“Making Connections: shadows, crossed ladders, couriers, Ceva and parallel sums”
Exploration of connections between apparently disparate elementary problems leads to a little known theorem in geometry and provides an opportunity to experience the role of shifts of attention in teaching and learning mathematics at Secondary level.
“The Sato-Tate conjecture for elliptic and hyperelliptic curves”
Consider a system of polynomial equations with integer coefficients. For each prime number p, we may reduce modulo p to obtain a system of polynomials over the field of p elements, and then count the number of solutions. It is generally difficult to describe this count as an exact function of p, so instead we take a statistical point of view, treating the count as a random variable and asking for its limiting distribution as we consider increasingly large ranges of primes. Conjecturally, this distribution can be described in terms of the conjugacy classes of a certain compact Lie group. We illustrate this in three examples: polynomials in one variable, where everything is explained in terms of Galois theory by the Chebotarev density theorem; elliptic curves, where the dichotomy of outcomes is predicted by the recently proved Sato-Tate conjecture; and hyperelliptic curves of genus 2, where even the conjectural list of outcomes was only found still more recently.
“Mathematics in Medicine: Enhancing your health” — ANZIAM Lecturer
Mathematical models use the language of mathematics to very effectively describe, understand and evaluate systems. Medical science was a comparative late starter in this process, largely because the two groups were not “talking together”. This has now changed and startling progress is being achieved worldwide. Mathematics-in-Medicine Study Groups are forming all around the world.
The term “Systems Biology” captures the activity that we are doing in the National Research Centre for Growth and Development which is one of New Zealand’s Centres of Research Excellence. We are currently involved in growth models, life-history model development, epigenetic models, developmental genetic models, cancer growth: to name just a few areas. Some of these will be described in this talk. Many of these do give rise to “New Maths”. Especially notable in this respect is the work on non-local ordinary differential equations. An overview of this work will be given in this lecture.
“The Square Root Problem of Kato for Elliptic Operators – a Survey with Emphasis on Related First Order Systems” — NZIAS Lecturer
The first order Cauchy Riemann equations have long been used in the study of harmonic boundary value problems in plane domains. The Dirac operator can sometimes be employed in higher dimensions.
First order systems provide insight into the solution of the Kato square-root problem for second order elliptic operators. I shall present a survey of this material, including the historical background, and recent progress.
"Realizing phylogenies with local information" — NZMS Research Award winner
Results that say local information is enough to guarantee global information provide the theoretical underpinnings of many reconstruction algorithms in evolutionary biology. Such results include Buneman's Splits-Equivalence Theorem and the Tree-Metric Theorem. The first result says that, for a collection $mathcal C$ of binary characters, pairwise compatibility is enough to guarantee compatibility for $mathcal C$, that is, there is a phylogenetic (evolutionary) tree that realizes $mathcal C$. The second result says that, for a distance matrix $D$, if every $4times 4$ distance submatrix of $D$ is realizable by an edge-weighted phylogenetic tree, then $D$ itself is realizable by such a tree. In this talk, we investigate these and other results of this type. Furthermore, we explore the closely-related task of determining how much information is enough to reconstruct the correct phylogenetic tree.
“The role of global manifolds in the transition to chaos in the Lorenz system”
The Lorenz system still fascinates many people because of the simplicity of the equations that generate such complicated dynamics on the famous butterfly attractor. This talk addresses the role of the stable and unstable manifolds in organising the dynamics more globally. A main object of interest is the stable manifold of the origin of the Lorenz system, also known as the Lorenz manifold. This two-dimensional manifold and associated manifolds of saddle periodic orbits can be computed accurately with numerical methods based on the continuation of orbit segments, defined as solutions of suitable boundary value problems. We use these techniques to give a precise geometrical and topological characterisation of global manifolds during the transition from simple dynamics, via preturbulence to chaotic dynamics, as the Rayleigh parameter of the Lorenz system is increased.
This is joint work with Bernd Krauskopf (University of Auckland) and Eusebius Doedel (Concordia University, Montreal).