at noon
in W866

(University of Lethbridge)

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.

at noon
in D631

(University of Lethbridge)

It is known that if the negative Pell equation $X^2-DY^2=-1$ is solvable (in integers), and if $(x, y)$ is its solution with smallest positive $x$ and $y$, then all of its solutions $(x_n, y_n)$ are given by the formula $$x_n+y_n \sqrt{D}=\pm (x+y\sqrt{D})^{2n+1}$$ for $n\in \mathbb{Z}$. Furthermore, a theorem of Walker from 1967 states that if the equation $aX^2-bY^2=\pm 1$ is solvable, and if $(x, y)$ is its solution with smallest positive $x$ and $y$, then all of its solutions $(x_n, y_n)$ are given by $$x_n\sqrt{a}+y_n \sqrt{b}=\pm (x\sqrt{a}+y\sqrt{b})^{2n+1}$$ for $n\in \mathbb{Z}$. We describe a unifying theorem that includes both of these results as special cases. The key observation is a structural theorem for the non-trivial ambiguous classes of the solutions of Pell equations $X^2-DY^2=\pm N$. This talk is based on the work of Forrest Francis in an NSERC USRA project in summer 2015.

at noon
in D631

(University of Lethbridge)

A Hadamard matrix $H$ of order $4n^2$ is said to be skew-regular if it is of skew-type and the absolute values of the row sums are all $2n$. It is conjectured that for each odd integer $\boldsymbol{n}$ there is a skew-regular matrix of order $\boldsymbol{4n^2}$.

A Hadamard matrix $H$ of order $m$ is said to be balancedly splittable if there is an $\ell\times m$ submatrix $H_1$ of $H$ such that inner products for any two distinct column vectors of $H_1$ take at most two values. It is conjectured that only (Sylvester) Hadamard matrices of order $\boldsymbol{4^n}$ are balancedly splittable.

The existence and applications of these two very interesting classes of matrices to Hadamard diagonalizable strongly regular graphs, maximal equiangular lines set, doubly regular tournament, and unbiased Hadamard matrices will be discussed in detail.

at noon
in D631

(University of Lethbridge)

In this talk I will consider the discrete moments \[ J_{k}(T) = \sum_{0 < \gamma < T} |\zeta'(\rho)|^{2k}, \] where $\zeta(s)$ is the Riemann zeta function, $\rho=\beta+i \gamma$ is a non-trivial zero of $\zeta(s)$, and $T >0$. In the 1980's Steve Gonek and Dennis Hejhal (independently) studied these moments and proposed a conjecture for the size of $J_k(T)$. I will give a survey of the known results towards the Gonek-Hejhal conjecture on $J_k(T)$. If time permits, I will present several new results.

at noon
in D631

We will discuss the Cohen-Lenstra heuristics, both in the classical and non-abelian setting. In particular we will make the connection between these heuristics and the problem of counting extensions of quadratic fields with fixed Galois groups, and when knowledge of the latter implies such heuristics. Finally we will discuss our recent work on the non-abelian case and counting unramified 2-group extensions of quadratic fields.

at noon
in D631

(University of Lethbridge)

A Digraphical Regular Representation (DRR) for a group $G$ is a directed graph whose full automorphism group is the regular representation of $G$. In 1981, Babai showed that with five small exceptions, there is at least one DRR for every finite group.

A Cayley digraph on a group $G$ is a group that contains the regular representation of $G$ in its automorphism group. So when a Cayley digraph is a DRR, its automorphism group is as small as possible given that constraint.

The question of whether or not requiring a certain level of symmetry (in this case, a Cayley digraph) makes having additional symmetry more likely is a natural one, particularly in light of results by Erdős and others proving that almost every graph is asymmetric.

In some 1981 and 1982 papers, Babai and Godsil conjectured that as $r \to \infty$, over all groups $G$ of size $r$, the proportion of Cayley digraphs that are DRRs tends to $1$. They proved this to be true for the restricted family of nilpotent groups of odd order.

I will talk about recent joint work with Pablo Spiga in which we prove this conjecture.

at noon
in D631

(University of Lethbridge)

Siegel zeros (or Landau-Siegel zeros) are potential counterexamples to the generalised Riemann hypothesis (for $L$-functions). Such zeros, if exist, have to be "very close" to 1 over the complex plane. In this talk, we will discuss some results concerning Siegel zeros and their applications. If time permits, we will talk about the corresponding "Siegel-Walfisz theorems" for certain $L$-functions.

at noon
in D631

(University of Lethbridge)

What happens when you add up the divisors of an integer? This seemingly innocuous question has motivated mathematicians across the ages, from antiquity to the present day. In this talk, we'll survey some conjectures and results on the functions $s(n)$, the sum of the proper divisors of an integer $n$, and $\sigma(n) = s(n) + n$, the sum of all divisors of $n$. The ancient Greeks derived religious significance from certain values of these functions; we'll do no such thing in this talk. However, by asking simple questions whose answers turn out to be very complicated – if they're known at all – we will see that an air of mystery continues to surround these two fascinating functions. The first half of this talk, at the very least, will be extremely accessible to everyone; tell your students!

at noon
in D631

L-functions associated with automorphic forms are vast generalizations of Riemann zeta functions and Dirichlet L-functions. Although the theory of L-functions play a fundamental role in number theory, it is still largely conjectural. If $\pi$ is an irreducible cuspidal automorphic representation of $GL_n$ over a number field $F$ and $\tilde \pi$ is its dual representation, it is conjectured that the Dedekind zeta functions $\zeta_F(s)$ (which is the Riemann zeta function when $F=\mathbb{Q}$) "divides" the Rankin-Selberg L-function $L(s,\pi\times \tilde \pi)$, i.e., the quotient $L(s,\pi\times \tilde \pi)/\zeta_F(s)$ (which is called the adjoint L-function of $\pi$) should be entire. For $n=2$, this conjecture was proved by Gelbart-Jacquet. In this talk, I will give a sketchy survey of constructions of some L-functions, including the Rankin-Selberg L-function $L(s,\pi_1\times \pi_2)$, and report our recent work on the above conjecture when $n=3$. This is a joint work with Joseph Hundley.

at noon
in D631

(University of Lethbridge)

The ternary Goldbach problem stated that every odd integer greater than $5$ can be written as sums of three primes. This was proved by Vinogradov for all sufficiently large odd integers and completely proved by Helfgott and Platt. On the other hand, by using Green's transference principle method one can extend Vinogradov's result to a density version. In this talk, I will talk about recent results in this direction.

In this talk, we present an introduction to p-adic numbers and their appearance in number theory. In particular, we give a proof of Ostrowski's theorem.

Vertex operator algebras (VOAs) as discussed in this talk are graded complex vector spaces with a collection of many bilinear operations satisfying some intricate identities. They first arose in mathematics and physics via the study of string theory and conformal field theories, in the study of the Monster group and moonshine, and in the study of the representation theory of affine Lie algebras. From their inception it has been known that VOAs are deeply connected with the theory of modular forms.

In this talk we will explain how results on vector valued modular forms can be used to classify VOAs satisfying certain finiteness conditions. Our classification for VOAs with exactly three simple representations rests on using arithmetic properties of a family of modular forms expressed in terms of generalized hypergeometric series. The classification of the integral specializations of this family relies on properties of the monodromy, and on results on distributions of primes in arithmetic progressions. No prior familiarity with VOAs will be assumed, and we will focus primarily on the number theoretic aspects of the problem.