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So. Well, Bryce, thank you very much for giving us this opportunity to discuss your life in mathematics.

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So were you always interested in mathematics as a child?

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I guess so. I mean, there was there was a strong tradition of mathematics in the family.

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I, I was born, I say, in Aberdeen and I went to Aberdeen Grammar School and, and this was in the sense the family school,

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because my father had been there and my grandfather, who was the first person to show any, as far as we know, any, any real mathematic ability.

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And he was the head of maths and science in this school.

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And that shows you how unimportant science was in those days, because science didn't even merit its own department.

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It was just part of maths and science. Anyway, he he had been he was just a farm boy.

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He'd been born in a little village in Carpathian, Perthshire.

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And and but it shows the the the the merits of the Scottish education system in those days.

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This is the 1860s because the Domini in the local school in Cape of the Domini recognised that this man had a

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certain facility for numbers and he told my great grandfather that this boy ought to go to university in Aberdeen.

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And so he went to university in Aberdeen and he got first class honours in mathematics and went on to be an assistant to the professor.

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Then and then in due course head of Maths and Science of the Aberdeen Grammar School.

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And that was all it was this tradition of mathematics.

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My father was an engineer. Another son did mathematics at Cambridge, although he he went into industry and a third son was a civil engineer.

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And I guess I just without that, I didn't think about it at all.

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But I suppose there was a an assumption that I might be some good at mathematics, but nobody ever made anything very much of it.

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And my only experience of my grandfather as a mathematician was rather humiliating one for me.

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At the beginning of the war, I was just transferring from primary school to secondary school.

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I knew no mathematics other than arithmetic. That's all I knew.

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And the, the, the schools in Aberdeen in order the government wanted to train soldiers in the,

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in the playgrounds of the schools, so they shut half the schools and they transferred and made the other schools double up.

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And as a consequence of my, my school grammar school had to go half time with another school.

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And my parents were that they were upset about this. They thought this would disrupt my schooling.

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There wasn't much they could do about it, except as far as mathematics concerned.

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They could send me to my grandfather. So they sent me to my grandfather and my grandfather was in his eighties.

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And, well, I think he no longer understood that a boy, if there were limits to what a boyfriend could do.

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And I although I had done nothing but arithmetic, he said, Well, start off with algebra.

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So the first lesson was algebra. And the first squadron out, we did linear equations.

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I mean, I wasn't at all sure what was going on, but X isn't Y's and Z's and things.

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And then he switched to quadratic equations. And this is after a quarter of an hour or so.

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And all I could he solved according to the equation.

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And then he wrote down the formula, the general formula for the solution for quadratic equations.

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And I had no idea what was going on. But I was I was very almost afraid of my grandfather, afraid to wrong what I had deep respect for my grandfather.

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And I didn't dare tell him that I couldn't understand what he was talking about.

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And so I remember going home after this first lesson, and I sat down and I read this formula again and again and again and again.

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And the next day, when I went back,

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I was enormously proud of myself because every equation he produced I could solve for him without of not understanding quite why.

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And but it taught me.

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It taught me that. It taught me that you had to think hard and work hard to get anywhere in mathematics.

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Anyway, after that, I went on to the through the grammar school and I was accelerated through the school.

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That was quite a common practice in Scottish schools in those days until I was ready to go to university when I was only 16.

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And this has happened in university and I.

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I took a first class honours degree in mathematics and natural philosophy.

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I was almost channelled into classics because the tradition in Scottish grammar

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schools was so strong on the classics that I was almost made to do Latin and Greek,

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but I escaped and did did mathematics.

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And at the end of those four years I got a scholarship from Aberdeen to go to Oxford for two years,

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and at this time I never entered my head that I was going to do research in mathematics.

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All the tradition in my family was school teaching, and I was going to be a schoolteacher.

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So if I was going to Oxford for two years, I better just take another undergraduate degree.

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There's no point in doing a research degree. And in any case,

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in those days it was still felt that Oxford and Cambridge was so much ahead of any

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other university in the UK that you could take an undergraduate degree in in Aberdeen,

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say, and still benefit from another two years of undergraduate work in Oxford.

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And, and I went down to, to Oxford to do this second undergraduate degree and I didn't regret it because there were lots of things I loved.

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And there I meant that for the my tutor at Christchurch was Theodore chanting.

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And there for the first time I met what I would describe as a real mathematical mind.

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I was I, I mean, my lectures lecturers at Oxford, at Aberdeen had been perfectly competent,

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but here was a man with a mind, which just the speed of it just, just left me standing.

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And his field was well, his field, his differential equations.

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And he he in many ways, he's most famous.

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I mean, he's kind of forgotten, which is a pity. He was he was a distinguished mathematician, a reader at Oxford.

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But the thing he was very, very famous for was the Delta operator.

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You know, the Delta operator is if X is you an independent variable, it's x, DPD X instead of DPD X.

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And this means that instead of X, this particular letter to E, to the x x x is particularly related to a power of x.

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And so if you want to solve a differential equation in powers of X, the thing to do is to understand the terms of the Delta operator.

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And then in many cases you can literally read off the solution.

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Now, this is something that this is I've never understood why?

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Why even nowadays you read books reasonably into books and differential equations and they still don't teach this.

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But this is one of this is one of John great strong points and he and he he had tremendous facility not only on that,

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of course, but in manipulating mathematics.

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And I would go for tutorials with him and I, I would sit there for an hour while he he had a very,

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a rather illegible squiggle and he would squiggle things on his on his on his manuscript.

