sum_{ \begin{smallmatrix} n \le x \\ {n \equiv a} \ (\mathrm{mod}\, {q}) \end{smallmatrix} } \Lambda(n) .$$We discuss new explicit bounds for $\psi(x; q, a)$, which provide an...

Optimal transport is a problem first considered by Monge in 1781. Given a distribution of mines over a landscape, producing (say) iron, a distribution of factories over that same landscape, consuming iron, and a transport cost (x,y) to move one unit...

`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...