A talk given at ACSME. Joint work with Joel Black, Naomi Borwein, Florian Breuer, Peter Ellerton, Jo-Ann Larkins and Malcolm Roberts.

The density of 1's in the Kolakoski sequence is conjectured to be 1/2. Proving this is an open problem in number theory. I shall cast the density question as a problem in combinatorics, and give some visualisations which may suggest ways to gain further insight into the conjecture.

This talk is a practice talk for an invited talk I will soon give in Indonesia, in which I was asked to present on Education at a conference on Graph Theory.

In 1929 Alfred North Whitehead wrote: "The university imparts information, but it imparts it imaginatively. At least, this is the function it should perform for society. A university which fails in this respect has no reason for existence. This atmosphere of excitement, arising from imaginative consideration, transforms knowledge. A fact is no longer a bare fact: it is invested with all its possibilities."

In the light and inspiration of Whitehead's quote, I will discuss some aspects of the problem and challenge of mathematical education as we meet it in Universities today, with reference to some of the ways that combinatorics may be an ideal vehicle for sharing authentic mathematical experiences with diverse students.

Brad Pitt's zombie-attack movie "World War Z" may not seem like a natural jumping-off point for a discussion of mathematics or science, but in fact it was a request I received to review that movie in "The Conversation" and the review I wrote that led me to be invited to give a public lecture on zombies and maths at the Academy of Science next week. This week's colloquium will largely be a preview of that talk, so should be generally accessible.

My premise is that movies and maths have something in common. Both enable a trait which seems to be more highly developed in humans than in any other species, with profound consequences: the desire and capacity to explore possibility-space.

The same mathematical models can let us playfully explore how an outbreak of zombie-ism might play out, or how an outbreak of an infectious disease like measles would spread, depending, in part, on what choices we make. Where a movie gives us deep insight into one possibility, mathematics enables us to explore, at all once, millions of scenarios, and see where the critical differences lie.

I will try to use mathematical models of zombie outbreak to discuss how mathematical modelling and mathematical ideas such as functions and phase transitions might enter the public consciousness in a positive way.

I am grateful to have been appointed in a role with a particular focus on First Year Teaching as well as a research mandate. The prospect of trying to do both well is daunting but exciting. I have begun talking with some of my colleagues who are in somewhat similar roles in other Universities in Australia and overseas about what they do. I would like to share what I've learnt, as well as some of my thoughts so far about how this new role might evolve. I am also very interested in input from the Maths discipline or indeed any of my colleagues as to what you think is important and how this role can benefit the maths discipline and our school.

This week the speaker in the Discrete Mathematics Instructional Seminar is Judy-anne Osborn who will be discussing Hadamard matrices.

This week the speaker in the Discrete Mathematics Instructional Seminar is Judy-anne Osborn who will be discussing Hadamard matrices.

Co-author: Thomas Prellberg (Queen Mary, University of London)

Various kinds of paths on lattices are often used to model polymers. We describe some partially directed path models for which we find the exact generating functions, using instances of the `kernel method'. In particular, motivated by recent studies of DNA unzipping, we find and analyze the generating function for pairs of non-crossing partially directed paths with contact interactions. Although the expressions involving two-path problem are unweildy and tax the capacities of Maple and Mathematica, we are still able to gain an understanding of the singularities of the generating function which govern the behaviour of the model.