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And I would have this pile of manuscript at the end of the hour,

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and I would go back to my room and I would spend hours trying to decipher what it was that he he'd been trying to tell me.

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And but I usually did in the end.

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And and I, you know, I just my concept of what you had to have to be the speed of mind that you had to

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have to to become a real mathematician was something that I learned from John De.

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Well, anyway,

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I not only learned that the other big difference between Oxford and Aberdeen was that the degree in Aberdeen mathematics and natural philosophy,

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the mathematics was pure mathematics, I mean, quite strictly pure mathematics.

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If, if, if applied mathematics was done at all, it was done as part of natural philosophy or physics, and it was elementary physics.

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I mean, the calculus played essentially no part. And when I came to Oxford, one thing that I did did Loughton, was I applied mathematics and and,

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and I my understanding and that direction was a great deal strengthened.

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Anyway, I got our first class to create that at Oxford and then I and I'm not quite sure how this happened,

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but I then got the offer of a Rotary Foundation fellowship from Aberdeen.

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I am not quite sure of the interplay that went on there, but.

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And this was. I could go anywhere in the world on this on this foundation fellowship to do research and presumably in mathematics.

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And where no matter how far I went, the Rotary Foundation Fellowship would cover my expenses.

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So I went to church one day and I said, Well, where should I go? And of course, the answer nowadays would be Harvard or Yale or something like that.

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But John, they said, John, they didn't didn't follow the sort of usual rules.

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John they said, well, he said if they are going to pay you to go wherever you go, you may as well go as far as possible.

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And the two further places you can go to on Australia or British Columbia.

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And I have a daughter in British Columbia, and so you better go to British Columbia.

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And so off I went to the University of British Columbia, and I had a marvellous year.

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Not that the University of British Columbia was outstanding in mathematics in those days.

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I don't think it was. But of course I had the Rotary Foundation Fellowship involved other things.

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Mean travelling around meeting Rotarians, giving talks about, not about mathematics.

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And I made some I made some very good contacts, in particular a chap called Tommy Hull,

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who we can come to later on called Tommy Hull, who gave some excellent lectures on integral equations.

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And I became very friendly with him. He had a charming wife.

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They would ask me around for a meal and that sort of thing. And I remained kept kept in contact with him for for for many, many years.

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Well, that takes us up to. I was in I was with John Day from 5252 and then in British Columbia at 253.

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And by this time at last, I had realised that I wanted to do research in mathematics.

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I mean I got a taste for it. So I came back to, to the UK to do research.

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But first of all I had to do my national service because everyone had to do that in those days.

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And, and I, I got a post as an education officer in the Royal Air Force and I taught for

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two years at the Royal Air Force Technical College in Henlow in Bedfordshire.

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And this was well, this was another it was a revelation in a different direction.

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I remember I was to lecture to them on on on elementary physics and mechanics.

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And I went in to give my first lecture, and I started off by writing down Newton's equations of motion.

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And I didn't realise it. But I saw as my lecture progressed, the eyes of this class just glazed over these one.

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These were people, these were technical officers. They they knew very, really, very little mathematics.

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And I was, without realising it, that I was just pitching things far too high, although not high as I thought it.

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But and I had to learn that you had to tailor your teaching to they to the attainment Sunday abilities and so on of of your audience.

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And that was a very good lesson to learn. And, and yet it was a good place to be to be because it was a technical college.

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And so what I was teaching was maybe in elementary terms, but I was teaching maths and physics and,

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and, and, and also again, I made some useful contacts there.

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One of the people that I got, many of the many of the people and they on the on the faculty what people like myself,

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young research students who were planning to go on to university, walk afterwards.

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And this was just an intermediary and. In particular, a chap called Byas Brown, who was a theoretical chemist.

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We became very good friends and remained friends for many years afterwards.

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And he in not actually at Henlow itself, but in some of our subsequent discussions,

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he introduced me to coagulation equations, something that I otherwise otherwise in my research would never come up across.

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Come, I can't come up against. But this is something that I did almost casually.

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And that was because of, of these contacts with bias. Browne.

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So it all was always little contacts that you didn't think are going to lead anywhere, but in the end did.

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Then you did become a research student. Yes. Now, this took us up until 55 and I finished my national service.

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And then I decided to go back to to to Oxford to to discuss with Titchmarsh who was the I mean, I my my my leanings were towards analysis.

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Titchmarsh was the civilian professor of analysis.

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And. Well, this again, was another shock.

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I. I remember the mathematical interest in those days was a very small affair.

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It stood at the corner of Keeble Road and Park Road and Titchmarsh, like the other professors,

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had a room on the first floor and his room looked across the park, showed to the Radcliffe Science Library.

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And.

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Well, the first day I went to see him, I went in and he was seated at this desk and he was looking out of the window at the Radcliffe Science Library.

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And I said Hello. And I guess he said hello. And there was another chair at the desk.

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So I sat down on that chair and then I thought, Well, no, Titchmarsh is going to tell me what to do, what, what are searching for do.

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So I sat there for about 10 minutes and Titchmarsh didn't say anything.

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So I thought, well, maybe, maybe I should say something. So I said, Hey, I had any suggestions to what I might work on.

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So there's another few minutes of silence. And then finally I said, Well, he said, I've just published a book.

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There must be something in that. And that was it.

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No, not a few minutes. And they thought, well, I better leave.

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So I left. So I got hold of the book.

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The book was I didn't have I Can Function Expansions Volume two.

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So I thought, well, I can't I can't do anything,

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but I can function expansions of volume two until I understand, I can function expansions, volume one.

