at noon
in E575

(University of Lethbridge)

Please bring your favourite (math) problems to share with everyone.

at noon
in E575

(University of Lethbridge)

In this talk I will look at some of Diophantine equations of the form \(x^p+y^q=z^r\). The oldest of this equation studied is probably when \(p=q=r=2\) (Pythagorean triples), and probably the most famous one is when \(p=q=r>2\) (Fermat's last theorem). However, many other variations exist, and in this talk I will present what is known about them, and what is believed to be true.

at noon
in E575

(University of Lethbridge)

A generalised n-gon is an incidence structure whose bipartite incidence graph has diameter n and girth 2n. Many of the known examples of generalised n-gons are highly symmetric, and in fact arise naturally from particular group actions.

I will give an overview of some of the things that are known about symmetries of generalised n-gons that we hope are leading toward classification of these objects, or at least to understanding the symmetry that they can have. My contributions to this problem are based on joint work with John Bamberg, Michael Giudici, Gordon Royle and Pablo Spiga of the University of Western Australia, done during my study leave.

at noon
in E575

(University of Lethbridge)

Modular forms play an important role in many branches of mathematics including number theory, arithmetic geometry, representation theory, and even theoretical physics. In this talk I will give a general introduction to modular forms and I will explain some of their fundamental properties.

at noon
in E575

(University of Lethbridge)

An old problem in analytic number theory is to show that an L-function possess simple zeros. Thanks to work of Levinson and Bauer, it is known that any degree one L-function has many simple zeros. For degree two L-functions there are fewer results known. Recently, the speaker and Milinovich have established a number of results concerning simple zeros of modular L-functions.

at noon
in E575

(University of Lethbridge)

One can conjecture the size of a certain arithmetic function: such a conjecture often implies the Riemann hypothesis and, courtesy of Ingham's theorem, that there are infinitely many zeroes the imaginary parts of which are linearly dependent. This is indeed bitter-sweet since, if one were keen to place a wager, one might venture to say that no zeroes are linearly dependent. Very little work has been done to find, computationally, 'near-zero' linear combinations of zeroes. The topic of this paper is to discuss what has been done by Bateman and co. in 'Linear relations connecting the imaginary parts of the zeros of the zeta function'.

at noon
in E575

(University of Lethbridge)

Two unit Hadamard matrices H,K of order n are called unbiased, if \(HK^* = \sqrt{n} L\), where L is a unit Hadamard matrix. Time permitting, I will do all or part of the following:

• If there are m mutually unbiased unit (real) Hadamard matrices of order n, then \(m \le n\) (\(m \le n/2\)).
• The above upper bound is sharp for n a prime power.
• Discuss the literature for composite orders n.
• Talk about mutually unbiased real Hadamard matrices and their applications to association schemes.

at noon
in E575

(University of Lethbridge)

Consider the irreducible polynomial \(f(x)=x^3-x-1\). Let p be a prime and consider this as a polynomial over the finite field of p elements. Over this field the polynomial is either irreducible, splits into three linear factors, or splits into a linear factor and a quadratic factor. Frobenius proved a theorem which asserts that these 3 cases occur with frequencies: 1/3, 1/6, and 1/2. Why do these fractions occur? The answer is related to the fact that the Galois group of f is the symmetric group of 3 letters. Moreover, he considered how an irreducible polynomial factors when reduced modulo p. In this talk I will explain Frobenius' theorem. This seminar will be accessible to undergraduate students who have taken Math 3400 (Group and Rings).

at noon
in E575

(University of Lethbridge)

Suppose one has managed to bound a certain complex-valued function, f(z) say, by another function g(z). How could one show that the bound g(z) is 'as good as it gets'? The study of so-called Omega-theorems is designed to answer this question. Dirichlet's theorem (which is not much more complicated than the statement that if k+1 students need to sit on k chairs then at least two of them must sit in the same chair) provides a good insight into Omega-theorems: in particular one may use Dirichlet's theorem to show that some bounds for the growth of the zeta-function are the best possible.

at noon
in E575

(University of Lethbridge)

In this talk we give a brief introduction to elliptic curves. We start by describing what they are, where they came from, and why number theorists are obsessed by them.

at noon
in E575

(University of Lethbridge)

We give an exposition of a paper of Jean-Pierre Serre with the same title (Bulletin of AMS, Volume 40 (2003), 429--440). Its abstract reads as follows: "The theorem of Jordan which I want to discuss here dates from 1872. It is an elementary result on finite groups of permutations. I shall first present its translations in Number Theory and Topology."

at noon
in E575

(University of Lethbridge)

Place a checker on some square of an m-by-n rectangular checkerboard. Asking whether the checker can tour the board, visiting all of the squares without repeats, is the same as asking whether a certain graph has a hamiltonian path (or hamiltonian cycle). The question becomes more interesting if we allow the checker to step off the edge of the board. This modification leads to numerous open problems, and also to connections with ideas from elementary topology and group theory. Some of the problems may be easy, but many have resisted attack for 30 years. No advanced mathematical training will be needed to understand most of this talk.

at noon
in E575

(University of Lethbridge)

We will talk about certain strategy games, in which the moves alternate between two players. Chess, checkers, and Go are some of the games we could discuss, but, to keep things simple, we will stick to easier examples. John H. Conway discovered that analyzing who will win from a given starting position has some very interesting consequences. In particular, we will see how to add two games (or subtract them, or multiply them), and we will encounter numbers that are infinitely large. No advanced mathematical training will be needed to understand most of this talk, but it would be helpful to have heard of "Dedekind cuts".

The main talk will be preceded by a short explanation of "Zero-Knowledge Proofs." These allow you to convince someone you know how to prove a theorem, without giving them any information at all about the proof (except how long it is).