Summer Scholarships 2010 Topics

From MathsDept

Claire Postlethwaite Rm 303.510, Ext 88817, homepage

Mathematical modelling of animal movement trajectories

The recent advent of GPS devices small enough to be carried by an animal has made available vast quantities of animal trajectory data, which has in turn led to the development of new methods by which to analyse the data. One question of interest is how to reliably infer an animal's "behavioural state" from the trajectory data. This project will involve developing and fitting movement models to previously collected data from possums, pigeons, fish and other animals.

Prerequisites: no specific courses, but knowledge of Matlab useful. Some statistics background would also be useful. An interest in biology is essential, but no specific knowledge is required.

Steven Galbraith Rm 303.501, Ext 88778, homepage

Pseudorandom sequences from elliptic curves

Algorithms to generate sequences of pseudorandom numbers have numerous applications in computing. There are several such algorithms which exploit operations in finite groups (such as elliptic curves over finite fields). It would be interesting to generalise Blum-Micali generator to elliptic curves and to consider whether existing results on the output size of these generators can be improved.

Prerequisite: MATH 328

Smooth integers in short intervals

An integer is smooth if it is a product of powers of relatively small primes. There are a number of theoretical and practical applications of smooth integers, especially smooth integers short intervals like [A, A + 4* Sqrt(A) ] where A is large. There is an algorithm due to Boneh to construct smooth integers in short intervals. It would be interesting to experiment with this algorithm and find ways to improve it. There is also hope to improve the theoretical results in this subject.

Prerequisite: MATH 328

David Gauld Rm 303.419, Ext 88697, homepage

Knots and Tangles

Have you ever wondered when tying your shoelaces whether tying over then under is different from tying under then over? Topologists have devised many so-called knot invariants to allow them to distinguish between different knots. One of the most famous is the Jones Polynomial, named for a former student (and now part-time Distinguished Alumni Professor) of this Department. You can study this topic algebraically and geometrically. It is helpful to know a bit about groups, but not much, and topological aspects of euclidean space, but again not much. If you have taken 333 you will already have more than you need but if you haven't and are interested have a chat with me before making up your mind.

Shayne Waldron Rm 303.415, Ext 85877, homepage

Bernstein operators on polytopes

In computer graphics (and the finite element method) surfaces are usually described as piecewise polynomial functions on a triangulation. Each triangle has natural coordinates called barycentric coordinates, and a corresponding (Bernstein) basis of polynomials of degree k. The coordinates of a function in this basis reflect the shape of the function, and function values can be efficiently computed from these coefficients via the de Casteljau algorithm. Recently, natural coordinates for rectangles, etc have been introduced. The student will investigate the corresponding Bernstein operators and their approximation properties.Prerequisite: Basic linear algebra (will use matlab, etc), and some analysis or numerical analysis.

Tight frames and their symmetries

If the one of the two cartesian coordinates of a battleship is lost, then its position can no longer be determined. It is possible to give three coordinates, so that its position can still be determined if one is lost (this is more efficient than repeating both coordinates twice). This is an example of what is called a tight frame. In addition to obvious applications to signal analysis, such as transmission which is robust with respect to erasures (loss of information), these tight frames offer natural generalisations of orthonormal bases which reflect additional symmetries of the space. The student will use MAGMA to construct and analyse some classes of tight frames which arise as the orbits of groups, and have numerous applications including making optimal quantum measurements. This is an ongoing project which has led to publications with summer students in the past. Prerequisites: The key idea requires an understanding of orthogonal expansions. Uses (elementary) mathematics from all areas.

Regular complex polytopes

The regular complex polytopes are generalisations of the platonic solids to complex Euclidean space. They consist of points, edges, faces, etc, which satisfy certain combinatorial properties (which have further been generalised to abstract regular polytopes). The regular complex polytopes have been completely classified via their symmetry groups, which are (Shephard-Todd) finite reflection groups (which have been further generalised to Coxeter groups). The points of the regular complex polytope can be constructed directly from the abstract symmetry group (by considering its irreducible representations). In the case of those which are real, the edges, faces, etc (and their incidence relationships) can be determined by considering the convex hull of the points. For the remaining regular complex polytopes it is an open question whether they are determined by their points (i.e., can the edges, faces, etc, be inferred) - if not this would be equivalent to new symmetries of the points (which are not symmetries of what are called the flags). The student will consider this question, and also in what sense are some highly symmetric configurations of points (which also come from finite reflection groups) semi-regular. Given the group theoretic nature of these questions calculations will be done with MAGMA.