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So I got all of that as well. But I hadn't done any real mathematics.

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I'll be safe in the audit for two years or thought I would rather slow reading getting into these books.

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And a week later I went back to Titchmarsh and he was sitting, looking out at the window, and I sat down beside him and he didn't say anything.

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And finally I said, Well, I said, I've got your books and I've been reading them, but it's rather slow going.

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I've only can only get through about four or five pages a day.

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And he said, That's pretty good. That's very good. And that was it.

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So I got up and I left again. And after another week, I.

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I thought I saw a problem. I thought I saw a problem like that.

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So I went back to see Titchmarsh. And once again, he was sitting in this chair looking out of the window.

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And I sat down beside him and I said, Professor, I think I've got a problem.

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Now, these are in fact expansions that deal with the equations minus double Y, double prime plus Q of x, y equals lambda y with boundary conditions.

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And you've got to talk about the eigenvalue parameter lambda, which can be a complex number, but always.

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Q Titchmarsh put on the condition that Q of X should be real.

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I thought, Well, if Lambda can be complex for my account to be complex too.

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There's a very good reason why it can't be. But I just thought that was a reasonable question.

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So I said to Titchmarsh, What happens if you make Q of X complex?

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And Titchmarsh had a bit of a silence.

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And then he said, That's a very good question, which I took it as being an invitation to, to, to start investigating this.

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What he didn't tell me. I mean, he must, must, must have known it.

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What he didn't tell me is, of course, this changes the problem completely.

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It takes it away from being a self adjoint operator with all sorts of nice properties that you can develop and so on into being a non self,

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a joint operative, which the, the theory in those days was simply, simply didn't exist and still is not completely worked out by any manner of means.

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And so I spent, I suppose, spent about ten and a half trying to discover what happened.

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When you bring a view of X complex and I made some trivial progress,

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but it and I finally allowed it, of course, that this wasn't really a very good problem to start on.

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But I learnt that not from Titchmarsh. Titchmarsh never, never mentioned self a joint or non-self a joint on anything like that.

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I learned it from some conversations with other students in the Institute.

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It was difficult actually to have conversations with students on my subject because I was.

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Titchmarsh is only research student. He never had many research students for perhaps fairly obvious reasons.

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But and. And I was the only one at that time.

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I another example of this, this was I did write a paper with Titchmarsh.

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And this again illustrates Titchmarsh. The one the one every week he gave a seminar on a Friday.

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Friday at five was the analysis seminar. Very frequently I was the only member in the audience, but occasionally, not ever.

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It came across. Not ever. It had just finished for Titchmarsh.

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The term before I started, he was no, it should have been among the staff at Shrivenham and he would come across quite regularly for the seminar.

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Anyway at the towards the conclusion how Titchmarsh around these seminars is this he he always taught himself there were no no no other speaker.

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He took the latest paper that he'd written and he simply worked through the paper on the blackboard.

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And one, one, one. When he finished one one afternoon, he said.

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Now I've proved this result about distribution of eigenvalues or something for, for certain functions.

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Q I don't know what happens if you generalise.

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Q In this way or that way. I thought, well, I think probably I can see how to do.

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So I went home and I thought about this overnight and I, I finally made some notes about how you could extend the results.

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And the next morning, I put these notes into Marcia's pigeonhole and, uh.

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Two days later, I went back and in my pigeonhole there was a complete and perfectly written up paper,

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complete distribution of eigenvalues by MacLeod and Titchmarsh.

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The whole thing. I mean, he'd taken my ideas and he'd broadened that out enormously.

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I need and Andy and I'd written this complete paper. The whole thing was finished.

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But there wasn't a lot of discussion. We hadn't discussed the problem at all.

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And so what I learned from Titchmarsh and I learned a lot.

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But what I learned from Titchmarsh, I didn't learn from his lectures and certainly not from conversations, but from his books and his papers.

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All the ideas where they are. If you read these and so on, then you've got you've got all sorts of ideas far, far, far from progress and so on.

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But that's where that's where you found them. You didn't find them in personal contact.

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And I don't want to I mean, I don't want to play it down.

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He was a very, very kindly man and and, of course, a magnificent mathematician.

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But he was not a particularly chill communicable soul, you would seem.

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Well, there I was. This is 1955.

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And I was with him until 1958 when I got my Ph.D. and at the end of the first year, the first year that I was with him,

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I had a Harmsworth senior scholarship that enabled me to live in Merton College, and that was very, very nice.

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But at the end of the first year and I got married so I had to live out and we had rooms in the city and Eunice and I well,

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that was when our first child was born in the in the years in 57.

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And then in 1958. I'd finished my Ph.D. and the question was, what happened next?

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And there was a lectureship advertised at the University of Edinburgh.

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And this seemed genius. And I to be smart.

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I mean, we we were looked forward to the idea of going back to Scotland again, to be nearer our families and so on.

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And so in 1958 we moved up to Edinburgh.

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We bought ourselves a nice house. We thought we would be here now for, well, not forever, but at any rate, for a considerable number of years.

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The Year of the International Congress in Edinburgh. That's right.

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The International Congress was in 1958 and that's the year I made the move from from Oxford to.

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And in fact, this house we bought, we bought when I was attending the Congress,

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I used some of the hours of attending the Congress to go and view houses in Edinburgh.

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Yes. And Titchmarsh. Titchmarsh was there. Also, he was giving one of the talks at the International Congress.

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So in 58 we moved to Edinburgh and had two very pleasant years and then well then things are beginning to change in Oxford.

