# Brian David Sleeman

### Quick Info

London, England

Dundee, Scotland

**Brian Sleeman**was an English mathematician who spent much of his career in Scotland. He specialised in applied analysis and mathematical biology.

### Biography

**Brian Sleeman**is the son of Richard Kinsman Sleeman and Gertrude Cecilia Gamble. Richard Sleeman (1904-2002) was born in Launceston, Cornwall on 6 October. He had two brothers, one of whom was killed in World War I, and two sisters. Gertrude Gamble (1904-1998) was born in Norfolk on 29 November.

Brian Sleeman was educated in London and attended the Battersea College of Technology (now the University of Surrey), being awarded a B.Sc. in 1963. On 7 September 1963 he married Juliet Mary Shea; they have three children, Elizabeth Anne Sleeman, Matthew Alexander Sleeman, and David James Sleeman.

Continuing his studies in London, Sleeman undertook research at the University of London advised by Felix Medland Arscott. Felix Arscott (1922-1996) served in World War II before obtaining a University of London degree through private study. He was awarded a Ph.D. from the University of London in 1956 for his thesis

*Ellipsoidal Harmonics and Ellipsoidal Wave Functions*and, after teaching in Aberdeen, was appointed to Battersea College of Technology. He was an expert in higher special functions. The reference [3] is Sleeman's obituary of his advisor Arscott.

Sleeman was awarded a Ph.D. in 1966 for his thesis

*Some Boundary Value Problems Associated with the Heun Equation*. His first paper was published in 1966, being

*The expansion of Lamé functions into series of associated Legendre functions of the second kind*. It appeared in the

*Proceeding*of the Cambridge Philosophical Society. He writes in the Introduction:-

In this paper, a study is made of the solutions of Lamé's differential equation as series of associated Legendre functions. The particular feature studied is the representation of the second solution corresponding to the case when the first solution is a Lamé polynomial.Sleeman's address on this paper, when it was submitted on 11 October 1965, was Department of Mathematics, Battersea College of Technology, London. A note on the paper gives his present address, presumably at the time of the acceptance of the paper, as Department of Mathematics, Queen's College, Dundee, Scotland. He ends the paper with the following acknowledgement:-

I am grateful to Prof F M Arscott and my colleague Mr D C Stocks for their valuable help in the preparation of this paper, and also to Mr R S Taylor for permission to use his unpublished results concerning the integral relations for Lamé functions of the second kind.His second paper

*The scalar scattering of a plane wave by an ellipsoid*was published in 1967 with his address given as Department of Mathematics, Queen's College, Dundee, Scotland. He gives the following Abstract to this paper:-

The problem of scalar Dirichlet scattering by a general ellipsoid is discussed. An exact solution of the wave equation is determined via the method of separation of the variables leading to expressions for the total field and the far field amplitude in terms of ellipsoidal wave function products. Particular attention is paid to the case when the ellipsoid is almost a prolate spheroid. Finally methods of numerical solution are discussed and two new results in ellipsoidal wave function theory are obtained.When Sleeman was appointed as an Assistant Lecturer in Mathematics at Queen's College, Dundee, he became a member of staff of the University of St Andrews since the College was part of the University of St Andrews until 1967. At the time of his appointment, Henry Jack was a senior lecturer in Mathematics at Queen's College, Dundee, Norrie Everitt was the Baxter Professor of Mathematics in Dundee and head of the Department of Mathematics while Douglas Jones was head of the Department of Applied Mathematics and held the Ivory Chair of Applied Mathematics. Brian Sleeman writes in [5]:-

On my arrival [in Dundee] I was immediately aware that teaching and scholarship in its widest form were of the highest priority. There was also no pressure on colleagues to write grant proposals but rather to pursue research for its own sake and to make original contributions. Douglas Jones never directed the research of young staff but was always there to give encouragement and offer ideas. In my case, after a couple of years, Douglas told me that he thought it would be a good idea if I spent a year at the Courant Institute of Mathematical Sciences at New York University. In particular he arranged for me to work in Joe Keller's group. So my wife, Julie, and I, together with two very young children, headed off to the 'Big Apple' and spent what for me was a momentous and exciting year, which had a fundamental influence on my career. There I met R (Richard) Courant and enjoyed seminars by P D Lax, L Nirenberg, J Stoker and E (Eugene) Isaacson as well as Joe Keller. With regard to teaching back in Dundee, Douglas assigned lecturing duties that on the one hand one would enjoy and on the other he thought would be 'good for the soul'. On my return to Dundee he assigned to me a new course on approximation theory, which was being offered to the first graduate students on the new Numerical Analysis and Programming Masters Course. I knew absolutely nothing about approximation theory and thought that Douglas had made a mistake with the assignment. So, plucking up courage I decided to go and discuss the matter with him. After knocking on his door and waiting for the red light to turn blue, indicating entry, I was ushered in. 'Professor Jones,' I said, 'you have assigned the Numerical Analysis and Programming course on approximation theory to me, but I know nothing about the subject.' His response was firm and short, 'Well you will do when you have given the course.'In [1] Sleeman writes the following about Joe Keller's influence on him during the year 1970-71 which he spent at the Courant Institute:-

