*Probability and Mathematical Genetics: Papers in Honour of Sir John Kingman*edited by N H Bingham and C M Goldie was published in honour of Kingman's 70th birthday. Kingman himself contributed an autobiographical chapter.

## A fragment of autobiography, 1957 - 1967

**J. F. C. Kingman**

It is a great honour to have been given this fine collection, and I am most grateful to the Editors and to my friends who have contributed to it. It is poor thanks to inflict on them this self-centred account, and I have tried to mitigate the offence by limiting it to a particular decade, which begins as a new student arrives at Pembroke College in Cambridge to start a degree course in mathematics, and ends with him as a professor in the new University of Sussex. It was obviously a crucial time in my own mathematical life, but it happens also to have been a very interesting period in the development of probability theory in Britain, in which I was fortunate to play a junior part.

With little reluctance I have restricted myself to the mathematical aspects of my life. The reader will seek in vain for any details of my transition from a schoolboy to a happy husband soon to be a proud father; the latter conditions have proved lasting.

Of course the story goes back long before 1957. It might be said to have started in about 1920, when Charles Kingman, a miner on the small but prosperous North Somerset coalfield, summoned his two sons William and Frank for a serious talk. Charles came of a family of Mendip villagers, his grandfather having been the carter of Ston Easton. Kingman is a commoner surname in the USA than in England, because most of the family emigrated to the American colonies in the 17th century, but obviously a few overslept and missed the boat.

The message that William and Frank heard from their father was that whatever they did in life they must not go down the mine. William became an excellent baker, and ironically his lungs suffered as badly from flour as they would have done from coal dust. My father Frank (born in 1907) won an award that enabled him to enter the young University of Bristol (chartered 1909), and for six years he commuted by train between the family home in Midsomer Norton and the University, ending up with a first class BSc in chemistry and a PhD in catalytic adsorption.

Armed with this qualification, Frank was able to move to the distinguished Colloid Science Laboratory in Cambridge, to undertake what we now call post-doctoral work under Sir Eric Rideal. It was an exciting environment, with colleagues from varying disciplines like H. W. Melville, who later founded the Science Research Council, and J. S. Mitchell, a future Regius Professor of Physic. Nowadays such an experience might well have led to an academic career, but university posts were scarce, and Frank entered the Scientific Civil Service, working at the Fuel Research Station in east London.

At first he lived with an uncle, but he soon met, and in 1938 married, a girl called Maud Harley who came of a London family, her father a successful gentlemen's hairdresser. They settled in the southern suburb of West Wickham, and produced two sons, myself and my younger brother Robert (who in due course followed his father in reading chemistry at Bristol). I was born six days before the outbreak of war in 1939, and therefore grew up in the grim time of war-torn London. The area around West Wickham was heavily bombed, but fortunately we were spared, and I started school towards the end of the war.

Because of my father's scientific occupation, he was not called up to fight, so that our family remained together. One of the values that my parents shared was a firm belief in education, and a concern that their sons should work hard to develop any talents they might possess. Thus I was taught to read and write at home before starting school, and thereafter was given every encouragement to take advantage of the teaching provided. My most vivid memory was of a lovely middle-aged teacher Mrs Underhay, who allowed her pupils to walk home with her after school to practise as we walked the multiplication table. I got to the 16-times table before graduating to the wonderfully complex weights and measures of that pre-metric age.

Shortly after the war, my father was transferred to the Fire Research Station, which was in Elstree north of London, so we moved in 1949 to the northern suburb of Mill Hill, and the next year I entered Christ's College Finchley, which had been founded in 1857 as a church school, but had later become a secondary grammar school funded by the local authority. Tragically my mother died of breast cancer in 1951, and my father was left to bring up two teenage boys on his own.

Christ's College was an excellent school of a type that has now almost disappeared. It was ruled by an old-fashioned headmaster H. B. Pegrum, an Oxford man who believed in academic excellence, team games, clean living and the Church of England. I found that I could share in the first of these, aided by a competitive instinct that found no expression on the sports field, where I was notably incompetent. The fourth was somewhat incongruous, since the school drew largely from the Jewish community of Finchley and Edgware, but this cultural mix was beneficial for me because I was pitted against boys with a strong family belief in learning.

I found that a good memory and quick thinking made me good at subjects that required no coordination of hand and eye, but mathematics was the one I enjoyed. The senior mathematics master was H. W. Turl, a man of broad interests who taught economics and law as well. He encouraged me to think of myself as a possible future mathematician, and he gained a powerful ally in the head of chemistry after I had dropped the Kipps apparatus and filled the laboratories with the smell of rotten eggs.