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When I was there as an undergraduate, and even when I was there as a graduate student, the situation was that each college.

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Had at most, one more student. Some colleges said no more students at all.

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But. The number of undergraduates is beginning to increase.

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Universities are opening up. That was more money for further education, so on.

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And one or two people and in particular I must mention Jack Thompson at Wadham College.

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Two people saw that the existing situation where you had just one mass shooter forever, forever in college just couldn't last any longer.

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And in particular, that you had to have children in pure mathematics under children and applied mathematics.

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And Woodham. He persuaded Wadham to be the first college to appoint two mathematicians.

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He was an applied mathematician himself, so he felt the broad definition of a pure mathematician.

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And I had I knew Jack because I had done teaching for him.

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I mean, what I'm farmed out teaching because they had too many students for the number of tutors, and I had done as undergrad as a graduate student.

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I had done teaching for what I'm at. I suppose I must have impressed Jack to some extent with my ability in that way, anyway.

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And Borah, who was the master of Warren at that time, Barbara,

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wrote to me in Edinburgh and asked me if I would like to apply for this new pure maths fellowship in Wadham and Eunice.

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And I agonised about this for a bit. We were settled in Edinburgh, we thought.

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But in the end I decided that Titchmarsh was an Oxford.

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Surely the sensible thing was to go back. And so in 1960.

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If we move back to move back to to Oxford now with two children, because our second child Callum had been born at Edinburgh and well they're,

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we've had for now for hopefully for a goodly number of years I had this fellowship in, in what I'm involved quite a lot of teaching in those days.

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When I first went back, you had to do 14 hours a week of teaching for the college and then a couple of hours of university lecturing on top.

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But it we we were very happy and I certainly enjoyed my bath, both the teaching and, and the research, which was gaining a certain amount of momentum.

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And then the next the next sort of development.

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Took place in 1962. There was the International Congress of Mathematics, this time in Stockholm.

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And so we went to Yunus and I, and they we packed the cup, the two kids into the back of the car.

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And we drove across Denmark and Sweden to Stockholm, and we exchanged our house in,

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in Abingdon with a house in Stockholm, and we'd have almost a month there.

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But the the interesting thing was that I met up again with Tommy Hulme, whom I mentioned as somebody that had known at UBC and Tommy Hull.

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I said, no. He said, the University of Wisconsin.

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Has just opened up a new mathematics asset centre and their interest is in well very much an applied analysis.

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You are sort of interest and the man in charge of the mountain called Langer,

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who was a very eminent man in the asymptotic analysis of differential equations and so on,

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and eigenvalue parameters and so and, and he said, I'm sure that they would like to have you and,

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and that you would benefit from, from a years sabbatical there, and I'm sure I can fix it, he said.

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So I hadn't got I wasn't I was just back in but just got back to Oxford.

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When comes this letter from Langer inviting me to spend a sabbatical year in in Wisconsin?

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Well, I couldn't. I mean, there were limits.

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I'd only come to it or two to Oxford in 1960 that well, I couldn't expect to get away in 1962, but I finally persuaded the college to let me have.

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And they're very, very generous of them to give me a year sabbatical in 1964.

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And so in 1964, 65, we took off with the two kids to go to the University of Wisconsin.

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And oh, that was that that that was an eye opener in all sorts of ways.

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I mean, here, of course, in Oxford, there was Titchmarsh.

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But here you are surrounded by applied analysts, people in differential equations and so on.

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So much going on.

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And not only at a mathematical level, but also of the social level, because they they they lang out had already had, in fact, retired.

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And the the new director, Butler, also had a wife, Anita, who was Anita, who was a marvellous social organiser and involved Eunice and the family.

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And I mean the whole thing, the whole sat centre was a marvellous place to be and we spent a great year there

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and I made all sorts of contacts and I met people that I hadn't met before and.

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At the end of that June and 65,

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I came back to Oxford and settled back down and again to the job of being a college tutor and that that the mother a couple of years,

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straightforward years. And then another development because in 1967, largely at the Hutchings, I think of Nora Everett,

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who was now professor in Dundee and Vivienne Sneddon in Glasgow, and of our TLG, who was by this time the professor in Edinburgh.

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The the ICRC was being urged to do something about applied analysis and differential equations.

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And so they organised this in this international meeting in Edinburgh for two weeks to get,

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try and get people in British mathematics who weren't interested in differential equations at this stage,

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but to try and get them interested in this and the way it was organised, there were two to last for two weeks.

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The second week was to be lectures by three international stars and one was Coddington, who was to lecture on spectral theory,

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the sort of things that up until that time I had been mainly interested in the spectral theory of differential equations.

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The second was Treves, who was going to be lecturing on generalised solutions,

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and the third was it was seven who was to lecture on the navier-stokes equations.

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That was the second week. The first week was to be introductory lectures to those three given by British mathematicians.

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And I was asked to give the lectures, the introductory lectures to Coddington.

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And. I think it was David Edmunds.

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I may be wrong about this.

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I think it is David Edmunds who gave the introductory lectures for travel, and Edward Frankel who gave the introductory lectures for seven.

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Anyway, it was an inspiring two weeks, but what inspired me most wasn't so much my own area.

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But the nonlinear problems that Seven was talking about the navier-stokes equation bounded linear equations, things like that.

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And in particular, in one of his lectures, he produced a number of similarity solutions.

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I just say you take a PD like the NAVIER-STOKES and you look for solutions, for actually solutions of all these similarities solutions.

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And they are very often indications, I mean, are simpler solutions than the general solutions,

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but they are very often indications of what general solutions are going to do.

