John Coates

0:09:07 Born Australia, about 200 miles north of Sydney, in 1945; mother's family were Irish and had come to the Manning Valley a little after the potato famine; grandfather was English, from Ilkley, and got land near the Manning River; father had been a school teacher but had to retire because of ill health and lived on part of my grandfather's property; I grew up there; my childhood was quite full of tragedy for my parents as my father had had a series of mental breakdowns so my mother had to manage the farm by herself; think that this was too much for her as she died when I was seven years old; my father had to take over our care but collapsed again six years later; this punctuated my childhood but the positive thing was that it was totally unspoilt bush with animals all around so marvellous for a little boy to grow up in; allowed complete freedom after being taught basic rules; we took it for granted; I was fortunate to be educated as were both my parents who were keen on education beyond the basic level; they were interested in literature so there were classic English and Irish books including Yeats and Synge; there was also a little public lending library where my father used to take us; have vivid memories of my mother and when she died I hardly knew my father as he had been away so much; he was tremendous after her death; only sad thing was seeing him become very ill in later life

7:02:17 I went to a very small school in a village about five miles away until I was eleven; when my brother went off to high school to Taree my father made big efforts to get me to Taree Primary School for my last year; at that time the rules of the State Education Authority were that no child could travel on a school bus that passed a public school, so that technically I could not travel to Taree some fifteen miles away; somehow my father got the local school inspector to agree; that was fortunate as the school was good and then the High School was excellent; my mother had one or two mathematics books and was probably quite interested, but my father not at all; really only in my second or third year at High School where I had an excellent teacher, Jack Gibson; my father had actually taught him French; the Scottish education system was still in force in New South Wales so we had a good curriculum; I learnt some calculus, for instance; from my childhood I was more interested in physics largely because of the natural world; at the back of my father's property there were quite high volcanic hills and as children we used to roll quite large boulders down; when I went to university I had no idea that there were professional mathematicians

11:19:11 I think I had a latent interest in music as a child but there was no possibility of learning; sport was the big thing at school; I played rugby league, there was no soccer at that time, and swam a lot in the summer, particularly in the sea; wonderful coast; later I chose to go to California thinking the coast would be like the Pacific coast of Australia, but was bitterly disappointed; mother used to have a house at Port Macquarie and I have happy early memories of it; after my mother's death spent some time living with my aunt at Cundletown; her husband had emigrated to Australia and joined the Australian Army during the First World War and been very badly wounded; he had been given a soldiers settlement farm there, but was really too ill to work it; in addition there was the problem of the Depression

16:13:20 The Irish side of my family were Protestants even though they were from southern Ireland; I can remember as a small child being upset when my mother went to play the organ in the little church and I wasn't able to sit next to her; they probably could not have survived in virgin bush without religion; that generation, who had nothing when they came, knew they would never go back, so religion was almost part of their English heritage; however, everything was changing as I grew up; the Pacific Highway was still a dirt road but the prosperity that came in the Menzies era was changing things; my mother's sister tried to bring me into the church and my brother ended up quite religious, but it was never a strong factor for me; I am not a believer in Christian religion but one of the really interesting things about number theory is that it is so full of mystery and unexplained beauty that you are conscious that there is something out there; I guess I am most attracted to Buddhism but it is because of Buddhist art and meditation, and it is thus much closer to mathematical work

21:06:03 I had hoped to go to university the year after High School but the problem was getting money; I also wanted to get away from the farm; I knew my father could not afford fees so I would have to get a scholarship; I took a summer vacation job at BHP Newcastle, a big mining company, and worked there until the leaving certificate results came out; it was a very interesting experience as they took about thirty on internships and then chose about two or three at the end to fund through university; I was not chosen but had applied for other scholarships, one to the Australian National University which had just begun to take undergraduates; I was very disappointed not to have got a Newcastle scholarship and was wondering how I would get to university when I got a telegram from Canberra inviting me for an interview; at that point A.N.U. had excellent funding; Canberra was the size of a small country town; think I was interviewed by Howard Florey; they brought down twenty people and were giving ten scholarships; luckily got one; at that time you either took a straight arts or science degree; I was going to read mathematics, physics and chemistry; thought I would probably then do physics, partly because it had a higher profile; Oliphant had a big accelerator that he had been building at A.N.U.; in my first year found that physics was not so well taught as mathematics and concentrated on the latter from then onwards; very fortunate that at that time they had just set up a Department of Mathematics in the research school and were recruiting people; very fortunate that Bernard Neumann came from Manchester and became head of the department in the Institute of Advanced Studies; his wife, Hanna Neumann, who worked with him on group theory, became the head of the undergraduate side; they also brought out another German refugee, Kurt Mahler; so in the second term of my first year he came and taught us a course on elementary number theory; it was so interesting that from then on felt I wanted to do mathematics