A. Rod Gover Rm 303.427, Ext 88792, homepage

The geoemtry of PDEs

Bill Barton Rm 303.505, Ext 88779, homepage

Mathematics in Different Languages

Different languages have different ways to express mathematical ideas. If you have a language other than English as your first language, you may like to make a systematic study of the differences, particularly at undergraduate level mathematics. We will explore what has already been written, the idea of mathematics as a language of its own, and prepare a glossary. Your project will result in a report on the differences, including recommendations for students who are learning mathematics in both your language and English.

Applications of Topics in MATHS 208

In MATHS 208 students are introduced to areas of mathematics that have many applications in many different fields. In this project you will focus on two or three applications, and find out as much as you can about the application context and how the mathematics is adapted to this use. You will talk to people in industry who use these mathematical ideas, explore extensions to the mathematics that is used, and write a report about each application.

Eamonn O'Brien Rm 303.401, Ext 88819, homepage

Automorphisms of abelian groups

We will study what automorphisms occur for abelian groups and consider whether their orders bound those for other classes of groups.

Prerequisite: Math 320

The automorphism group of the free group

The automorphism group of the free group is little understood. Many questions about its subgroups of finite index are intriguing and can be explored computationally. We will look at descriptions of the automorphism group and investigate some of these questions.

Prerequisite: Math 320

The "tadpole" algorithm for permutation groups

Most permutation groups are "small base": elements can be described by their action on a small number of points. The "tadpole" algorithm is a of reducing from a "large base" to a "small base" group by constructing a homomorphic image. We will study the algorithm.

Prerequisite: Math 320.

James Sneyd Rm 303.519, Ext 87474, homepage

All my summer students work in groups, with postdocs and PhD students. They use a lot of computation, and have to learn a lot of physiology (as well as how to construct models). Each project listed below is part of a major research effort, and so every summer student has access to a lot of help and advice from others in the group. If 1) you don't like biology, 2) don't like computations, or 3) don't like working in groups, then my summer projects are not for you.

Airway smooth muscle and asthma

Contraction of airway smooth muscle causes asthma, but we don't really have a very good understanding of what causes this muscle to contract. It's controlled by calcium, we know, but there's a lot we don't. Models of calcium dynamics in smooth muscle are an important part of the overall lung project. These models use the theory of nonlinear ODE's, reaction-diffusion equations, excitable systems, and a lot of computation.

Cystic Fibrosis

When particular chloride channels break down, they cause the disease called cystic fibrosis. But how do these channels work? How do they control fluid secretion? What can models say about this? We use nonlinear ODE's and a lot of computation to study the function of these channels, and the overall control of fluid secretion.

Water transport in epithelial cells

how do epithelial cells transport water? Why do they? Do we need to use three-phase models to describe this? Do we need to use PDE's at all? These are good questions. I'm glad you asked them.

Neurons and bursting oscillations

calcium controls electrical spiking in neuroendocrine cells. But how? Are spatial aspects important? We use the theory of nonlinear dynamics and geometric bifurcation theory to study these questions, but we also look closely at experimental data.

Mike Thomas Rm , Ext n/a, homepage

An International Comparison of Teaching of Antideriviative

This project considers the classroom presentation of 4 modules of work on graphical antiderivative in schools in Italy, Israel and NZ. The scholarship student would be involved in helping to analyse NZ data from the project. This involves characterising two NZ teachers in terms of their resources, knowledge and goals in four lessons and analysing how these influence their in-the-moment teaching decisions.

A Teaching Framework for Linear Algebra

This project considers the teaching of linear algebra to stage 2 students based on a theoretical framework that considers embodied, symbolic and formal thinking. The scholarship student would be involved in helping to analyse data comprising the lecturers stated aims, post-lecture reflections, the resources, knowledge and goals he sought to apply and the outcomes in terms of student learning.