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So if you really understand this similarity solutions, you're a long way along them anyway,

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setting it up on the board five of these equations and he said he had no idea of how

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the these are how these equations could be solved or what you could do with them.

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And I, I looked at them. Some of them were more complicated than others, but the first two were just single, ordinary differential equations.

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And I thought, it can't be all that difficult to say something about these.

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So I went back to my room and I thought a little bit, and I finally decided that I could solve I mean, or say useful things about these two equations.

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And I took this decision the next morning.

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And Sam was delighted. And that that that that opened another huge can of worms, if you like, because it not only led to collaboration with seven,

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but it also, and perhaps more importantly, led to a long, long standing friendship.

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Seven was at that time visiting under the auspices of the CRC, visiting at Sussex University of Sussex.

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And so I had him come across to Oxford, give us some lectures,

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and he brought his wife Barbara and Eunice and Barbara got on like wildflower on like wild fire.

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And, and, and the two families have remained.

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Close friends didn't remain close friends until Jim's death, only just a year ago.

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And that was a tremendous a tremendous advance.

307
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And then you. Then you started visiting Wisconsin regularly.

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Right. Well, now what? Well, I should have mentioned in.

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But one of the things that happened during my year to Union Wisconsin in 1964, apartment in mathematics is that we we had twins and our two twins,

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Patrick and Erin and Bridget were born in Wisconsin and became American citizens and so on.

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So by this time, we now had four, four, four children and.

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Well, yes, the contacts with Wisconsin were strong and they were strengthened.

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Well, I think if I take things chronologically.

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The next thing that happened was that the ICRC was now.

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They're being very generous if I wanted to have people to Oxford I notable American mathematicians and so on then I could do that.

316
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And I think it was in about 69 that I invited Carter to come to Oxford.

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Now, Carter was mainly at that time I wasn't a linear mathematician.

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I mean, I invited him because he was interested in spectral theory and things like that and.

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He came and. Just before he came in there was the the applied mathematicians in Oxford.

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I had a little, little difficulties with them because they weren't much interested in proving things, and I did want to prove things,

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but they were very active and they had had a they had had a workshop on,

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on applications of, of, of of of mathematics, particularly to industrial problems.

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And in one problem that had a that came up was the problem of the then what's the

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name of the what you put on top of a of an electric car to touch the rails up above.

325
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Yes. They tell you is not a cantilever, but the Pentagon.

326
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The Pentagon. Yes. And they had a problem which had come up from British Rail, I guess, about this a graph.

327
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And it involved an on a sort of this kind of difference delay equation slightly.

328
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And they they they wanted me to try and say what the solutions of this equation looked like.

329
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And it was it was nothing I'd ever worked on before, but it sounded interesting.

330
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And I took it away, took it away and thought about it. And just at that time, Khatam arrived.

331
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So I mentioned it to Cato, and so we worked on that together.

332
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And that again was a very successful collaboration and we hadn't quite finished it.

333
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When he went back to California. And I finally walked out how the last of it could be done.

334
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And I. I sent it to Carter so the paper could be written up and published.

335
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And I. One of my proudest moments was when I got a letter back from Carter.

336
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I mean, Carter was a giant in mathematics.

337
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I got a letter back from him saying I did write.

338
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Thank you very much for your for your letter. How on earth did you think of that?

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And, I mean, I. I was immensely proud that Carter should think that I had thought of something that was not entirely obvious, so to speak.

340
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Anyway, in this way, my contacts with. With America increased.

341
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And then in 1970, 1970, I had my second sabbatical.

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And I went back, of course, to Wisconsin with the family, and we were greeted like a long lost friends by all Annette Rosser and all.

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And once again, we slotted in like, like so, so easily.

344
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And the. I made more contacts now.

345
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Now going more into the non-linear reactions.

346
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I mean, now I know that now that Saturn had opened my eyes to the possibilities of nonlinear mathematics, more of my work was becoming non-linear.

347
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And I was going and I met met people who like Paul Fife and Avner Friedman, who were nonlinear mathematicians.

348
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I made contacts with these people and then, well, that lasted for a year,

349
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came back to Oxford and I guess 72 and things began to ease in Oxford a little bit because I was given a university

350
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lectureship and this meant that I could cut my college teaching almost in half at the cost of doing a little more lecturing.

351
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But not, not, not, not a great deal. And this gave me more time for research.

352
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And and I was able to have people like in the in the in the in the seventies have people like like Paul Fife and and

353
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I have not the CRC would fund visits by these people and I was able to collaborate with them in Oxford during the,

354
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during the year and then in the delightfully long sabbatical.

355
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The long. The vacations?

356
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Yes, vacations and the delightfully long vacation.

357
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So Doctor gave me, I could I could go back to the States and where I went very often was to Wisconsin.

358
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But they were they said they they invited me. They said they would like me to come regularly every summer.

359
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And throughout much of the seventies. Then I would spend my summers in Wisconsin.

360
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And as this became a regular fixture, I thought, Well, I don't want to go renting a new apartment.

361
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Every time I go, why don't I why don't I buy myself a mortar caravan?

362
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Because, of course, motor caravans in America in those days were, well, injurious things.

363
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I mean, they are more luxurious here now, too, but they were very luxurious.

364
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And you could live in one of these a single person live in one of these in great luxury.

365
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And then, of course, the family came out to I mean, once soon as the schools were out, the family would come across as well.

366
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And we all got into the motor caravan and we would go off for a tour of the states out to the west or down to the south of every that might be,

367
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and doing doing mathematics. Doing mathematics.