30:39:21 Number theory is the study of deep and hidden properties of the most basic of all mathematical objects which are the integers and the rational numbers; of course, a lot of mankind’s evolution in mathematics has kept enlarging the number systems but true that the deepest and most interesting and mysterious properties of all in mathematics are there in the integers; because it is dealing with these very basic objects it is certainly the oldest part of mathematics, its only competitor being astronomy; it is still true today that in some of the most important discoveries are being made in number theory by people looking at numerical examples; historically this is how it all started, people would look at numerical data and notice surprising things going on; it certainly started in Asia long before the Greeks, probably in India, but there was an enormous interchange of ideas throughout Asia, as we know through Buddhism; it is true that in the West we tend to think of mathematics as starting with the Greeks; they did make a great contribution, for instance it was they who emphasised the idea of an abstract proof; in number theory even today the goal is to prove these basic properties that people have noticed about integers, most of which are still unproven; the sort of mathematics I have worked on, the classic example is a problem that was studied numerically by Arab mathematician a thousand years ago, and it turns out to be a beautiful example of the modern theory; it has practically nothing to do with Greek mathematics; a lot of number theory in the popular mind has been in connection with prime numbers but I have worked more in what we call diophantic equations Diophantus of Alexandria, the study of equations that need integers; there you have these very ancient problems that we still do not know how to solve today; Swinnerton-Dyer and Birch made a wonderful conjecture linking these problems to something else, but I do believe there has been a little bit of progress on their conjecture and in the end it will be solved; I think we are all missing something rather simple in the case of this problem, but someone has to find it

35:47:01 Distribution of prime numbers still one of the great mysteries of mathematics; long known that there are infinite prime numbers but how can one know how many will occur which are less than a given large number; turns out to be of real practical importance today; we do know now that it is very roughly x/ln(x) but that is not really the full answer; you really want to know how much the number of primes up to x differs from x/ln(x); this turns out to be a problem equivalent to Riemann's hypothesis, who was the person who saw that there was a connection between this problem and the distribution of zeros of an analytic function which we call the Riemann Zeta-Function; in fact it was Gauss earlier who had made a guess which is closely related to the Riemann Hypothesis that in fact π(x) was very close to what we call the logarithmic integral function; another problem concerns prime pairs with a difference of two and whether there are infinite prime pairs; everyone believe that there is but it has never been proved

39:38:21 In my second year at university I may have dropped chemistry, certainly in the three years I took progressively more mathematics, and in the fourth year did only mathematics; I was indeed very fortunate that I got interested in number theory through the lectures of Kurt Mahler; I was able to start research in my fourth year, doing an honour's project, and he gave me an old unpublished paper of his; it was quite difficult because it was in German but it really gave me confidence; one of the real difficulties of number theory is that it is a vast subject, there is so much to learn, and the interesting problems are so hard, that as a student you have to have a lot of courage; what gave me the confidence to go on was precisely the time I spent in that year and the help given me by Kurt Mahler; I also met my wife who had arrived a year after me at A.N.U.; I also got a little bit involved in student politics; the University was tiny at that time and Robert Menzies, the Prime Minister, used to come along to the hall of residence, Bruce Hall, which he'd been instrumental in building, for meals; my wife's father was also a Member of Parliament; she was studying arts and concentrated on politics; at the end of my third year I decided that I didn't have the time to involve myself in politics

45:16:20 A.N.U. had a travelling scholarship and I got it; Kurt Mahler and Hanna Neumann (a group theorist) suggested that I go to Paris where there was a vibrant school of abstract algebraic geometry; Mahler wrote to some of the mathematicians in Paris and I was given a place as a foreign student at the École Normale Superieure for a year; it was quite a shocking experience for me in some ways; I had never lived in a big city before; I could read French fluently because of my father but I had never spoken very much, but I found I did not have the mathematical background to cope with the type of research that these people wanted one to do; I felt very frustrated and wrote to Ian Cassels in Cambridge; Mahler had spoken to him about me; told Cassels that I was out of my depth in this abstract world and asked if I could come to Cambridge; he agreed and I went there after spending a year in Paris where my wife had joined me