Turl had a sardonic style that commended him to cynical schoolboys. Faced one day with some laboured calculation I had produced, he advised me: "Kingman, let me tell you, mathematicians are lazy". This injunction, to find the right way to solve a problem, if possible without grinding through avoidable algebra, was probably the first piece of real mathematical insight I encountered, and half a century of experience has proved its worth.

It was assumed, by both the school and my father, that I would go on to a university to read mathematics. The most popular universities for science students from Christ's College were Imperial College London and Bristol, but Pegrum persuaded me and my father that I should drop all subjects except mathematics and physics in the sixth form, take the final examinations a year early, and then try for a Cambridge scholarship. My father, who had nostalgic memories of his time at Cambridge, found this advice congenial. I had little idea of what would be involved, but when I looked at past papers for the scholarship examination, I found them much more interesting than those I had seen in school examinations. However, both he and I were more than surprised when a telegram arrived from Pembroke College telling me that I had been awarded a Major Scholarship.

Thus it was that, in October 1957, I presented myself at the porter's lodge of Pembroke to study mathematics. Cambridge is full of splendid ancient structures, some like King's College Chapel built of stone, some of flesh and blood, and others less material but no less important. Of the last group, the Mathematical Tripos is pre-eminent. Founded in 1748 as probably the first written degree examination in Christendom, it has graded some of the finest British brains into its three classes of Wrangler, Senior Optime and Junior Optime. Until 1909 the three classes were each listed in order of merit, and to be Senior Wrangler was to be hailed by the whole university. Even after the order ceased to be published, it was known informally, and there was fierce competition between and within colleges.

This competitive aspect has attracted criticism, but I am not convinced that its effects were altogether bad. It encouraged examiners to set questions that demanded more than the regurgitation of standard proofs and fashionable jargon, and put a premium on solving hard problems. If one wanted to do well in the Tripos examinations, one attended the lectures and tested one's understanding by trying to answer questions in recent past papers. If this failed, one took difficulties to the weekly supervision, at which the problem might or might not be resolved. It was a demanding regime, but one that made allowance for different levels of ability, and that challenged the better students to develop their abilities to the full.

At that time the Mathematical Tripos was divided into three parts. Most students took Part I at the end of their first year, and Part II at the end of their third. Success in Part II allowed them to graduate BA, after which they either left Cambridge or spent a fourth year taking Part III. It was however considered that someone who had gained a college scholarship had probably covered most of the Part I syllabus, and could start at once on Part II, taking that examination at the end of the second year. It was not allowed to graduate in less than three years, so the third year could be spent on Part III or indeed on some other subject.

The lectures for Part II were very variable both in style and in competence. Some of the lecturers were world class mathematicians Hoyle, Atiyah, Cartwright for instance. Others were unknowns who had published nothing since their PhD theses. Some were brilliant lecturers, while others would never have survived the modern world of teaching quality assessment. The correlation between these two classifications was not statistically significant. The syllabuses were old-fashioned, because any change had to be agreed by the whole faculty. Lecturers had to stick to the syllabus, because they did not in general set the examination questions. Thus it was very difficult to introduce new approaches, still less to introduce new' branches such as functional analysis.

Much therefore depended on the supervisions, which were organised on a college basis. The normal pattern was for the students to be grouped in pairs of similar ability. Each pair would have two supervisions each week, one in pure and one in applied mathematics. These might be given by a fellow of the college, or a research student, or by one of a diffuse cloud of mathematicians whose only source of subsistence was to offer such teaching to colleges in need.

I had chosen Pembroke for personal reasons unconnected with its teachers. In fact the only teaching fellow was a delightful old gentleman Robert Stoneley, a geophysicist who was an applied mathematician of a very old school. He regarded vector calculus as a very dubious invention, and tensors as unfit for civilised discourse. For pure mathematics, we were sent up the Newmarket Road to a charming man who was probably a better teacher than we realised, but who certainly did not inspire.

At the end of my first year, my luck changed. The Master of Pembroke, Sir Sydney Roberts, retired and was succeeded by Sir William Hodge, the Lowndean Professor of Astronomy and Geometry. Hodge had been a professorial fellow for many years, and as such had not taught in the College, but he had been instrumental, with the Tutor Tony Camps, in raising the academic standards and introducing such things as research fellowships. He now made a decisive move by attracting his former student Michael Atiyah from Trinity. It is impossible to understand how Trinity, of which Atiyah later became Master, allowed this to happen, but it was a stroke of good fortune for Pembroke mathematics and for me in particular.