Sleeman was a visiting member of Joe's group and was greatly influenced by Joe's enthusiasm and ideas which led to his investigations into the rigorous justification of the geometrical theory of diffraction, an interest in nerve impulse transmissions as well as inverse problems.In 1967 Queen's College, Dundee became the new University of Dundee. In the same year Sleeman was promoted to Lecturer in Mathematics, then promoted to Reader in Mathematics in 1971. He spent the academic year 1976-1977 as a visiting professor at the University Tennessee, Knoxville, USA. A series of lectures he gave at the University of Tennessee at Knoxville in the spring of 1977 became the basis for his book

*Multiparameter spectral theory in Hilbert space*(1978) (see below). In 1978 he was promoted to Professor at the University of Dundee.

One of Sleeman's major contributions was assisting in the running of conferences on the 'Theory of Ordinary and Partial Differential Equations'. The first, held in Dundee from 28 to 31 March 1972, had Proceedings published in the Springer Lecture Notes in Mathematics series with Norrie Everitt and Brian Sleeman as editors. They write in the Preface:-

These Proceedings form a record of the lectures delivered at the Conference on the Theory of Ordinary and Partial Differential Equations held at the University of Dundee, Scotland during the four days 28 to 31 March 1972. The Conference was attended by 140 mathematicians from the following countries Belgium, Canada, Denmark, France, Germany, Ireland, Italy, The Netherlands, Poland, Sweden, Switzerland, the United Kingdom and the United States of America. ... The Conference was organised by the following committee: W N Everitt (Chairman), J S Bradley, B D Sleeman and I M Michael. Dr Sleeman and Dr Michael acted as Organising Secretaries for the Conference.Sleeman was an editor of further conferences in the same series: the Conference held at the University of Dundee, Dundee, 26-29 March 1974; the Fourth Conference held at the University of Dundee, Dundee, 30 March-2 April, 1976; the Sixth Conference held at the University of Dundee, Dundee, 31 March-4 April, 1980; the Seventh Conference held at the University of Dundee, Dundee, 29 March-2 April, 1982; the Eighth Conference held at the University of Dundee, Dundee, 25-29 June, 1984; the Ninth Dundee Conference held at the University of Dundee, Dundee, 30 June-4 July, 1986; the Tenth Dundee Conference held at the University of Dundee, Dundee, July 1988; the Eleventh Conference held at The University of Dundee, Dundee, 3-6 July, 1990; and the Twelfth Conference held in honour of Professor D S Jones at the University of Dundee, Dundee, 22-26 June, 1992.

Up to 1980, Sleeman's research was mostly in applied analysis, multiparameter spectral theory, direct and inverse scattering theory. Examples of his papers on these topics are

*Integral equations and relations for Lamé functions and ellipsoidal wave functions*(1968),

*Multi-parameter eigenvalue problems in ordinary differential equations*(1971),

*The three-dimensional inverse scattering problem for the Helmholtz equation*(1973), (with G F Roach)

*Generalized multiparameter spectral theory*(1976), and

*Klein oscillations theorems for multiparameter eigenvalue problems in ordinary differential equations*(1979). In 1978 he published the book

*Multiparameter spectral theory in Hilbert space*. Gary Roach writes [2]:-

This is only the second book to have been written which is devoted entirely to topics in the theory of multiparameter problems. As the title indicates this particular book is concerned more with spectral theory than with general multiparameter problems; nevertheless it makes a significant contribution to an area of research which over recent years has seen a considerable renewal of interest. Clearly written, in a readable and persuasive manner, it collects together most of the recent Hilbert space developments which have occurred in the subject. The book is divided into nine chapters. The first gives an introduction to multiparameter problems and provides some of the motivation for subsequent chapters. The second introduces certain basic notions and techniques which are necessary for the theoretical developments given in Chapters three to eight. The final chapter is devoted to a consideration of open problems.For reviews and further information about this book, see THIS LINK.