50:02:06 Thoughts on why the French were so good at pure mathematics; their emphasis on abstract thought is now taken for granted; again I was very fortunate because at the same time that all this major abstract work was going on in Paris there had been this great number theoretic discovery made here in Cambridge by Birch and Swinnerton-Dyer; it happened in a very different way as this discovery was made in no sense by big abstract machinery as was going on in Paris but by numerical experiments on the first computers; I went back to France later on and taught in Paris for eight years; one of the reasons why French pure mathematics has been so good has been the École Normale Superieure which undoubtedly has been a great training ground, particularly since the Second World War

53:21:14 We arrived in Cambridge in mid-summer and I was a student at Trinity; they gave us a nice flat above 'The Blue Boar' until October but then we had a delightful flat in Westminster College, above the Master's Lodge; Mrs Burkill ran the Society for Visiting Scholars in Botolph Lane who found us that flat; as I only had two years of funding left decided that the only practical thing for me to do was abandon the work I had been trying to do in Paris and write a Ph.D. thesis much more closely related to work I had done under Kurt Mahler; I didn't have much time but am infinitely grateful for the nice atmosphere in the Department then; my supervisor was Alan Baker though I was working on algebraic number theory and was already interested in the Birch and Swinnerton-Dyer conjecture and I attended Swinnerton-Dyer's lectures; he was the Dean of Trinity and he had the room over the gate at the back of the college; every afternoon any student could go there and have a drink with him; I used to go about once a week which I much appreciated; at that time he was doing a lot of mathematics and did work that was subsequently very important, but he was also getting heavily involved in University administration; it is true that in terms of real mathematical work that three or four hours a day is the maximum; but there are all the chores of the academic world which occupy time; in my case I often found that running would wake me up though now I am too tired to do so
Second Part

0:09:07 In 1969 remember that a group of us put our money into buying wine in bulk for Christmas and then the whole operation collapsed, it was a scam; somehow Swinnerton-Dyer got word of this and to our absolute amazement he handed us a cheque for the whole sum; a very charming man; previous holder of Sadleirian Chair was G.H. Hardy who was at Trinity, a number theorist; he and Littlewood were great mathematicians but against the great German school of number theory, they were in the same class and created the subject of number theory in England; the German school destroyed by Hitler though a great many went to the U.S.; Hardy's discovery, Srinivasa Ramanujan, was a number theorist and a remarkable mathematician; he and Mordell, another Sadleirian Professor, were the opposite of Hardy and Littlewood and the German school in that they made really great discoveries without using so much mathematical machinery; Mordell proved his celebrated theorem, which came out of Fermat's work, which is central to the study of diophantine equations and the conjecture of Birch and Swinnerton-Dyer is about certain aspects of that; I have spent my life working on that

5:32:04 In Ramanujan's case it was probably his life and education which gave a religious aspect to his work; it is true that there are occasions where you get an idea that is very important for your work that seem to come out of the blue; if you are a mathematician trying to solve a problem there us a whole body of theory that you have been through, and when you can't solve it there are certain inconsistencies; it is true that by taking a new idea or another way of looking at the problem that it suddenly does fit together; you could not have this experience if you had not spent time thinking of all the detailed mathematics round the problem; in my case it has often been chance remarks of other mathematicians that have been important for me; I tend to run in the evening and often towards the end of the run an idea comes, some lemma or inconsistency I have been worried about is resolved; it is very individual, like all creative work

