# Research - Mathematics

## Research Areas

Staff in the Department of Mathematics carry out research on a wide variety of topics in pure and applied mathematics, work that calls on mathematical knowledge from many fields such as algebra, analysis, number theory, differential equations and numerical analysis. A typical graduate programme includes papers from several of these areas.

For a PhD or MPhil, which involve original research, supervisors will only consider topics closely related to their own research. Otherwise there is a risk of repeating work already published, or which is of little interest.

Other projects may also be possible provided a suitable supervisor can be arranged. In some cases, this might involve a team which includes staff outside of the Department of Mathematics. For details consult the Graduate Adviser.

### Elementary Number Theory

The proof of Fermat's Last Theorem, together with a growing need for encryption within e-commerce, has rekindled interest in the techniques and outstanding problems of number theory. For example, smart cards sometimes include elliptic curve encryption algorithms. The use of the computer has also improved our ability to test conjectures and devise hypotheses based on real numerical data. In this project a problem from prime, algebraic or applied number theory will be considered: reading the background history and theories, looking at related results, carrying out computer experiments, testing some plausible conjectures etc will all lead up to the main goal – an attack on the outstanding problem or application itself.

**Supervisor:** Professor Kevin Broughan

### Sieve Theory

The twin primes conjecture has long been regarded as a suitable problem which could be resolved using sieve theory, but so far the approach has failed. There are problems which have been solved. This project includes a study of the work of Henrich Iwaniec on sieves and might include an extension of his recent theorem "p=x^{2}+y^{4} for an infinite number of primes p".

**Supervisor:** Professor Kevin Broughan

### Zeta Functions

Modern analytic number theory includes the study and application of zeta and L-functions in a variety of settings, including number fields, groups and graphs. This is an active area of research and the aim of the project is to bring the student to a level (through a study of the works of Peter Sarnak, Dorian Goldfield and others) where one of the many unsolved problems might be attacked. The Waikato work has a strong computational flavour.

**Supervisor:** Professor Kevin Broughan

### Combinatorics

Combinatorics is a branch of discrete mathematics, which in turn is a branch of pure mathematics.

Within combinatorics, Dr Nick Cavenagh does a lot of work on latin squares, latin trades or bitrades and graph decompositions. Latin trades connect with many branches of pure mathematics including geometry (eg partitioning an integer-sided triangle into smaller, integer-sided triangles), finite field theory (in particular Weil's theorem has been useful), group theory (some latin trades may be defined in terms of a group with specified properties) and linear algebra.

A latin square of order n is an nxn array of symbols 1,2,....,n such that each symbol occurs exactly once in each row and once in each column. Note that a completed Sudoku puzzle is a type of latin square of order 9.

Problems in combinatorics are often easy to state but sometimes hard to solve. Those with an aptitude and disposition for finding patterns and solving puzzles often enjoy research in combinatorics.

**Supervisor:** Dr Nick Cavenagh

### Astrophysics

### Magnetic Field Line Reconnection

A problem of great interest in astrophysics is magnetic reconnection. The central idea is to release magnetic energy bound up in the topology of solar and stellar plasmas. Although it is known that reconnection is the only mechanism which allows topological change in the magnetised plasma, the real challenge for astrophysicists is to demonstrate a mechanism that can explain the explosive release of a solar or stellar flare.

**Supervisor:**Professor Ian Craig

### Arithmetic Geometry

Number theory is as relevant today as it was 2,500 years ago, with the advent of high-powered computing and cryptography. Dr Daniel Delbourgo's research interests lie in the area of arithmetic geometry, which uses tools from geometry and cohomology to study rational solutions to equations. As a famous example, Fermat's Last Theorem asserted that there are no (non-trivial) integer solutions to the equation x^n+y^n=z^n when n>2, yet its eventual proof by Andrew Wiles was found only after a mere 350 years of concentrated effort

by numerous great mathematicians!

Dr Daniel Delbourgo's work in this area applies ideas from classical Iwasawa theory and Galois representations, to study the arithmetic behaviour of invariants arising from these objects. He is also interested in the special values of these L-functions, and there is a rich vein of conjectures connecting these values with elements in K-groups.

Some of his recent efforts involve extending what we know over abelian extensions of the rationals, to some brand new non-abelian examples.

**Supervisor:** Dr Daniel Delbourgo

### Generalised Sylow Theorems

Sylow's theorem is one of the most useful tools in a group theorist's toolkit. It has now been generalised in a multitude of ways. The problem today is one of classifying the different generalisations and seeking a better understanding of the underlying principles that give rise to various categories of generalised Sylow theorems.

**Supervisor:** Dr Ian Hawthorn

### Solvable Group Theory

The composition series structure within a solvable group equips the group with a kind of a 'scaffold'. This allows us to employ inductive arguments. Hence solvable group theory has quite a distinct flavour from the more difficult theory of finite groups in general. I have particular interest in the area of Fitting classes of solvable groups where there are a number of unsolved problems of current interest.

**Supervisor:** Dr Ian Hawthorn

### Lattice Rules

Lattice rules are used for the numerical integration of multiple integrals in hundreds or even thousands of variables. There has been much recent work on lattice rules and one of the main results is that the generating vectors for these lattice rules may be constructed by using a component-by-component algorithm.

