### MY MATHEMATICAL LIFE

My first school was called Adwalton. The pupils were all boys, aged from six to twelve and about 90 in number when I was there. The headmistress had herself established the school and the teachers were all women. My one mathematical memory of Adwalton is being told that multiplication must precede addition. No reason was given and it was years before I found out that it was just a convention which saved writing a lot of brackets.

My elder brother introduced me to algebra by revealing its role in the following piece of magic. Think of a number, add 5, double it, subtract 6, divide by 2, take away the number you first thought of: and the answer is 2! While I was still a student in primary school he gave me as a birthday present H E Dudeney's

*Amusements in Mathematics*. One of its problems, alleged to have come from an ancient manuscript about the Battle of Hastings, required the solution of a particularly difficult example of Pell's equation.

At the age of ten I was sent by my parents to Geelong Grammar, a boarding school that was reputed to be the Eton of the southern hemisphere. It was then for boys only. When, after three years, I completed junior school I was advanced to the second year of senior school. My first mathematics master in senior school did not have a university degree, only a teacher's certificate, but I learnt more from him than from many of my masters in other subjects who did have a degree. He introduced me to trigonometry.

My second mathematics master in senior school was also my housemaster. He introduced me to differential calculus and integral calculus. (I was very impressed by the ease with which these made it possible to solve problems that the greatest of ancient Greek mathematicians had solved only with difficulty.) His lessons were always prepared with care and I became proficient at solving examination questions. We were not given other types of problem and I did not learn then that problem solving was not my forte.

I spent much of my free time in the school library. The books that I read included Karl Pearson's

*The Grammar of Science*, Oliver Lodge's

*Pioneers of Science*, Bartky's

*Highlights of Astronomy*, W W Sawyer's

*Mathematician's Delight*, E T Bell's

*Men of Mathematics*and J W Young's

*Fundamental Concepts of Algebra and Geometry*. My housemaster lent me his own copy of Rouse Ball's

*A Short Account of the History of Mathematics*. I decided that I would study mathematics if I went on to university.

The Mathematics Department of Melbourne University was then part of the Faculty of Arts and a prerequisite for entry to that faculty was to have passed an examination in a foreign language at matriculation level. I qualified by passing in Latin, the only foreign language I studied at Geelong Grammar.

At that time the courses for the Honours degree in mathematics were completely separate from the courses for the Ordinary degree and a prerequisite for entry to the Honours programme was to have passed an additional mathematics examination. This meant an extra year's study of mathematics at school, which few schools were able to provide. I was fortunate to be at one of them and I passed the additional examination. Because of the extra year it required I entered university at a normal age.

My mother had spent the happiest years of her life at Janet Clarke Hall, which was then a part of Trinity College that admitted only women. She wanted me, her favourite son (as she herself told me), to be equally blessed. Unlike my two brothers, I therefore spent my three years at Melbourne University as a boarder at Trinity College. University lectures were very different from lessons at school. Students spent the lecture period busily writing down notes. Why didn't the lecturer just hand out a set of notes? I believe that a lecture can be a good way of introducing a topic to students, but that there is a law of diminishing returns with a course of lectures.

Besides lectures there were also classes (later called tutorials) in which students were given problems to solve and, after first year, also given instruction in numerical methods. I suspect that this instruction followed the practice at Edinburgh University, whereas the lecture syllabus for the Honours degree was modelled on the syllabus at Cambridge.

One problem we were given in first year was about iterating the values of the function $a^{x}$. My experience with this and other examples led me to write down in a notebook the conjecture that the sequence of iterates of any value of an arbitrary continuous function $f(x)$ would always converge, provided there did not exist distinct values $a$ and $b$ such that $f(a) = b$ and $f(b) = a$. For future reference I will call this conjecture $C$.

At the end of my first year there was an evening get-together for all staff and honours students at which a representative of each of the three honours years gave a lecture. I had been chosen to represent the first-year students and my lecture was entitled

*The converse of Toeplitz's theorem*. Toeplitz's theorem is a very general result which says that, when a sequence tends to a limit, many associated sequences will tend to the same limit. It is proved early on in Konrad Knopp's splendid book

*Theory and Application of Infinite Series*.