11:15:50 I was lucky I was able to work in parts of number theory where you were able to break new ground but the key ideas I had when I was very young, within four or five years of my Ph.D.; think it is much harder today as so many people are now working on number theory so there is a huge body of theory for people to learn; there has certainly been an argument going on for a long time whether we should give the Fields Medal to someone under forty or whether to raise the age, precisely because the subject was getting so big; I still think that if someone is going to do really original work in mathematics some of it will emerge within four or five years of a Ph.D.; after my Ph.D. I wanted to go back to some of these more algebraic problems like the conjecture of Birch and Swinnerton-Dyer; a conjecture is a number theoretic fact or even a general fact that we think is always true but we cannot prove; number theory is full of conjectures, but just the discovery of this conjecture changed the whole of a large part of number theory; I came across the person who had a big influence in a concrete mathematical way in my life; I was looking for a job and applied for Cambridge research fellowships and got none; I knew John Tate of Harvard who was a leading proponent of algebraic number theory in the U.S.; he did come to Cambridge in my last year and gave some lectures; I applied for a post-doctoral position at Harvard and was offered a job; that was the other great stroke of luck of my early career; we moved to Boston after certain visa difficulties; fortunately my father-in-law was able to intervene as he knew the Australian Ambassador who spoke to the American Ambassador and somehow they were able to intervene and arranged to waive visa restrictions; fortunate as we were expecting our first child; had three wonderful years there; Tate was a great teacher and he started running a seminar on an interesting conjecture he had made which was simpler that the Birch and Swinnerton-Dyer conjecture but clearly related to it, in fact he had made it with Birch; he gave this tremendous seminar during which he made a chance remark on which I have spent the rest of my life more or less developing

20:28:21 In algebraic number theory there were two cases which have gone hand in hand throughout the twentieth century; one of them is much easier than the other, it is more geometric, and he found a proof of this conjecture; his chance remark was that a certain Japanese mathematician, Kenkichi Iwasawa, at Princeton had found an analogue of the thing he was using in the number field case and that it might be worth looking at; Iwasawa had already done quite a lot of work in that direction but he was only then publishing it; within two or three days of Tate's chance remark it was obvious that if you looked at Iwasawa's work there were some things there that the proof could go on; Iwasawa was very helpful in the sense that though he hadn't published too much but he sent me his lecture notes; from then on I've spent the rest of my life developing these ideas; I was fortunate that another student of Tate's, Steve Lichtenbaum, who was by then at Cornell, quite independently had the same idea as I did; Tate introduced us so we were able to start working together; with regard to working with Japanese, language is quite irrelevant in mathematics; what really matters is if a person has grown up in this environment as number theory is highly evolved, using a lot of machinery, really everything that grew out of this great German school of mathematics; thus the Chinese were at a tremendous disadvantage; the Japanese were very lucky because in the Meiji era they sent one of their young people to Germany to study and he went back and created this great school of number theory that has flourished ever since; Tokyo University was the centre until the Second World War; Iwasawa was a product of this school and was an assistant professor there during the war which was probably why he was not in the army; by the end of the war he was very ill, probably through malnutrition; like so many other Japanese mathematicians of that age group he ended up in the U.S. and stayed there until he retired back to Tokyo; the person I have worked closest with is in Kyoto, but there is a whole school now spread out over the whole country

26:52:16 In 1976, through Iwasawa, I was invited to a conference in Kyoto at the peak of springtime; it swept me off my feet; Kyoto at that time was very unspoilt and from that time I became interested in Japanese art and literature; I had read people like Proust earlier but I came to Japanese literature 1978-9 and discovered Waley's translations and it opens whole new worlds; read Waley's translations of the tankas of the Heian period and the only sadness is that it has spoilt my appreciation of Western poetry; it was only when I finally came back to Cambridge that I met a shop owner who was able to get hold of Japanese ceramics; once you see a few of the really good pieces you realize that it is great art and want to get hold of it; England is unique in that way, it would not be possible to make collections of early Japanese ceramics in almost any other country unless you were very wealthy; it is unknown why ceramics of the period 1650 to 1750 should have arrived here; it is mysterious as there was no official trade; some came through the Dutch but I'm not sure I am convinced; I suppose the East India Company was buying some in China and that there was trade from Nagasaki to China; there has to be something like that as there are many types of early Japanese porcelain that you find in England that you do not find in Europe; it is true that in mathematics there is intellectual beauty that appeals to one; but there is intellectual beauty in other things such as Waley's translation of 'The Genji' and I certainly think that a lot of the early high quality Japanese porcelain gives one a sense of intellectual beauty too; that is part of the charm of Oriental porcelain that somewhere in it is a real intellectual thought that the artist has put into it