368
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En Route II, we went to the Grand Canyon one year and I remember and of course Paul Fife was down there in Arizona and so able to.

369
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So it was a it was a blissful combination of of all by vacation and and and and mathematics.

370
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And. This went on. Through the more or less through the seven days.

371
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I think until. Yes, I think until almost in the seventies.

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And then and then the next sabbatical I took, which was Amendment 79 or 78, something like that, I spent didn't spend that actually in Wisconsin.

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I spent that in in Minnesota with Jim Saturn.

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And on my contacts, my general contacts were widening.

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One person whom I and that in that particular year whom I, I had met him before, indeed he had visited Oxford.

376
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But in that particular year we collaborated in,

377
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in a paper on a panel of a to the second panel of it and sent this and this was Stuart Hastings who who had been a student of Levinson,

378
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another of the great men in the and it is of spectral theory and differential equations

379
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on he'd been a student of Levinson's and we collaborated on this paper and once again.

380
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It wasn't. I mean, it was a nice mathematical collaboration.

381
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But the other thing and the more perhaps the more important thing was that we became very friendly there.

382
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He wasn't, actually. I was in Minnesota.

383
00:45:16,010 --> 00:45:24,580
He wasn't, actually. I don't think he was actually doing a sabbatical in Minnesota, but he came up to Minnesota for a couple of weeks, stayed with us.

384
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And once again, Eunice and and. Uh.

385
00:45:31,960 --> 00:45:37,600
Stewart Dining. Stewart's my unit, and I stuck it off very well.

386
00:45:38,020 --> 00:45:43,540
And and we remained and became even closer friends with them later on.

387
00:45:43,540 --> 00:45:55,430
But that was the start of that friendship. So that took me through to the end of the seventies.

388
00:45:57,920 --> 00:46:01,760
And well. I don't know.

389
00:46:02,180 --> 00:46:14,180
And of course, I know one of these visits I would get offers to go and permanently to an American university in Kentucky or SUNY in Buffalo and so on,

390
00:46:15,920 --> 00:46:25,730
all of which I, uh, well, I took seriously, but in the end decided against it largely.

391
00:46:26,180 --> 00:46:26,920
Not entirely.

392
00:46:26,930 --> 00:46:34,909
I mean, I largely because the children were still in school and I didn't understand that at a reasonably critical period in their schooling.

393
00:46:34,910 --> 00:46:38,690
And I didn't want to disrupt them from British school and American schools.

394
00:46:39,290 --> 00:46:42,680
And partly because I, I was very happy in Oxford.

395
00:46:44,360 --> 00:46:50,750
I mean, I wasn't getting much support from the university, the faculty board in mathematics.

396
00:46:51,440 --> 00:47:00,440
I think it has to be said that it just wasn't interested in applied analysis and I was ploughing something of an own furrow.

397
00:47:01,820 --> 00:47:05,240
But on the other hand, the CRC were very supportive.

398
00:47:05,720 --> 00:47:12,830
I could have visitors when I wanted them. Every summer I could go to the States and recharge my batteries and I enjoyed other aspects.

399
00:47:12,830 --> 00:47:17,540
I mean, I enjoyed my teaching at Oxford, I enjoyed the life and what and then so on.

400
00:47:18,170 --> 00:47:21,440
And so I wasn't unhappy.

401
00:47:23,750 --> 00:47:30,170
But there was always the feeling that Oxford wasn't really supporting this subject as it should.

402
00:47:32,590 --> 00:47:36,900
And. Well, I think the critical.

403
00:47:37,890 --> 00:47:44,800
The critical thing came. Well, everything came together about 1986, I think.

404
00:47:48,570 --> 00:47:56,399
I again, I had to add another sabbatical and I had spent part of it after I spent part of it with Ivan Friedman,

405
00:47:56,400 --> 00:48:04,770
but part of it at the University of Pittsburgh, where I had various contacts and where I had been invited.

406
00:48:07,410 --> 00:48:14,520
And just before going off on this sabbatical, there'd been a college meeting in Wadham.

407
00:48:16,650 --> 00:48:23,250
At which we were discussing future plans for the college and ten year plan, as it were.

408
00:48:23,790 --> 00:48:29,490
And then someone someone said, Well, no, what about mathematics?

409
00:48:30,270 --> 00:48:34,770
Because when MacLeod retires, we'll have to. And I thought to myself.

410
00:48:35,250 --> 00:48:40,880
MacLeod retires. And I realised that I only had about six years more to go.

411
00:48:41,670 --> 00:48:48,329
And I did not feel like retiring. And I realised while I had known for some time, I suppose,

412
00:48:48,330 --> 00:48:55,230
but I realised that if I went to the States there was no retirement age and I could go on for as long as I wanted.

413
00:48:56,160 --> 00:49:01,770
So he had all sorts of things came together. The children finished, were finishing at school.

414
00:49:02,520 --> 00:49:08,420
I had this I got a very nice offer from the University of Pittsburgh of our research professorship.

415
00:49:10,490 --> 00:49:19,880
I was going to have to leave Oxford quite soon anyway, but whether I liked it or not, I could go to talk to America and work on I Pittsburgh.

416
00:49:20,690 --> 00:49:24,829
I knew people in the. There were two universities, the University of Pittsburgh.

417
00:49:24,830 --> 00:49:27,140
I knew several people on the faculty there.

418
00:49:27,470 --> 00:49:33,200
And then there was Carnegie Mellon, which again was a hive of activity and differential equations and that sort of thing.