Converse theorems are about cases where an associated sequence tending to a limit implies that the original sequence tends to the same limit. I used a theorem of this type more than a decade later in a paper on the Falkner-Skan equation of boundary layer theory. It states that if a function plus a positive constant multiple of its derivative tends to a limit, then the function itself tends to that limit (which implies that the derivative tends to zero).

After the last lecture of the evening a staff member, Hans Schwerdtfeger, came over and told me that he had studied under Toeplitz. He added that he would like to talk with me and we fixed a time to meet. Two useful things came out of that meeting. He told me about

*Mathematical Reviews*, which was then not held in the university library. He said that another staff member, Felix Behrend, subscribed to

*Mathematical Reviews*and that his personal copy was available in the staff room.

Linear algebra was not then taught at Melbourne University and Schwerdtfeger also said he would obtain for me (on loan for a week from the library of Adelaide University, where he had taught before moving to Melbourne) a copy of the first edition of Paul Halmos's

*Finite-Dimensional Vector Spaces*. It seems odd now that Hardy and Littlewood knew nothing about linear algebra.

Towards the end of my second year, when I should have been preparing for exams, I succeeded in proving conjecture $C$. The Head of the Mathematics Department was T M Cherry. Because he was on study leave throughout my third year, the only lectures I received from him were two lectures he gave because the appointed lecturers were unwell. They were both memorable. In one he said that in his opinion the most useful of the many tests for series convergence was the integral test. In the other he used continued fractions to prove that, if $x$ is irrational, the powers exp$(inx)$ of exp$(ix)$ are everywhere dense on the unit circle.

In my third year I attended an extracurricular course on almost periodic functions that was given by Schwerdtfeger. I have written two papers involving almost periodic functions. The first consisted of several applications to ordinary differential equations and the second gave a simple solution to a problem of Hahn.

Throughout my three years at Melbourne University I spent much of my free time in libraries (usually the university library, sometimes the college library and for a while the public library in the city). I read with delight Lagrange's supplementary chapter on continued fractions in the French translation of Euler's

*Algebra*and Jacobi's lectures on elliptic functions in his

*Gesammelte Werke*. French gave me little difficulty, as I had studied it for two years at Adwalton, but reading German word by word with the aid of a dictionary was troublesome. I could not find many of the words which began with 'ge'.

Some books which especially impressed me were de La Vallee Poussin's

*Cours d'analyse infinitesimale*, Widder's book

*The Laplace Transform*and Nörlund's

*Vorlesungen über Differenzenrechnung*. My introduction to number theory came from Uspensky and Heaslet's

*Elementary Number Theory*.

I learnt also that important mathematics which had appeared in journals many years earlier was still unavailable in books. For example, the papers on linear integral equations by Erhard Schmidt (1907) in

*Mathematische Annalen*and by F Riesz (1916) in

*Acta Mathematica*. I read many other books and papers that were not so notable, but through them I developed my own taste and learnt that I could become proficient in at most a small area of mathematics. Later, my own audience would be similarly limited.

Melbourne University kindly gave me a Bachelor of Arts degree with first-class honours, even though my applied mathematics was certainly below that standard. I was then asked by Schwerdtfeger to bring him any original work I had done. On seeing my axioms for the field of complex numbers he told me of the much neater topological characterization by Pontryagin. I was very disappointed, however, by his reaction to my proof of conjecture $C$. He said that it was an "im grossen" result and that what one needed was an "im kleinen" result. The whole point was that the usual textbook treatment was the latter and what one needed was the former.

The Australian academic year ends in December and the English academic year starts in September. In early March 1951 I arrived in England with a sealed envelope, which contained a reference for me from Professor Cherry and which was addressed to Professor Besicovitch at Trinity College, Cambridge. 'Bessy' arranged for me to meet two potential PhD supervisors, Frank Smithies and Philip Hall.

I told Smithies that I had read Widder's book and he brought me up to date by telling me about Gelfand's use of maximal ideals to prove a theorem of Wiener on absolutely convergent Fourier series. But his idea for a thesis topic, "dualizing" some work of Laurent Schwartz on mean periodic functions, had no appeal.