There is now a need to do numerical testing of these lattice rules to see how they perform. Besides standard test problems, these lattice rules could be tested out on integrals arising from practical situations such as those from financial models.

Lattice rules are usually constructed for integrands over the unit cube. However, there are some applications in which one wants to approximate integrals where the integration region is all of Euclidean space. A question that arises is whether to use lattice rules for the unit cube and then do some mapping to Euclidean space or whether to use lattice rules designed for Euclidean space in the first place.

Of course, there are many other unanswered questions on lattice rules (such as those to do with their structure) and these are worthy of exploration as well.

**Supervisor:** Associate Professor Stephen Joe

### Perturbations and Stability in General Relativity

Professor Ernie Kalnins is interested in the theory of perturbations in the vicinity of compact astrophysical objects such as black holes, and the stability of such structures with respect to such perturbations.

In addition to these studies the solution of Einstein's equations for bounded rotating masses is being actively pursued. In particular, the gravitational field in the vicinity of such configurations both classically and relativistically is under study.

Affiliated to these ideas is the study of atoms in high magnetic fields and the relation to quantum chaos. These are important quantum mechanical problems to be solved here in an astrophysical sense.

**Supervisor:** Professor Ernie Kalnins

### Quantum Groups and Special Functions

Quantum groups and quadratic algebras is the study is of actual quantum mechanical and classical mechanical systems which admit explicit solution and have definite algebraic properties. Also associated with this study are the properties of the special functions that arise in the solution of these problems and the consequences for the corresponding algebra. Of particular interest are generalisations of ellipsoidal harmonics in the case of quantum algebras.

**Supervisor:** Professor Ernie Kalnins

### Inhomogeneous Cosmological Models

Dr Woei Chet Lim is interested in the evolution of inhomogeneous cosmological models according to general relativity. The goal is to build an inhomogeneous model of the universe consistent with observational data, and to find any new relativistic phenomena.

Dr Lim is currently studying the spike solution (in vacuum, with matter, or with an electromagnetic field), and the void model. The vacuum spike solution describes recurring inhomogeneous sheet-like gravitational distortions that occur during the chaotic BKL (Belinski-Khalatnikov-Lifshitz) phase shortly after the Big Bang; the void model describes the evolution of a relatively empty vast space. Sheets or bubbles of spikes are conjectured to intersect and interact with each other in filaments and points, and cause matter to gravitate towards these sheets, filaments and points to form large scale structures, leaving behind relatively empty regions that become voids. The inhomogeneous paradigm conjectures that the accelerated cosmic expansion, presently attributed to hypothetical dark energy in the homogeneous standard model, is an apparent effect of averaging the different expansion rates of the voids and the large scale structures.

The Einstein field equations of general relativity are a set of hyperbolic partial differential equations. Dr Lim generally solves them numerically using finite difference methods. In special cases such as the spikes, he finds the exact solution using solution-generating transformations. He also use analytical approximations and qualitative dynamical systems methods to study the evolution of the models.

**Supervisor:** Dr Woei Chet Lim

### Astrophysics

Dr Yuri Litvinenko is interested in developing theoretical models for a wide range of astrophysical processes – from energy release in flares on the Sun to the acceleration of galactic cosmic rays. The work is motivated by observations that put strong constraints on the theories, so developing models and making quantitative predictions is usually an interesting but challenging job.

**Supervisor: **Dr Yuri Litvinenko

### Turbulent Flows

Associate Professor Sean Oughton's current research interests centre on understanding the behaviour of turbulent flows. Physically we all have a good understanding of what a turbulent flow is. For example, white water rapids are clearly turbulent, whereas a (stationary) jar of honey is not. In fact, on the earth most flows, at most times, areturbulent. Mathematically, one might say that a turbulent flow is characterised by motions which occur over a broad range of length (and time) scales and that these motions interact nonlinearly. It is this nonlinear nature of the problem that makes it simultaneously so rich and so challenging.

A particular interest is magnetofluid turbulence, where the fluid is electrically conducting so that one must consider not just the behaviour of the fluid's velocity, but also that of its magnetic field. Examples of magnetofluids include liquid metals (eg mercury) and plasmas (eg the sun, the solar wind, the working fluid in nuclear fusion devices). Most of the matter in the universe is thought to be in the plasma state, that is, the atoms have been ionised. One way to study conducting fluids is using magnetohydrodynamics (MHD). This is the marriage of the equations of fluid dynamics with those of electrodynamics, and provides a good approximation to the behaviour of various parts of the solar system (or heliosphere). Important dynamical features of MHD include waves, turbulence, plasma heating, and particle acceleration. The work involves a mixture of theory (including statistical mechanics and modelling) and computer simulations of the governing equations. Associate Professor Oughton is happy to supervise PhD and masters topics on fluids and MHD, particularly solar wind/solar corona/turbulence.

**Supervisor:** Associate Professor Sean Oughton

### Algebra of Partial Maps

One of Dr Timothy Stokes' main research interests is to generalise this correspondence to other situations. There are connections with the theory of relation algebras, of importance in Computer Science.

### Radical Theory

**Supervisor:**Dr Timothy Stokes

### Free Surface Problems

**Dr Timothy Stokes**

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