University of Lethbridge

We will discuss a graph that encodes the divisibility properties of integers by primes. We prove that this graph has a strong local expander property almost everywhere. We then obtain several consequences in number theory, beyond the traditional...

In 1944, Linnik showed that the least prime in an arithmetic progression given by a mod q for (a,q)=1 is at most cq^L for some absolutely computable constants c and L. A lot of work has gone in computing explicit bounds for c and L. The best known...

It is a classical problem to understand the set of Jacobians of curves among all abelian varieties, i.e., the image of the map Mg→Ag which sends a curve X to its Jacobian JX. In characteristic p, Ag has interesting filtrations, and we can ask how the...

In his famous '86 paper, Andrews made several conjectures on the function σ(q) of Ramanujan, including that it has coefficients (which count certain partition-theoretic objects) whose sup grows in absolute value, and that it has infinitely many...

Given coprime integers $a,b$, the classical identity of Bezout provides integers $u,v$ such that $au-bv = 1$. We consider refinements to this identity, where we ask that $u,v$ are norms from a quadratic extension. We then find ourselves counting...

In 1896, the prime number theorem was established, showing that π(x) ∼ li(x). Perhaps the most widely used estimates in explicit analytic number theory are bounds on |π(x)-li(x)| or the related error term |θ(x)-x|. In this talk we discuss methods one...

I will describe a version of the prime number theorem for arithmetic progressions that is uniform enough to deduce the Siegel-Walfisz theorem, Hoheisel's asymptotic for short intervals, a Brun-Titchmarsh bound and Linnik's bound for the least prime...

Given a Galois extension of number fields L/K, the Chebotarev Density Theorem asserts that, away from ramified primes, Frobenius automorphisms equidistribute in the set of conjugacy classes of Gal(L/K). In this talk we report on joint work with D...

The exciting and rapidly-growing field of topological materials has brought with it unexpected new connections between physics and pure mathematics. As the name suggests, topology has played a significant role in understanding and classifying these...

Mathematical transcendence refers to objects (usually numbers or functions) that do not satisfy any polynomial equation, that is, they are not algebraic. The numbers $\pi$ and $e$ are famous transcendental numbers, and $e^x$ and the Gamma function...