Philip Hall asked me what I knew about group theory and I said that I had read Ledermann's

*Introduction to the Theory of Finite Groups*. He asked if I had any ideas for a thesis topic and I proposed proving, without the use of group characters, Burnside's theorem that a finite simple group cannot have a conjugacy class of prime power order. He said he knew people who had tried without success to do precisely that. He asked if I had any ideas for going about it and I said that it would follow from a general conjecture about factorisable groups. He then agreed to be my supervisor and said he would let me work at this for a year.

After I had found accommodation in Cambridge, I added a Concluding Theorem to my conjecture $C$ paper and submitted it to the

*Quarterly Journal of Mathematics*(Oxford Series). In due course they returned it with a note, saying they regretted that in the present state of publishing they were unable to accept it. As they did not suggest that I send it elsewhere, I put the paper away in a folder. Towards the end of my time in Cambridge, when I started applying for a job, I asked Philip Hall to submit the paper to the

*Proceedings of the Cambridge Philosophical Society*and it was published there in 1955.

After my paper had appeared, conjecture $C$ was proved again by the Ukrainian mathematician Sharkovsky. Later he proved a remarkable generalization, which has attracted wide attention, of the key Preliminary Theorem in my paper. Some years later I further developed Sharkovsky's work in a paper that was intended for a volume celebrating the eightieth birthday of Philip Hall. When he died, the volume was abandoned and the paper appeared instead in the

*Proceedings of the Cambridge Philosophical Society*. This paper started a collaboration with Louis Block, which resulted in a joint paper in the

*Transactions of the American Mathematical Society*. In 1992 Louis Block and I gave an account of the subject, complete with proofs (a few of which had not previously appeared in print and many of which had not been available in English), in

*Dynamics in One Dimension*, a volume in the

*Springer Lecture Notes*series. It has already had over a thousand citations.

I had applied for a Rouse Ball research scholarship soon after I arrived in Cambridge. My application was successful and the scholarship supported me for three years at Trinity College without my having to be reassessed during this period.

After one year I had made no progress with my search for a character-free proof of Burnside's theorem. To this day a character-free proof has not been found, although there now exists a very involved one for the corollary that a finite group is solvable if its order is divisible by at most two distinct primes.

Philip Hall then told me about L E Dickson's finite analogue of the simple Lie group $G_{2}$ and he proposed that I find finite analogues for other simple Lie groups. I obtained a set of generators for the symplectic group that would carry over to finite fields. He said that he liked my work, but did not return it. Soon afterwards André Weil visited Cambridge and in the course of a lecture he mentioned that Chevalley, in work that was not yet published, had constructed finite analogues for all the simple Lie groups. I rushed to see Philip Hall, who said that Weil had visited him at his rooms in King's College and had told him about it.

Cambridge provided opportunities for buying mathematics books that were in stock nowhere else in the U.K., except possibly at Blackwell's in Oxford. I financed new acquisitions by selling old ones. Two books I particularly remember are B. Segre's lecture notes, Forme differenziali e loro integrali, and Differentialgleichungen: Lösungsmethoden und Lösungen (Band 1), a compendium of information assembled by Kamke.

After leaving Cambridge I married my fiancée in her home town of Verona. When we returned to Verona after our honeymoon I bought in a bookshop there two books on ordinary differential equations, Tricomi's

*Equazioni differenziali*and Sansone's

*Equazioni differenziali nel campo reale*.

My first academic position was at Birmingham University. It was called a Research Fellowship, but it also involved giving one undergraduate course of lectures. The university had two mathematics departments: C A Rogers was Head of Pure Mathematics, to which I belonged, and Rudolf Peierls was Head of Applied Mathematics. Peierls attracted research students from all over the world, one had even left Dirac at Cambridge to work with him.

After I had been in Birmingham for some months I was told, for the first time, that I was liable for English National Service. Instead of serving in one of the armed forces I obtained a position in a reserved occupation with the English Electric Company. The National Service obligation was normally for two years, but it ceased at the age of 26, so I spent only thirteen months with the English Electric Company.

I then obtained a position as Assistant Lecturer at Birkbeck College in the University of London. At that time Birkbeck College was an evening college and the students, some of them middle-aged, had daytime jobs. Because of this the lecture load was light and one could do research during the day without interruption. After three years I was promoted to Lecturer. Altogether, I spent five years and one term at Birkbeck College.