419
00:49:33,950 --> 00:49:37,660
And everything seemed to come together. I decided this was the time to go.

420
00:49:38,300 --> 00:49:46,640
And so we moved to the University of Pittsburgh and I, I must say I never regretted it.

421
00:49:46,880 --> 00:49:51,860
We regret 20 glorious years. We regretted it through laughter.

422
00:49:52,970 --> 00:49:58,820
Nice of you to say that. But I had 20 glorious years of testimony and.

423
00:49:58,820 --> 00:50:05,020
And I did not want to end and didn't lose my contacts in this country.

424
00:50:05,030 --> 00:50:09,510
I mean, there were many people in this country, of course, with whom I wanted to keep contact me.

425
00:50:09,800 --> 00:50:18,470
And and and I know what happened was that I would spend eight months a year in Pittsburgh and then four months back on this side of the Atlantic.

426
00:50:18,770 --> 00:50:21,950
And I tried and we didn't sell our house. We kept our house in Abington.

427
00:50:22,460 --> 00:50:26,740
And I tried as much as possible to keep my contacts alive on the side, too.

428
00:50:27,740 --> 00:50:34,250
Yes. And so that was that takes you up to sort of the end of the eighties.

429
00:50:42,190 --> 00:50:47,319
So what I was going to say was in 1992, you elected a fellow of the Royal Society.

430
00:50:47,320 --> 00:50:52,660
So we all thought that this was a wonderful thing and let the thought too late.

431
00:50:52,660 --> 00:50:58,810
But but it was a great recognition for not just your work, but what you've done for British mathematics.

432
00:50:59,250 --> 00:51:07,680
When I was a research student, you were really one of the very few major figures, and it was it was quite a small subject nationally.

433
00:51:07,690 --> 00:51:11,910
So yes, it was. Yes. Do you see changes now because the last few years you've been back.

434
00:51:12,490 --> 00:51:17,260
Oh, yes, oh, yes. Oh, yes. I mean, mathematics definitely applied.

435
00:51:17,260 --> 00:51:24,460
And also different degrees is no longer a poor relationship. Well, just look at look at you as a unit, the Oxford PD.

436
00:51:24,760 --> 00:51:31,750
I mean, that shows that shows that could never have existed never existed in the Oxford that I that I knew.

437
00:51:32,170 --> 00:51:39,579
Yes. And differential equations and applied analysis has blossomed.

438
00:51:39,580 --> 00:51:44,330
Certainly. Certainly. And.

439
00:51:45,550 --> 00:51:53,480
Uh, yes. I don't think I don't think I have any fears for the future of the subject in this country.

440
00:51:53,510 --> 00:52:02,709
No, no, not at all. So you've got a very interesting style of mathematics or the sense of that among nonlinear analysts that you are very,

441
00:52:02,710 --> 00:52:09,670
very problem oriented, that you're that you're you're not somebody who seems to be very interested in big theories or something.

442
00:52:09,670 --> 00:52:14,140
You're always interested in specific problems, very different ones and lots of different areas.

443
00:52:14,890 --> 00:52:20,130
So how do you see that's true? Well, I guess that's just the way my mind works.

444
00:52:20,140 --> 00:52:25,450
I, I, I never I could never get very interested in generalities.

445
00:52:25,450 --> 00:52:33,069
I mean, a big class of differential equations about which you, of course, you can see things,

446
00:52:33,070 --> 00:52:41,650
but somehow you can't get into the the details because the subject to general and always.

447
00:52:43,530 --> 00:52:53,129
Always. What appealed to me was a particular perhaps a particular equation when I thought I could say things which wasn't obvious and so on,

448
00:52:53,130 --> 00:52:56,550
but when I thought I could say things which would be interesting.

449
00:52:57,030 --> 00:53:01,890
I mean, and. Why was I attracted to particular problems?

450
00:53:03,060 --> 00:53:09,420
Well, just because some quirky, quirky thing about the man, perhaps the best example.

451
00:53:11,300 --> 00:53:16,280
Is the is panel of the panel of equations. Now, these are ordinary differential equations.

452
00:53:16,700 --> 00:53:21,800
They are second order, nonlinear only differential equations.

453
00:53:22,040 --> 00:53:32,180
And there doesn't seem to be any very special about them. But I was introduced to Pound Levy to about 1978 by by Stuart Hastings and Stuart Hastings.

454
00:53:33,350 --> 00:53:46,060
Well, the. I mean, if I if I sponsored a method for dealing with nonlinear differential equations, it was the shooting method you had.

455
00:53:46,450 --> 00:53:50,350
You wanted to solve a differential equation with boundary conditions at two ends.

456
00:53:51,370 --> 00:53:56,950
You had if you'd had all the conditions at one end, you'd have known exactly what the solution was.

457
00:53:57,550 --> 00:54:05,140
But because they were to two ends, you couldn't do that. And so what you had to do was you had to introduce an extra condition at the at

458
00:54:05,140 --> 00:54:10,450
one end so that you could specify the equation in terms of this extra constant,

459
00:54:10,450 --> 00:54:19,480
this extra condition, and then work with that and and argue that you could choose this extra constant source to do the right thing at the other end.

460
00:54:19,810 --> 00:54:28,210
That's the essence of a shooting method. And when you've just got one constant to play with, it's pretty straightforward on the whole.

461
00:54:28,690 --> 00:54:34,630
One of the nice things that in and I did and this goes back to these lectures in 1967,

462
00:54:35,080 --> 00:54:42,490
we looked at places where you shot with more than one parameter, and this makes the topology of the situation rather more complicated.