Despite most of his papers being on these topics in applied analysis before 1980, there were also a couple of papers on mathematical biology, for example

*FitzHugh's nerve axon equations*(1975). In 1981 he published the paper

*Analysis of diffusion equations in biology*and in 1983 the book, written jointly with Douglas Jones,

*Differential Equations and Mathematical Biology*. The publisher gave the following information about one of the editions of the book:-

The conjoining of mathematics and biology has brought about significant advances in both areas, with mathematics providing a tool for modelling and understanding biological phenomena and biology stimulating developments in the theory of nonlinear differential equations. The continued application of mathematics to biology holds great promise and in fact may be the applied mathematics of the 21st century. Differential Equations and Mathematical Biology provides a detailed treatment of both ordinary and partial differential equations, techniques for their solution, and their use in a variety of biological applications. The presentation includes the fundamental techniques of nonlinear differential equations, bifurcation theory, and the impact of chaos on discrete time biological modelling. The authors provide generous coverage of numerical techniques and address a range of important applications, including heart physiology, nerve pulse transmission, chemical reactions, tumour growth, and epidemics. This book is the ideal vehicle for introducing the challenges of biology to mathematicians and likewise delivering key mathematical tools to biologists. Carefully designed for such multiple purposes, it serves equally well as a professional reference and as a text for coursework in differential equations, in biological modelling, or in differential equation models of biology for life science students.For reviews and further information about this book, see THIS LINK.

In 1989 Sleeman published the paper

*Complexity in Biological Systems and Hamiltonian Dynamics*. To obtain a feel for some of the topics which interested him we quote from the paper [4]:-

It is well known that in biology and the life sciences in general there are numerous instances of complex behaviour arising from apparently deterministic processes. One is familiar with the 'fractal' nature of the shapes of leaves and snowflakes. Such shapes may be described as the outcome of simple deterministic evolutionary dynamics. In another area the stochastic or chaotic response of stimulated cardiac nerve cells is well known and can be modelled by nonlinear deterministic systems of ordinary differential or difference equations. In most examples describing complex behaviour in biological systems the underlying models are either ordinary differential or difference equations leading to an analysis of temporal behaviour and its dependence on certain parameter values such as growth rates, generation times or reaction constants. In this paper we explore both the temporal and spatial nature of complex phenomena in biological systems. The examples involve partial differential or partial difference equations and are drawn from models of population genetics, mollusc shell patterning and excitable systems. In each situation we demonstrate aspects of complexity and show that non-integrable hamiltonian dynamical systems play a crucial role. This brings into the realms of biology such concepts as structural stability and Kolmogorov-Arnold-Moser (KAM) theory [named for Jürgen Moser], which lie at the heart of current developments in the theory of dynamical systems.In 1995 Sleeman left Dundee when he was appointed as Professor at the University of Leeds. He retired in 2004 and was honoured with a conference:-

The University of Dundee and the Mathematics Biomedical Network held this two day meeting in honour of Professor Brian Sleeman to mark the occasion of his 65th birthday and to celebrate his contributions to many areas of applied mathematics over the last 40 years. The meeting took place on Monday 11th and Tuesday 12th October 2004 at The West Park Conference Centre, University of Dundee. We organized a programme of 12 plenary talks given by internationally renowned speakers and spread over the two days with plenty of morning coffee, afternoon tea and lots of informal discussions. There was a special dinner on the Monday evening. We invited all of Brian's former PhD students and post-docs and many of his research collaborators from the last 40 years and many of them were able to attend. The plenary speakers were drawn from people whom Brian had worked with over the years or whom he had known as colleagues over the years, or who were active in some of the main areas of research Brian worked in. The programme was divided into two broad areas: Mathematical Biology and Applied Analysis.Among other honours given to Sleeman note he was elected a Fellow of the Royal Society of Edinburgh in 1976. He was founding editor of the journal

*Computational and Mathematical Methods in Medicine*which adopted this title in 2006 having been founded in 1997 as the

*Journal of Theoretical Medicine*.

Finally we note his interests in choral music and hill walking.

### References (show)

- J R Ockendon and B D Sleeman, Mathematics in the spirit of Joe Keller,
*European Journal of Applied Mathematics***28**(2017), 708-715. - G F Roach, Review: Multiparameter spectral theory in Hilbert space, by B D Sleeman,
*Proc. Edinburgh Math. Soc.***24**(1981), 231-232. - B D Sleeman, Felix M Arscott 1922-1996,
*Mathematics Today - Bulletin of the Institute of Mathematics and its Applications***33**(4) (1997), 119-121. - B D Sleeman, Complexity in Biological Systems and Hamiltonian Dynamics,
*Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences***425**(1868) (1989), 17-47. - B D Sleeman and I D Abrahams, Douglas Samuel Jones MBE. 10 January 1922 - 26 November 2013,
*Biogr. Mems Fell. R. Soc.***61**(2015), 203-224.

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Honours awarded to Brian Sleeman

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Written by J J O'Connor and E F Robertson

Last Update November 2019

Last Update November 2019