Current research projects
Research projects that are currently being undertaken in the Department of Mathematics and Statistics on the Bendigo campus of La Trobe University are described below. For further details, please contact the researchers directly (contact details are on the Staff page).
Descriptions of recently-completed projects are available on a separate page.
Axiomatic Foundations of Space-time – Dr John Schutz
John's current active interests in this area include the axiomatic foundations of Euclidean and non-Euclidean geometries and the space-time structure of Einstein's theory of general relativity.
One of the aims of mathematics is to express theories in the simplest possible form, both in terms of the intuitive acceptability of the primitive concepts and in the assumed properties which are stated as axioms. Recently John has proposed an axiomatic system for hyperbolic geometry that appears to be simpler than other such axiomatic systems, although full proofs of the mutual independence of the axioms are yet to be established. The axiomatic system uses a recently developed characterisation of ellipsoids in projective space. This work has appeared recently in two papers in the Journal of Geometry.
John's earlier research on the foundations of relativity has been published in several journal articles and in the monograph Independent Axioms for Minkowski Space-Time.
Graph Theory – Dr Christopher Lenard and Prof Terry Mills
Graph theory is full of interesting unsolved problems. One such problem is the "3 longest paths problem". It is well known that every pair of longest paths in a graph must share a common vertex. Is the same true for every set of three longest paths? Christopher Lenard is investigating this question.
Christopher Lenard and Terry Mills are collaborating with Graeme Byrne and Angela Pezic to develop models from graph theory to describe aspects of internal migration in regional Australia. This project will demonstrate a new tool to demographers and other social scientists, and lead to an interesting application of graph theory in an Australian context.
Christopher Lenard and Terry Mills are collaborating with Lakoa Fitina (Divine Word University, Madang, PNG) on a project dealing with cartesian products of graphs.
Hilbert's Sixth Problem: Foundations of Physics – Dr John Schutz
John is interested in the general problem of the axiomatisation of the
theories of physics, which is one of the still-outstanding Hilbert
problems.
A project of current interest is the investigation of the foundations of the two major theories of the electromagnetic field in order to ascertain how and why they differ, especially in relation to their descriptions of the vacuum state.
History of Mathematics – Dr Christopher Lenard, Prof Terry Mills and Ms Lex Milne
This research program explores the history of mathematics. Current projects are described below.
Bougainville: Terry Mills is collaborating with Lakoa Fitina and Deane Arganbright (both at Divine Word University, Madang, PNG) in studying the contribution of Louis-Antoine de Bougainville (1729-1811) to mathematics.
Fibonacci and Finger Counting: Tina Fitzpatrick, Diane Itter (Faculty of Education, LTU), Christopher Lenard, Terry Mills, Lex Milne and Annie Mona (LTU student) are investigating Fibonacci's approach to one of the most basic mathematical activities – finger counting.
Fibonacci and Problem Solving: Tina Fitzpatrick, Diane Itter (Faculty of Education, LTU), Christopher Lenard, Terry Mills and Lex Milne are investigating Fibonacci's approach to problem solving in his seminal work "Liber Abaci" (1202).
Mathematical Models in Health Care – Dr Robert Champion, Dr Christopher Lenard and Prof Terry Mills
The aim of this research program is to apply ideas from mathematics, statistics and operations research to problems that arise in health care. This program involves collaboration between La Trobe University (LTU) and Bendigo Health (BH). Two current projects are described below.
Streaming: An innovative model for emergency care in a regional hospital: This project applies operations research techniques to describe a new approach to managing the accident and emergency department of a hospital. The research team for this project is led by Leigh Kinsman (LTU) with support from June Dyson (BH), Paulett Thomas (BH), Geraldine Lee (LTU), Robert Champion (LTU) and Terry Mills (LTU). The project is supported by an Industry Collaborative Grant 2006, Faculty of Health Sciences, LTU.
Models for delivery of cancer services: In collaboration with researchers at Loddon Mallee Integrated Cancer Service, Terry Mills is involved in a number of research projects around the theme of improving the delivery of cancer services in Victoria, with a special emphasis on regional Victoria.
Numerical Methods in Statistics – Dr Robert Champion, Dr Christopher Lenard and Prof Terry Mills
A team consisting of Christopher Lenard, Robert Champion, Terry Mills and Chris Matthews (Computer Science & Computer Engineering, La Trobe) is studying applications of cross-validation methods to a variety of problems in applied mathematics.
Oscillation of a Mass Suspended on a Spring – Dr Robert Champion
The oscillatory motion of a mass suspended on a spiral spring is a standard application of second order linear differential equations in undergraduate mathematics subjects, and a commonly used illustration of simple harmonic motion and damped oscillations in physics texts. This topic is often a significant milestone for mathematics students in particular, in that it opens the door to the world of modelling with differential equations. The so-called spring-mass system is conceptually simple, and the elementary theory provides at least a qualitative understanding of observation. However, if one seeks answers to the question "How well does elementary theory actually compare with measurement?", a wealth of hidden, but fruitful, theoretical and practical problems are revealed.
Polynomial Interpolation – Dr Simon Smith
The major emphasis of Simon's on-going work in polynomial interpolation has been on the classical Lagrange and Hermite-Fejér interpolation
methods, and on their generalization to so-called