Vivien Kirk Rm 303.525, Ext 88812, homepage

Dynamical systems with noise

This project will look at the effect on solutions to certain nonlinear differential equations of adding small noise to the differential equations. It is known that sometimes the addition of noise smears out the intricate dynamics of a system but that other times noise reinforces the dynamics. This project will look at what happens when noise is added to systems with special kinds of periodic solutions.

Prerequisites: A pass in Maths 260 and Maths 250.

Nonlinear dynamics in models of calcium dynamics

Many mathematical models of intracellular calcium dynamics have complicated oscillatory solutions, called mixed mode oscillations. This project would look at a variety of these models, to see if there are differences in the way the models respond when a large pulse is applied to one of the model variables.

Prerequisites: A pass in Maths 260 and Maths 250.

Arkadii Slinko Rm 303.509, Ext 85749, homepage

Competition of Political Parties under Different Parliament Choosing Rules

Prerequisite: A+ in 250. Although the work is theoretical, ideally the student should be able to do simple computer simulations.

Tom ter Elst Rm 303.229D, Ext 86901, homepage

Sobolev spaces on arbitrary subsets

n/a Rm n/a, Ext n/a, homepage

Topics in Geometric Algebra

Requirements: computing experience, elementary knowledge of abstract algebra (eg. from 255, 320, 328, 333 or 340) (1 or more students)

Steve Taylor Rm 303.523, Ext 86622, homepage

Transmutation operators for PDEs

The project involves investigating transmutation operators for PDEs. Such operators convert one type of PDE to another (for example, the wave equation to the heat equation), allowing the construction of the solution of a PDE from the solution of another. Prerequisite: Maths 361 or Maths 363.

Marston Conder Rm 303.417, Ext 88879, homepage

Symmetries of discrete structures

I have a number of possible projects for a summer student, all to do with symmetries of discrete objects (graphs/networks, maps, polytopes, surface tessellations, 3-manifolds, block designs, etc.). Use of the MAGMA computation system would be helpful. Prerequisite: Maths 255, 320, 326 or 328, and some computing experience.

n/a Rm n/a, Ext n/a, homepage

Numerical Solution of Stochastic Differential Equation

Stochastic differential equations (SDEs) are used to model phenomena with random events, for example, the fluctuations in the stock market. We investigate the application of known numerical methods used in solving ordinary differential equations to SDEs and consider examples in financial mathematics.

Prerequisite: A strong background in mathematics and Matlab or equivalent computing experience and some knowledge of statistics would be useful.

John Butcher Rm , Ext n/a, homepage

Connes-Kreimer Hopf algebra

This algebra (sometimes known as the Butcher-Connes-Kreimer Algebra) has applications in physics and geometry as well as in numerical analysis.

Prerequisites: Strong background in Algebra

Sina Greenwood Rm 303.421, Ext 88776, homepage

Generalised Inverse Limits

Inverse limits have been studied and widely applied by many mathematicians for many years. Generalised inverse limits have only recently been introduced, and they are attracting a lot interest. They involve infinitely dimensional spaces whose coordinates are found by working backwards through bonding functions. They are fun to learn about in themselves and there are many open questions available to think about.

Generalised Brunnian Links

A Brunnian link can be thought of as a finite number of linked rubber bands. If you pick up one of them the rest will come with it. On the other hand if you cut one of them, all the others fall apart and no two of them remain linked. It is possible to describe Brunnian links by algebraic words and it is quite challenging to consider what the shortest possible word might look like. Links are associated with braid groups, and elements of a braid group can also be represented by words. There are some interesting questions involving braids, generalised links (e.g. what if you must remove two bands before the rest become separated) and possible applications within cell biology.

Warren Moors Rm 303.425, Ext 84746, homepage and Julia Novak Rm 303.235, Ext 84747, homepage

The combinatorics of choosing sets

The project involves investigating the combinatorics of choosing sets of different sizes. For example, if we have a set X of 10 elements and we consider the family of all pairs of X and call this set Y1, then consider the set of all triples of Y1 and call this Y2. Does Y2 have more or less elements than the set Z2 where Z1 is the family of all triples of X and Z2 is the set of all pairs of Z1?

Prerequisites: Strong background in Combinatorics and Analysis