In the first year I had been paired with Raymond Lickorish, whose forte was (and is) geometry, but Atiyah taught him alone, and I teamed up with John Bather, later to be my professorial successor at Sussex. Supervisions with Michael Atiyah were a quite remarkable experience, and the opportunity for close contact with one of the greatest mathematicians of our time (though we did not then know it) was an immense privilege. He had a habit of saying, when we took him a tricky question from a past Tripos paper: "Of course, the examiner hasn't really asked the right question". Only gradually did we realise that, as well as being a way of avoiding an awkward issue, Michael was educating us in mathematical priorities and developing our mathematical taste.

Where was probability in all this? Nowhere in the lecture list for the first year of Part II, but there was a Part I course in Statistics given by a Mr Lindley. The splendidly introverted convention of the Cambridge lecture list was that this prefix meant that the lecturer had no Cambridge doctorate. Had he been a Cambridge PhD he would have been listed as Dr D. V. Lindley. This would distinguish him from a Dr Lindley with no initials, denoting the holder of the higher ScD degree. Anyway, I had heard that Lindley was a good lecturer, so I went to this course although it would not contribute to my Part II.

The lectures, on basic probability and the standard statistical tests and estimators (Lindley had not yet met Bayes on the road to Damascus) were first rate, clear, well organised and full of interest. Lindley worked in the Statistical Laboratory, which meant that unlike most of the lecturers he had access to a duplicator and was able to issue duplicated notes, an unheard-of innovation.

Since I was not particularly well off, I thought I should try to get a job in the summer vacation after my first year. Through a friend of my father in Mill Hill, I heard that the Post Office Engineering Research Station at Dollis Hill, not far away, took on university students in the summer in a very junior temporary capacity, and I successfully applied for a job lasting some eight weeks. I was allocated to RE1, a small research group giving advice on mathematics and computing to the engineers working on a range of problems in the telecommunications field. After proving that I was not good at making the tea, I was given a handbook of common Fourier series which was known to be riddled with errors, and which I was told to revise. This had the useful effect of giving me one Tripos question which I could answer at sight, since there was always a function whose Fourier series we had to compute, and I had all the easy examples by heart.

After a couple of weeks, I graduated to some simple modelling the group was doing of the demand for telephone cable pairs to homes. In those days not every family had a telephone; if you applied for one you were lucky if there were enough cables in your road. Our models were very simple, and involved probability calculations little more advanced than those in Lindley's lectures. But it was enough to rouse my interest, and when I went back to Cambridge I looked at the available lectures to see if I could discover more about this sort of mathematics.

The word probability' did not figure in the list. Later I discovered that this was at the insistence of Sir Harold Jeffreys, who had just retired from the Plumian chair and who had laid down that the only Theory of Probability that could be taught was that contained in his book with that title. For Jeffreys, probability was a sort of objective subjective concept, Bayesian inference where the prior distribution was determined by considerations of invariance; no one but Jeffreys wanted to teach that. However, there was a course entitled Random Variables', which seemed promising not least because it was given by Mr Lindley.

It proved even better than Lindley's Part I course. From an axiomatic foundation which was a much watered down version of Kolmogorov, it proceeded to Markov chains and some of their important applications. In particular, Lindley used examples from the theory of queues. What his audience did not know was that Lindley had been the first referee for the paper that D. G. Kendall read to the Royal Statistical Society (RSS) in 1951, and that in proposing the vote of thanks to Kendall, Lindley had announced his famous identity which became the main basis of subsequent work on single server queues.

John Bather shared my enthusiasm for this course, and we asked Stoneley and Atiyah whether it counted as pure or applied for the purposes of supervision. Neither, was the unhelpful response, and when we protested they arranged for Dennis Lindley himself to give us a few special supervisions. These were in their own way as fascinating and challenging as Atiyah's, and they gave both of us a lifelong interest in probability.

At the end of my second year I returned to Dollis Hill, and told the head of RE1 (H. J. Josephs, whose main interest was the life and work of Oliver Heaviside) of my interest in queueing theory. He saw this as an opportunity to keep me out of mischief, and told me to go to the library and read the run of the Bell System Technical Journal, where I would find much queueing theory in its historic context of teletraffic theory. Thus I encountered the work of authors such as Pollaczek, Khinchin, Jensen, Palm, and others in the teletraffic world, as well as mathematicians such as Kendall, Lindley, Taka cs and the like.

