Department of Mathematics

DR A.D. OSBORNE

MacKay Building 2.13, tel: 01782 583265, e-mail: a.d.osborne@keele.ac.uk.

RESEARCH

My fields of interest can be conveniently categorised according to the following headings.

General Relativity

My focus of interest lies in the qualitative properties of geodesic equations for space-time, which are nonlinear ordinary differential equations. The general aim is to investigate properties of space-times via the properties of such equations. There are many different areas of mathematics involved in this work, including topological techniques, techniques of real analysis applied to differential equations and tensor analysis. My doctoral thesis, "Gravitation and Dynamical Systems" set out to find a viable, non-singular alternative to Einstein's theory of gravitation by associating dynamical systems with the flows of geodesic equations. This work was very novel for the time and, of course, was and still is, rather controversial.

Partial Differential Equations

While still a research student, I became interested in systematic methods for solving nonlinear partial differential equations. I had papers published in this non-controversial area while also working on my thesis. The importance of obtaining exact solutions of such equations, using analytical techniques, is in revealing unexpected phenomena, as in soliton theory, and in generating new principles such as conservation principles. My particular area of interest lies in separable solutions of such equations and the relationship between those and similarlity solutions. The study of the connection between the structure of solutions obtained from such distinct methods involves a variety of techniques and knowledge of group theory.

Number Theory in Algebra

Towards the end of the 80's, I became interested in the use of number theory in algebra. This interest arose from the problem of finding orthonormal bases of R with integer coordinates and integer lengths. I worked with H. Liebeck, now retired from the department on this and related problems. We solved this and certain related problems in a number of publications and then moved on to the more general and difficult problem of rational congruence of matrices. In a certain sense, we have solved this problem, but our results depend on an existing theorem from number theory which is very difficult to prove. My hope is that obtaining our result in a different way may lead to a simpler proof of this difficult theorem. More generally, I believe that there is a lot of scope for using results in number theory to study particular problems in algebra.

Philosophical Relativity

I have worked on the philosphical side of relativity with N.V. Pope over a number of years. We have been concerned with a new philosophical approach to Special Relativity in particular, which has adressed a number of issues of current concern, such as the EPR paradox on action-at-a- distance. We have had problems in finding an outlet for this work since it is difficult to convince physicists that philosophy is important and traditional philosophers would argue that it is not their place to question physics. However, our work is summarised in our Physics Essays papers, which have been well-received by certain groups, particularly in Canada. We still have a lot of work to do in this particular area.

Staff | Mathematics Department | Keele University

This page modified 20th September 1999. Send comments, corrections and complaints to a.looms@keele.ac.uk