Near the end of my time there I attended an excellent international conference on nonlinear oscillations in Kiev. Among the many Russians who took part were Bogolyubov, M G Krein and Pontryagin. The American delegation, led by Lefschetz, included a youthful Smale. Mary Cartwright was one of three people from the U.K.

In a paper (with an obvious misprint) which I wrote while I was at Birkbeck I showed that an algorithm proposed by Lin could be used to converge to a real root for all cubic polynomials with real coefficients. In 1987 McMullen found an algorithm that converges to a complex root for all cubic polynomials with complex coefficients. He also showed that no algorithm can converge to a complex root for all complex polynomials of degree $n$, if $n$ is greater than 3.

To save it from premature oblivion I mention that, in a paper on a differential equation which is of interest in the theory of elasticity, I discovered purely by chance that the incomplete lemniscate integrals of both the first and second kinds can be expressed by hypergeometric functions.

My paper on the Falkner-Skan equation (the longest paper I have ever written) was also a product of my time at Birkbeck. For future reference I will call it my $W$ paper, as in it I generalized a result of Hermann Weyl and greatly simplified the proof. I also obtained much additional information. I submitted it to the Proceedings of the London Mathematical Society, but they rejected it. I then asked my Head of Department, who was an F.R.S., to submit it to the Philosophical Transactions of the Royal Society and they did accept it. Many years later somebody told me that they had refereed the paper when it was first submitted to the London Mathematical Society and had strongly recommended its publication.

In 1950 an Institute of Advanced Studies began operating in Canberra (Australia's capital) with four separate research schools, one of which was the Research School of Physical Sciences. Undergraduate education was already being provided at Canberra University College. In 1960 the two institutions amalgamated to form the Australian National University. It was decided to create a Department of Mathematics in the Research School of Physical Sciences and Bernhard Neumann was appointed as its Head.

I was interviewed by Neumann for a Research Fellowship at a café in Swiss Cottage (he was visiting his parents in nearby Hampstead). I obtained the position and early in 1962 I arrived with my family in Canberra. Robert Edwards was already there. John Miles, the Professor of Applied Mathematics, arrived soon after me, followed by Zvonimir Janko. Because he was in the United States on study leave from Manchester University, Neumann himself came later.

The department was accommodated at first in a bungalow on the campus. An adequate mathematics library had already been built up by Pat Moran, the Head of the Department of Statistics in the Research School of Social Sciences from its foundation.

In my $W$ paper I pointed out that my results would hold for a larger range of parameter values if a certain quadratic system of differential equations had at most one limit cycle. This led me to search the literature for information about quadratic systems. I summarized my findings in a survey article in the

*Journal of Differential Equations*, founded by La Salle the previous year. A Russian mathematician subsequently proved the result that I needed for my $W$ paper. In a paper in

*Dynamics Reported*(which contained also other results) I later gave a simpler proof and there is now a simpler proof than mine.

A new topic I worked at for some years was exponential dichotomies, which Massera and Schaffer defined and investigated in a series of papers in the

*Annals of Mathematics*. I found an elementary proof for one main theorem, which they had proved using the Open Mapping Theorem from functional analysis. My book

*Dichotomies in Stability Theory*, also in the

*Springer Lecture Notes*series, was written to provide an accessible account of the subject.

Another area that I worked in for some time was the theory of Linear Systems. This had been developed by electrical engineers (Kalman in particular) who specialized in control theory. I became aware of their work through Roger Brockett's book

*Finite Dimensional Linear Systems*. I single out for mention my paper

*Matrices of rational functions*, because its determinantal denominators could also have applications in number theory.

My privileged working conditions enabled me to expand my interests in this way. In order to retain them I had turned down offers of Chairs in Australia and in the United States, although they would have greatly increased my salary. Research is a roller coaster and some of my papers exist only because I felt under pressure to publish. I found it more rewarding to write books that were additions to the literature (but not textbooks).

After existing for twenty-five years my Department of Mathematics was closed down, on the recommendation of a committee of overseas experts. I transferred to the Department of Theoretical Physics, which continued to exist in the Research School of Physical Sciences. I remained there until I retired at the end of the year in which I turned 65, as was then mandatory.