Reporting back to Josephs, I was told that much of the mathematical work was irrelevant to the real world. For instance, the assumption that customers are served in order of arrival was not applicable to calls arriving at a telephone exchange, because the equipment could not store waiting calls in order. So I tried to solve simple queues with service in random order, and eventually found a formal solution for M/G/1 for this discipline. Although the solution was pretty uncomputable, it did have a useful approximation in heavy traffic, when the traffic intensity is just less than its critical value. This made me wonder whether there was a general phenomenon of robust approximation in heavy traffic, and so it turned out.

Back to Cambridge and Part III of the Mathematical Tripos. Part III is a complete contrast to the staid Part II. Every lecturer gets the opportunity to lecture on a topic of his or her own choosing, and to set the questions for the summer examination, and the result is a wonderful array of courses across the whole sweep of current mathematics. Dennis Lindley, Morris Walker and Violet Cane lectured on statistics, and Peter Whittle, fresh from New Zealand, on stochastic processes. But these made up only a fraction of the number of courses one was expected to take, so I found myself listening to Hodge on differential geometry, Fred Hoyle on relativity, George Batchelor on fluid dynamics and, most remarkable of all, Paul Dirac on quantum mechanics. To hear Dirac explaining how he had predicted the spin of the electron was an experience not to be missed. Even that, however, did not convert me to mathematical physics, and I remained firm in my preference for probability.

One other course, which I added only as an afterthought, had consequences much later. Sir Ronald Fisher, the father of modern statistics, had for many years been the Arthur Balfour Professor of Genetics, with his own department unconnected with the Mathematics Faculty. He had however given a Part III course on mathematical genetics, and when he retired he gave his voluminous lecture notes to his colleague A. R. G. Owen. George Owen was a nervous man whose main interest was poltergeists, and he gave the impression that he was always frightened of something (possibly Fisher). He lectured very badly, by turning over Fisher's notes and picking out topics at random for our delight.

One such topic was the Mandel--Scheuer inequality, which asserts in the spirit of Fisher's Fundamental Theorem of Natural Selection that, in a large randomly mating population evolving under selection at a single locus, the mean fitness increases from one generation to the next. It is not biologically important because it does not generalise to multilocus selection, but it is a nice bit of mathematics, and Owen dropped a hint that it would appear in the examination. The proof was however too complicated to memorise, and I managed to find a much simpler one, which I could remember. After the examination it was suggested that I might publish my proof, and this drew me to the attention of population geneticists in Australia and California, with whom I was later to collaborate.

My assumption however was still that I would embark on a PhD in stochastic processes, and my Part III performance was good enough to win me a grant, so I asked Lindley to be my supervisor. He declined because he had just been appointed a professor in the University College of Wales in Aberystwyth, which he thought would be a less good environment than Cambridge. I therefore joined the Statistical Laboratory in 1960, as a PhD student of Peter Whittle, and set to work to build on the research I had started at Dollis Hill.

Peter was a supervisor with a light touch, who led by example. He rarely gave direct advice, but to see the way he attacked the most difficult problems, with a combination of intuition and rigorous analysis, was to learn how to do applied mathematics at the highest level. I liked too the way he visibly enjoyed mathematics, not a chore but a delight.

I was fortunate too in my fellow students. The Laboratory was housed, for reasons it would take too long to relate, in a basement of the new chemical laboratories on Lensfield Road. Two rooms were allocated to research students, and we decided that one would be for smokers and one for non-smokers. The latter was occupied by Roger Miles (just finishing his thesis on Poisson flats), Bob Loynes (who had proved beautiful theorems about general queueing systems) and me. The other room was so full of pipe smoke that its occupants were not visible, but Hilton Miller and John Bather sometimes came up for air. It was a small enough group for us to learn a lot from each other, and I benefited particularly from Bob's very perceptive approach to queues.

Like most probabilists of my generation, I had been greatly influenced by William Feller's *Introduction to Probability Theory and its Applications*, of which Volume 1 had appeared in 1950. I was particularly taken by his use of recurrent events to analyse discrete time Markov chains, an approach which D. G. Kendall had used in his influential 1951 and 1953 papers on queues. But my own work on queues had been much more concerned with continuous time, and I wondered whether Feller's technique could be made to work with continuous time chains, such as those I was reading about in Kai Lai Chung's 1960 book *Markov Chains with Stationary Transition Probabilities*. In fact, a formula analogous to Feller's fundamental equation could be found in Maurice Bartlett's *Stochastic Processes*, and I had used it for the random service problem.