34:07:14 After three years at Harvard I looked around for a permanent job and was offered one at Stanford; in many ways the Department was very kind to me but the problem was there was no one in my subject; as you get older in mathematics you feel less need for having really close collaborations; at that time it influenced me and I felt unsettled at Stamford; another factor was that it was very expensive to buy a house and I had a small family; at that time I had an offer of a university lectureship in Cambridge; I had just got a Sloan Fellowship and had gone to Paris; my wife wanted to spend a year in Paris but we came to Cambridge; difficult two years as I had my best mathematical ideas then but I had taken on a college teaching post at Emmanuel and had a heavy lecturing load in the Department; found I did not have time to write down my ideas; in the end I left after three years; my best research student there was Andrew Wiles; Peter Swinnerton-Dyer had taught Andrew for Part III of the tripos but was heavily involved with university administration so he asked me to take him on as a research student; it was perfect for me as Andrew was very talented; we had a conference in Durham in the early summer and Iwasawa was there and as soon as we got back I said that now we were going to apply these ideas to the Birch Swinnerton-Dyer conjecture; he was very bright and almost immediately began to make some progress and then we really were able to collaborate fully; by the time I went to the Japan conference we announced some results that we were sure we had proven but the first real theoretical result in the direction of Birch Swinnerton-Dyer happened after we returned; I was only here for one more year when we worked further in this direction but I became totally fed up with having too much teaching so I went back to Canberra for a little over a year, after which I accepted a job in Paris and Andrew went off to Harvard; he then started on another aspect of Iwasawa's work which he was able to extend by using some ideas of Barry Mazur's; many people were working on aspects of Fermat's Last Theorem but Andrew saw that you could really prove the whole thing completely where everyone else was looking at the parts; seven or eight years later when I had come back to Cambridge from Paris, Mazur came to give a lecture and I was working with another research student on a related issue; Mazur said in his lecture if only you could do this type of thing, but Wiles did just that; lottery of the Fields Medal

44:09:06 I was in Paris for eight years as a professor in the University of Paris and had expected to stay there; one day I got a call from one of my colleagues who asked me if I'd be interested in the Sadleirian Chair; Ian Cassels had taken early retirement; thought about it but could not resist it in the end; there is a packet of papers that every Sadleirian Professor hands on to his successor, in which is part of the will of Lady Sadler who left the money to the University in 1710; it was not a professorship initially but paid for the lectures in mathematics, in algebra and geometry; she gave such detailed instructions and had such technical knowledge of mathematics in the will that someone who knew how mathematics was going in the world and how it should be taught at Cambridge had advised her; we do not know who that was; the only connection with the academic world was that her first husband was a man called Croon who was a fellow of Emmanuel College and was one of the founding fathers of the Royal Society; they still have a Croonian Lecture; it is a big mystery why Cambridge in the 1620's started to teach mathematics systematically; exactly the opposite happened in the University of Paris, for example; as far as I know the first people to teach mathematics systematically were the Jesuits in Rome, and Ricci, who went to China, was basically taught mathematics; I am told there are detailed programmes of what the Jesuits taught in the Vatican; even politically it is a very curious thing why the University should have begun to favour mathematics; for a long time it was the only examinable subject, and certainly people like Newton grew out of this school; the first products were people like John Wallis; the Sadleirian donation is somehow tied up with that, but it is a big mystery; Cambridge is a good place to do mathematics; it is remarkable that we can still teach it in the same systematic way we always have; one of the problems that struck me when I came back was the tendency to be cut off from the world, but that has changed; we have solved the physical problems because before the 1960's mathematics had a couple of offices in the Arts School; the quality of students we get is outstanding; Part III has gone from strength to strength in the last fifteen years; the number of foreign student has gone up and up; the numbers from the Far East are growing but there are still relatively few scholarships for students from China

51:40:10 The college system is fundamental to our teaching; we could not teach to the same level that we do for such a large number of student without it; we probably rely on college teaching as much as any subject, perhaps more; what students get out of Cambridge when compared to even top American universities is the contact with a supervisor or director of studies; it is fundamental in ironing out misunderstandings so think it is an absolute cornerstone of our teaching; lectures cover the syllabus; it is a pleasure to lecture to such bright students and I enjoy it; valuable also to be able to systematically going through a proof or argument when lecturing, even in parts of mathematics that are well known; I enjoy the college system and have been genuinely interested in things outside my subject; as an undergraduate teaching institution I think we can claim to be as good as anywhere else in terms of producing original thinkers; at the research level in some sense we have not had the resources until recently to hire the most outstanding people at a permanent level but that is changing; one thing that I am very proud to have pushed for is that we have now adopted the American system that all of our tenured faculty teach exactly the same amount; mathematics consumes long hours but the rewards are great, and the world needs mathematicians.