The part of Hilbert's 16th problem about the number of limit cycles of a polynomial system of differential equations remains unsolved even for quadratic systems. Quadratic systems with four limit cycles are known and it seems likely that no quadratic system has more than four limit cycles. Because of the fame of Hilbert's problems, I get asked why I restrict myself to quadratic systems. I do so, not because they occur most frequently in applications, but because there is a body of results for quadratic systems, not one of which holds for systems of higher degree. Quadratic systems are a subject, whereas even cubic systems are not.

Poincaré's work on polynomial systems was extended by Bendixson to continuously differentiable systems. On account of this it took me some time to realize that Poincaré's plan of studying polynomial systems in the projective plane, and in particular their critical points at infinity, was useful for my work on quadratic systems.

Quadratic systems have been studied in China more than in any other country, which meant that I needed to read mathematical articles in Chinese. I managed, but I never became as competent as Kurt Mahler, the colleague at the Institute of Advanced Studies whom I most admired. As professional translators often have difficulty with technical terms, I compiled

*A Chinese-English Mathematics Primer*to help mathematicians translate by themselves articles in Chinese.

Some might think that, because mathematics is an exact science, there is no place in it for differences of opinion. There should be a place, but authority carries as much weight in mathematics as anywhere else. Euclid's axioms for geometry remained sacrosanct for over two thousand years, until Pasch showed that further axioms were needed. One of the additional axioms he proposed was more cumbersome than the others and Peano used instead two simple axioms that had the advantage of holding in Euclidean spaces of arbitrary dimension. Because Hilbert restricted himself to three-dimensional space in his

*Grundlagen der Geometrie*, and because Peano used his own symbolic notation, Peano's contribution was not as well-known as it deserved to be. Rectifying this was one aim of my book

*Foundations of Convex Geometry*.

After my retirement I remained for six more years in the Department of Theoretical Physics as an unpaid Visiting Fellow. I used this time to complete the book

*Number Theory: An Introduction to Mathematics*, for which I had long been collecting material. My reasons for writing the book are set out in its Preface. To see if others shared these views I self-published a small number of copies, but the book still obtained a review in

*Mathematical Reviews*. I then approached commercial publishers and, after three rejections, it was accepted by Springer. It was their decision to issue the two volumes separately, but they changed their minds and the second edition appeared more conveniently as one volume.

It should be clear by now that books have played a large part in my mathematical education and that I have tried to repay my debt. Because corrections, improvements and additions can be incorporated so easily with a computer, e-books are replacing published books. For subjects that are being actively developed they certainly speed things up, but they are less helpful for the legacy of the past. I believe that it is still worth publishing, in hard copy, classics like Kato's

*Perturbation Theory for Linear Operators*and Helgason's

*Differential Geometry, Lie Groups, and Symmetric Spaces*, but that now they should be accompanied by a website for updates.

Why did I choose mathematics as a way of earning a living, when to so many people it is an anathema?

One attraction of mathematics for me is what Wigner called its "unreasonable effectiveness". A simple example is the discovery by Rutherford that the rate of radioactive decay is described by the exponential function, which in the late 1940s led to radiocarbon dating. A famous example is Kepler's use of Apollonius's work on conics in formulating his three laws of planetary motion. The scope of mathematics was greatly enlarged when Newton showed that all three laws follow from his law of universal gravitation. A more recent example is the application of the Radon transform to tomography.

Another attraction of mathematics for me is its universality. Contributions have been made by people of many nationalities and of differing social classes, religions and political ideologies. Mathematics, also, is not (like Literature) simply a creation of the author's imagination.

Other ways in which it appeals are its beauty, which outsiders have difficulty in understanding, and that (unlike Classics) it continues to evolve. I wanted to contribute to this evolution.

Even though I did not have a doctoral degree, for almost all my working life I have been free to pursue mathematics in my own way. It is not for me to say if this was merited, my task was to make the most of the opportunity.

I have spent much of my time on exposition. Research is rightly more highly valued, as it is both more difficult and riskier, but exposition (along with editing journals, funding scholarships and much else) is a part of the whole enterprise. I believe one should concentrate on what one does best, and my expository work has had an element of originality by breaking with tradition when this seemed to me appropriate.

I rate mathematicians who spend their time on non-mathematical activities below those who apply themselves to mathematical problems that seem to me of little interest, as often what eventually emerges is more significant than the original problem.