I was pursuing such thoughts alongside more concrete problems when Peter Whittle took me by surprise by announcing that he was to move to Manchester to succeed Bartlett in the statistics chair there. His departure, following that of Lindley, would leave a very small statistics group in Cambridge, and clearly I should have a change of scene. I would have been happy to follow Peter to Manchester, but he suggested that I should take the opportunity to ask David Kendall if I could work with him in Oxford.

With some daring, for DGK was an acronym to conjure with, I wrote to him, and received a charming and welcoming reply. I could pursue my Cambridge PhD for at least a year in the other place', or transfer to an Oxford DPhil. But fate had another surprise in store. One summer evening after dinner I was playing croquet on the Pembroke lawn with college friends, when I became aware that Sir William Hodge was standing in gown and mortarboard watching our game. The standard of play was not such as to justify such an exalted spectator, but after a few minutes he called me over to tell me that I had just been elected to a research fellowship of the College. I missed the next shot and lost the game.

I had not realised that I was being considered for such a position, and was more than surprised by the news. I explained that I was committed to a move to Oxford, but the Master was not put off. Not only would the College allow me to hold my fellowship at a distance, but Pembroke had a sister college in Oxford which would treat me as one of its own research fellows.

Thus it was that in the autumn of 1961 I arrived in Oxford, with a desk in the Mathematical Institute, high table rights at The Queen's College, and David Kendall as my PhD supervisor.

The Institute was something of which there was no equivalent in Cambridge, offering space to mathematicians of all varieties, including the professors and other teaching staff, but also to research students and visitors from other universities. It was ruled in theory by E. C. Titchmarsh, the Savilian Professor of Geometry who carried the wonderful title of Curator of the Institute, but in practice by a formidable administrator Rosemary Schwerdt. Rosemary told me I had to visit the Curator to be given a key; the interview took place in his room which resembled a rather fusty summerhouse, and it would be hard to say which of us was the more shy and tongue tied.

My desk was in an annexe which seemed to be used as a quarantine station for new arrivals from other places like Cambridge. In particular, there was my old teacher Atiyah, who had moved to a readership in preparation for his inevitable succession to the Savilian chair. I also found to my delight two probabilists visiting from the USA, Steven Orey and Don Iglehart, with whom I enjoyed fruitful collaborations.

David Kendall however, like other mathematicians whose teaching posts were primarily college based, had no place in the Institute, and worked in his rooms in Magdalen College, with beautiful views over the deer park. He was very welcoming, but also frightening, and I looked on my weekly visits to Magdalen with both delight and trepidation. His standards of both substance and presentation were high, and he decided in particular that my mathematical writing was sloppy and imprecise.

He also made it clear that his enthusiasm for queueing theory had waned. He thought that the decade since his RSS paper had seen an explosion of papers giving useless formal solutions of routine problems. I hope he found my efforts a little above the general level, but he certainly did not encourage me to continue to concentrate on queues. He was much more impressed by my thoughts on extending Feller's recurrent events to continuous time, and he made a number of excellent suggestions which helped me enormously in formulating what became the theory of regenerative phenomena.

I knew of course of his famous collaboration with Harry Reuter on the general theory of continuous time Markov processes with a countable infinity of states, and of the deep work that their student David Williams had done in developing their methods. David had left Oxford to work with Harry in Durham, but he returned for his DPhil viva by David Edwards and Steven Orey. I was allowed to attend this as a fly on the wall (but in proper academic dress), and I realised how much I had to learn before I could apply my ideas to general Markov theory. DGK told me that Kolmogorov had posed the problem of characterising the diagonal transition probabilities of Markov chain; I knew that they were standard p-functions (which I could characterise), but the question of which pfunctions could arise from Markov chains was much more delicate. I was sure that my methods were the right ones to answer the question, but it took me nearly another decade to achieve the result.

Once a week, at David Kendall's suggestion, he and I would join Steven and Don for an austere supper in Halifax House, the Oxford attempt at a social centre for postgraduates. One of the issues we discussed was how slow British mathematicians had been to embrace the rigorous probability theory that had grown out of the work of Kolmogorov and spread to the USA, France and elsewhere. There were active groups of applied probabilists in London under Bartlett and Cox, in Manchester with Whittle, Birmingham with Daniels, and elsewhere, but nothing comparable on the pure side. David was determined to spread the Kolmogorov gospel, and he and Harry invented the Stochastic Analysis Group (StAG) to give some coherence and visibility to the scattered disciples.

