John Coates interviewed by Alan Macfarlane 25th February 2008

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 mankindís 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 …cole 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 …cole 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