at noon
in C756

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 C756

(University of Lethbridge)

Andries Brouwer is 65 and a special issue of Designs, Codes and Cryptography is issued to celebrate the occasion.

Professor Brouwer maintains an elegant public database of existence results for all possible strongly regular graphs on $n\le 1300$ vertices. In a very nice paper, Cohen and Pasechnik implemented most of the graphs listed there in the open source software Sagemath and obtained a graph for each set of parameters mentioned in the database. In their initial version of the paper, they mentioned 11 cases as missing values. A number of the cases were related to my work with professors Janko, Tonchev, and Ionin. I tried to help out with these cases and four cases were resolved quickly, after I sent detailed instructions. However, there was a problem with the case of SRG$(765,192,48,48)$. This talk relates to this special case and a nice application of generalized Hadamard matrices.

To make the talk accessible to general audiences, I will provide many examples illustrating the concepts involved.

at noon
in C756

We discuss the possibility that the Riemann zeta function has a zero $\sigma +iT$ to the left of the classical zero free region. We will show how the existence of this zero forces the function to have many more zeros in the vicinity of $\sigma+iT$ or/and $\sigma +2iT$.

at noon
in C756

(University of Lethbridge)

Place a checker in some square of an $m \times n$ rectangular checkerboard, and glue opposite edges of the checkerboard to make a projective plane. We determine whether the checker can visit all the squares of the checkerboard (without repeating any squares), by moving only north and east. This is joint work with Dallan McCarthy, and no advanced mathematical training will be needed to understand most of the talk.

at noon
in C756

An association scheme is a coloring of a complete graph satisfying certain regularity conditions. It is a generalization of groups and has many applications in algebraic combinatorics. Every association scheme yields a special matrix algebra called the Bose-Mesner algebra of a scheme. A scheme is called commutative if its Bose-Mesner algebra is commutative. Commutative schemes were the main topic of the research in this area for decades. Only recently non-commutative association schemes attracted the attention of researchers. In my talk I'll present the results about non-commutative association schemes of the smallest possible rank, rank $6$. This is a joint work with A. Herman and B. Xu.

at noon
in C756

(University of Lethbridge)

Hardy and Littlewood initiated the study of the $2k$-th moments of the Riemann zeta function on the critical line. In 1918 Hardy and Littlewood established an asymptotic formula for the second moment and in 1926 Ingham established an asymptotic formula for the fourth moment. In this talk we consider the sixth moment of the zeta function on the critical line. We show that a conjectural formula for a certain family of ternary additive divisor sums implies an asymptotic formula for the sixth moment. This builds on earlier work of Ivic and of Conrey-Gonek.

at noon
in C756

(University of Lethbridge)

We describe a theorem of M. Mourtada and V. Kumar Murty on the distribution of values of the logarithmic derivative of the $L$-functions attached to quadratic characters. Under the assumption of the generalized Riemann Hypothesis they prove the existence of a density function that gives the distribution of values of the logarithmic derivative of such $L$-functions at a fixed real point greater than 1/2. Following classical results of Wintner, we also describe how this distribution can be described as an infinite convolution of local distributions.

at noon
in C756

(University of Lethbridge)

In this talk, we describe a method for studying the value-distribution of $L$-functions based on the Jessen-Wintner theory. This method has been explored recently by Ihara and Matsumoto for the case of logarithms and logarithmic derivatives of Dirichlet $L$-functions of prime conductor and by Mourtada and V. K. Murty for the case of logarithmic derivatives of Dirichlet $L$-functions associated with quadratic characters. We show how to extend such results to the case of cubic characters. In fact, we describe a distribution theorem for the values of the logarithms and logarithmic derivatives of a certain family of Artin $L$-functions associated with cubic Hecke characters. This is a joint work with Amir Akbary.

at noon
in C756

The use of graphs to study permutation groups goes back to Higman who first introduced the orbital graphs, and used them to characterise the primitive groups. Since then, graph theory and permutation group theory have become intertwined, with many beautiful results. In this talk, I will discuss some problems which lie across the boundary of permutation group theory and graph theory (or at least algebraic graph theory), such as how to characterise a new class of permutation groups that includes the primitive ones - the so called semiprimitive groups.

at noon
in C756

A permutation group $G$ on $\Omega$ is transitive if for every $x, y\in\Omega$ there exists $g\in G$ mapping $x$ to $y$. The group $G$ is called primitive if, in addition, it preserves no nontrivial partition of $\Omega$. Let $\Gamma$ be a vertex-primitive digraph, that is, its automorphism group acts primitively on its vertex-set. It is not hard to see that, in this case, $\Gamma$ cannot have two distinct vertices with equal neighbourhoods, unless $\Gamma$ is in some sense trivial. I will discuss some recent results about the case when $\Gamma$ has two vertices with "almost" equal neighbourhoods, and how these results were used to answer a question of Araújo and Cameron about synchronising groups. (This is joint work with Pablo Spiga.)