StAG was to be a very informal group of like-minded enthusiasts, sharing ideas and meeting as opportunities arose, for instance as splinter groups at the annual British Mathematical Colloquium. The London Mathematical Society was persuaded to hold one of its Durham instructional conferences on probability, at which the lecturers included Kendall's student David Edwards, by now specialising in functional analysis but still interested in probability, and the London analyst James Taylor. James and I were encouraged to write a book suitable for final year undergraduates, basing probability firmly on measure theory.

The existence of StAG gave great encouragement to young mathematicians taking their first steps in the field, and gave a more visible presence to the rigorous approach. For myself, I was excited by these new developments, but I still felt an affinity with the more pragmatic approach in which I had been brought up. Kai Lai Chung once said that mathematicians are more interested in building fire stations than in fighting fires, but I enjoyed using the hose as well as designing the headquarters.

Among the interesting mathematical figures I came to know in Oxford, one of the most memorable was George Temple, the Sedleian Professor of Natural Philosophy. He was a Fellow of Queen's, and the egalitarian world of the high table made it easy for a young research student to talk with a great authority on applied mathematics. George had a very broad range of mathematical interests, and by that time he was lecturing mainly on generalised functions, which to the pure mathematicians were distributions in the sense of Laurent Schwartz. But he was worth listening to on almost any subject, and I learned much from his genial wisdom. Many years later, I was privileged to attend the service in Quarr Abbey on the Isle of Wight, at which George took his final vows as a Benedictine monk.

All this time, I was of course in touch with my Cambridge college, and heard news of stirring developments. The applied mathematicians were at last tired of being scattered around the University, and were forming themselves into a Department of Applied Mathematics and Theoretical Physics (DAMTP) under an energetic leader, George Batchelor, who was determined to find a building in which his colleagues could have proper offices and facilities. The pure mathematicians could see the likely outcome as being that they would lose out, and were inching reluctantly towards a departmental structure in self-defence. Meanwhile, the University was under pressure from the RSS, which pointed out that Cambridge was the only major UK university that had never had a professor of statistics, and which helped to raise some funds to establish a chair.

What I did not know was that the two RSS representatives on the board of electors to the new chair, Maurice Kendall (no relation) and Egon Pearson, had seen the importance of the first professor being someone of real mathematical weight, and had set their sights on the appointment of DGK. Thus it came as a double surprise when David told me that he had agreed to move to Cambridge to head the Statistical Laboratory, and that he wanted me to go with him as an Assistant Lecturer. (Things were done like that in those days, with no nonsense of applications or interviews; the post to which I was appointed was the lectureship which had remained vacant when Whittle left, downgraded because of my junior status.)

So October 1962 found me back in Lensfield Road, with a room of my own and with all the jobs which neither David, nor the much more senior lecturers Morris Walker and Violet Cane, wanted to do. One of these was to give the Part I Statistics lectures. Most courses were given in parallel by two lecturers, but this was a luxury the statisticians could not afford, and I had all the first year mathematicians, well over 200, in Lecture Room A of the Arts Schools. Mercifully, I had kept my copy of Lindley's duplicated notes

Another job was rather different. The University Press was moving to new buildings out by the railway station, and Batchelor had persuaded the University to allocate part of the vacated site, on Silver Street, to DAMTP. Meanwhile the pure mathematicians had persuaded Sir William Hodge to head a new Department of Pure Mathematics and Mathematical Statistics (DPMMS), and Hodge applied his considerable political skill to finding a similar home for his department. He found an unexpected ally in his fellow Scot, the chemist Sir Alexander Todd. If the statisticians could be extruded from Lensfield Road, the space would be filled with chemists.

Thus it was that the paper warehouse of the Press was allocated to DPMMS, and Hodge set up a committee to plan the new home, with himself and two colleagues Frank Smithies and Ian Cassels, offering Kendall a place to represent the statisticians, which David filled with his new Assistant Lecturer. Here it was that I first learned the lessons of academic politics, especially that les absents ont toujours tort'. Of course it helped that I was a fellow of the college of which Hodge was Master, and he was less ruthless with me than if I had been from another college.

The conversion of the warehouse took a long time, and I did not stay long enough to benefit from my labours. The quarters in chemistry soon filled up with the visitors attracted by the new Professor Kendall, and his enthusiasm and vigour breathed life into what had been a demoralised group. He had had few research students in Oxford, but now very able mathematicians came to study for their PhD in pure and applied probability, Daryl Daley from Melbourne, John Haigh, Peter Lee, David Mannion, Jane Speakman. I shared with David in their supervision.

One casualty was my own PhD. Not only were the tasks of the junior member of the staff somewhat heavy, but my research studentship was superseded by the university post, and I should have had to meet the supervision and examination fees myself. I was reconciled to remaining Mr Kingman, partly because the doctoral title had always in my family been attached to my father. And so it remained until much later my daughter qualified in medicine.

Among the overseas visitors at a more senior level the one I remember best was Alfred R enyi from Budapest. He came with a great reputation in both probability and number theory, and proved to be a great source of ideas, problems and anecdotes, not least from his collaboration with Paul Erdős.

One day in early 1963 a knock on my door introduced someone I knew well by reputation but had never met, Joe Gani over from Canberra. His opening words were "How would you like to go to Western Australia?". This turned out not to be a threat of transportation; he had been commissioned to find a young statistician who would spend the (English) summer in the Perth winter teaching in the mathematics department of the University of Western Australia. I could give a graduate course on whatever I liked, so long as I also gave a final year undergraduate course on either sampling or experimental design. Joe dismissed my protestations that I knew nothing about either subject, so I found myself enjoying the Australian winter and instructing bemused students in the finer points of balanced incomplete block designs. At a party a real statistician delivered the killer question in broad Australian: "Have you ever designed an experiment?".

The most valuable aspect of my Australian trip was the opportunity to visit other universities, in Adelaide, Melbourne, Canberra and Sydney, on my way home. It was particularly good to meet Pat Moran at ANU, whose work on genetics as well as other areas of probability and statistics I much admired. He knew of my proof of the Mandel--Scheuer inequality, and suggested some other genetical problems. It was through him that I later came to collaborate with Warren Ewens and Geoff Watterson at Monash on genetic diversity in large populations, work that led to the useful concept of the coalescent.

Continuing to the east, I landed in San Francisco and spent a few days in Stanford. There I called on Sam Karlin, who like Moran talked about genetics as well as his work on Markov chains. I was to return to Stanford for a longer stay in 1968, to enjoy Sam's hospitality and that of Kai Lai Chung.

The early 1960s were of course the great period of university expansion in Britain, when newly established universities were scrabbling for academic staff. One such was the University of Sussex, sited between Brighton and Lewes in a beautiful fold of the South Downs. Its first Vice-Chancellor, John (later Lord) Fulton, had taken full advantage of the opportunity to recruit staff of the calibre of David Daiches and Asa Briggs, and the science side was led by the Oxford physicist Roger Blin-Stoyle. They saw the need for a strong mathematical presence, and enlisted the help of George Temple in finding possible leaders. He recommended Bernard Scott and Gilford Ward as professors of pure and applied mathematics respectively, but statistics proved more difficult.

Perhaps George remembered his conversations with the young Cambridge probabilist, for in 1964 Bernard Scott approached me to ask if I would want to join his group. What Bernard did not know was that my future wife Valerie Cromwell, a Fellow of Newnham College, was being offered a lectureship in history by Asa Briggs, so that his proposition had unusual attractions. Both Kendall and Hodge advised me against the risks of joining a new university, but their advice seemed to me not entirely disinterested, and was ignored. I became Reader in Mathematics and Statistics on April Fool's Day 1965, and Professor a year later.

Sussex was naturally a great contrast to both Cambridge and Oxford. In an ancient university you do what you did last year unless there are overwhelming reasons for change. (The Oxford don asked to support a measure of reform: "Reform, aren't things bad enough already?") In Sussex there were no precedents, and one always had a blank sheet of paper. It was easier to strike out in a different direction than to copy the practices of more traditional universities. As a result many good ideas were put into effect, but so were many bad ones, and the University was not always good at telling the difference.

One of the new ideas was that all students in the School of Social Sciences must take and pass a course on elementary statistics, which it was my job to deliver. In the year before I arrived the course had been given by Walter Ledermann, an algebraist who is I think the best teacher of mathematics I have encountered. Walter assured me that it was a very difficult assignment, and told me of the female student who had come to him in tears, sure that she would fail and have to leave the University. He had tried to discover her problem, and eventually she sobbed "Oh, Dr Ledermann, it is so difficult, I was away from school the day they did decimals". I did my best to temper the mathematical wind to the shorn lambs, but I doubt if I matched the Ledermann skill, or his gentle courtesy.

My main task at Sussex was to try to build up a group in statistics to complement the already lively groups of pure and applied mathematicians. We were very fortunate to attract John Scott from Oxford to lead the more applied statistics, and he proved a delightful colleague. He was to remain at Sussex, rising to be Pro-Vice-Chancellor, until he emigrated to Australia in 1977 as Vice-Chancellor of La Trobe University. We were joined by my Pembroke contemporary John Bather, and on the probability side by John Haigh and Charles Goldie.

John Fulton retired as Vice-Chancellor in 1967, and as a member of Senate I was involved in the choice of his successor. This turned out to be a foregone conclusion, since it had always been intended that Fulton would be succeeded by Asa Briggs. I was allowed to record my dissent, because I believed that the University would have benefited at that stage from a fresh look by an outsider not so intimately involved in the founding decisions. The subsequent history of Sussex does I think bear out that view, and the University has perhaps not altogether lived up to its early promise.

As it transpired, however, I was not to be directly involved in that history. The University of Oxford had at last registered the loss that it had sustained when David Kendall moved to Cambridge. Encouraged by Maurice Bartlett, who had become Professor of Biomathematics in 1967, it had agreed to create a new chair dedicated to the mathematics of probability. I did not know anything of this in 1967, or indeed until I was told that I had been elected as the first professor. Still less did I know that I only had some twelve years left as a full time mathematician, and that this would be followed by twenty years in other posts, when mathematics would become a hobby and an aid to sanity.

Meanwhile probability theory was establishing its place in the mainstream of British mathematics. I had left Cambridge too soon to be involved in the flowering of the subject there under David Kendall's enthusiastic leadership. In 1965 Rollo Davidson started his PhD, and the remarkable five years of research in stochastic analysis and geometry cut short by his tragic death. David Williams was in Cambridge from 1966 to 1972, when he left for a chair in Swansea before returning in 1985 to succeed David Kendall. And Peter Whittle's return in 1967, to the newly established Churchill Professorship of Mathematics for Operational Research, balanced the pure and applied aspects, particularly when he was joined by Frank Kelly.

A similar alliance between pure and applied probability took place in London in 1965, when Harry Reuter moved from Durham to join David Cox's already strong group at Imperial College. That same year Joe Gani came to Sheffield from Canberra, and founded both the joint school with Manchester, and the Applied Probability Trust, whose journals have established themselves as major publications on the world stage. In fact the distinction between the pure and applied approaches, between the spirit of Kolmogorov and the spirit of Bartlett, has largely disappeared. Queueing theorists talk happily of martingales and Lévy processes, and the Itô calculus forms the basis for applications from biology to mathematical finance.

Moreover, sophisticated probabilistic ideas have proved their usefulness in the development of statistical methodology, and excellent probabilists like Peter Green and Bernard Silverman have made important advances in modern statistics. In this they stand in the DGK tradition, for David's research in Cambridge was largely in problems of inference, in particular in his development of Shape Theory.

This phenomenon of confluence is a characteristic of modern mathematics, where boundaries between different areas have come to be regarded as increasingly irrelevant, and even as obstacles to the advance of the subject and its many applications. That is why I was shocked to find, when I returned to Cambridge in 2001, that the divide between DAMTP and DPMMS was as real as when I had left 36 years earlier. Most British universities had either, like Oxford, never had such an institutional apartheid, or had abandoned it in favour of a recognition of the essential unity of mathematics.

My own attitude has always been to enjoy mathematical problems both for their own sake, and because they can often be applied to a wide range of other disciplines. William Morris famously told us to "have nothing in your houses that you do not know to be useful, or believe to be beautiful". In mathematics, usefulness may take decades or even centuries to manifest itself. On the other hand, beauty or elegance can often be appreciated at once, and shared with like-minded colleagues. We should therefore invert the Morris dictum, and argue that we should have nothing in our mathematics that we do not know to be beautiful, or believe to be useful.

**A note on sources**

Detailed references would be out of place in an informal history such as this, and the reader will easily link what I have said about my own work with the bibliography on page 1. I have written at greater length about David Kendall in the *Biographical Memoirs of the Royal Society* **55** (2009), 121 - 138. Something of the story of my interest in genetics can be found in [M102]. Finally, I strongly recommend Peter Whittle's lively history of the Cambridge Statistical Laboratory, which can be found on the Laboratory's website together with a fascinating series of group photographs of the members through the years.