463
00:54:43,030 --> 00:54:50,799
And and and what we were able to do was to develop a routine for dealing with at least two coupled equations.

464
00:54:50,800 --> 00:54:53,950
So you had two parameters and so on. However.

465
00:54:55,240 --> 00:55:00,260
Taking a back to the pound levy to what Stuart?

466
00:55:00,260 --> 00:55:04,940
When Stuart first introduced it to me, it seemed to be a simple shooting shooting problem.

467
00:55:06,020 --> 00:55:11,270
Which we sold you wanted you at a certain behaviour at one end and you wanted a behaviour at the other end.

468
00:55:13,040 --> 00:55:20,120
But the strange thing was that the behaviour on the other end involved a constant, and this constant was one.

469
00:55:21,380 --> 00:55:25,220
And there was absolutely no reason why this concern should be one.

470
00:55:25,550 --> 00:55:29,150
I mean, it suggested that there were some.

471
00:55:30,260 --> 00:55:33,400
Particular solution or something that produced this gun.

472
00:55:33,410 --> 00:55:40,850
But there was nothing. There was just this constant was one. And it was discovered that it was one by numerical methods, not by analysis.

473
00:55:42,020 --> 00:55:46,130
And I was just intrigued. Why was this constant one?

474
00:55:46,580 --> 00:55:50,640
Could I prove that it was one? And. And eventually I could.

475
00:55:50,820 --> 00:56:00,540
And and that's the sort of thing that's the sort of problem that that these are the sort of problems that I and that interested me again,

476
00:56:01,620 --> 00:56:09,090
uh, with Paul Fife on working on convergence to working on, on nonlinear diffusion.

477
00:56:09,840 --> 00:56:17,579
Again, I wasn't the, I wasn't interested very much in general solutions and nonlinear diffusion,

478
00:56:17,580 --> 00:56:25,110
but similarity solutions again because from these similarity solutions you could tell they were simpler solutions,

479
00:56:25,380 --> 00:56:33,150
but you could tell a great deal about more general solutions. And and Paul of Fife and I worked on this.

480
00:56:33,510 --> 00:56:37,200
I starting, I think, with one of his visits to Oxford.

481
00:56:37,200 --> 00:56:45,929
That's where it began. And we had eventually able to prove that, prove the general solutions do converge to these similar solutions.

482
00:56:45,930 --> 00:56:51,870
But it's a question of looking at the similarity solutions rather than the more general thing.

483
00:56:52,380 --> 00:56:56,430
Yes. I don't know why. I mean, I many people would say that it's a weakness.

484
00:56:56,940 --> 00:57:00,540
What do I mean? What's the point of looking at particular problems? Look at the general ones.

485
00:57:00,540 --> 00:57:07,470
These are the one. That's what covers the whole scope. But somehow I've just my mind my mind just doesn't work that way.

486
00:57:07,920 --> 00:57:13,889
So I have I've always been very happy immersing myself in in special problems.

487
00:57:13,890 --> 00:57:19,860
Yes. So over the last few years, we've been delighted to have you as a member of the Oxford Centre for Nonlinear PDFs.

488
00:57:19,870 --> 00:57:23,100
You've been very active and and you in fact wrote a book with.

489
00:57:23,220 --> 00:57:26,730
Yes, the last two or three years.

490
00:57:26,810 --> 00:57:34,110
A good deal of my time were taking up writing a book with, with, with Stuart Hastings on well on.

491
00:57:34,380 --> 00:57:45,150
I mean, again, I think it mirrors very much what I've just been saying, looking at a number of I mean, there are a number of problems,

492
00:57:45,150 --> 00:57:52,620
special problems, hopefully, while I think important problems, but special problems and the techniques that you can bring to bear on these problems,

493
00:57:52,980 --> 00:57:59,460
of course, you hope and it's true that these techniques can be broadened and used for more general things,

494
00:57:59,970 --> 00:58:05,130
but our primary interest would be to use them to solve the problem, which we've been given.

495
00:58:05,640 --> 00:58:11,760
And once that's done, then, then, then very often I want to go off and try something different.

496
00:58:12,690 --> 00:58:14,670
Leave, leave generalisations to other people.

497
00:58:15,240 --> 00:58:26,580
So finally, if if a young person was considering spending the career in mathematical research, which what would you say to them?

498
00:58:27,180 --> 00:58:31,080
Well, I would say above all, have fun. Mathematics is fun.

499
00:58:33,720 --> 00:58:41,730
I mean, whether it's delving into the mathematics itself or talking to other mathematicians about it, it's fun.

500
00:58:42,270 --> 00:58:46,290
And and and if you can't have fun doing it, then probably you shouldn't do it.

501
00:58:48,600 --> 00:58:55,890
And, and, and that and in my coming back to your previous question, in my experience,

502
00:58:56,380 --> 00:59:05,940
that fun comes from not getting hold of one problem and spending your life digging deeper and deeper and deeper into that problem.

503
00:59:06,360 --> 00:59:12,239
It lies in keeping your mind open to what other people are doing to problems other

504
00:59:12,240 --> 00:59:20,940
people are having and and and and conversing with them and and working with them and,

505
00:59:20,940 --> 00:59:29,280
and and developing new ideas with them. I mean, I have I have to say that I've had a marvellous life.

506
00:59:30,030 --> 00:59:31,980
I would never have wanted to do anything else.

507
00:59:32,610 --> 00:59:41,580
It's so marvellous that I've been able to spend my life doing something that I love doing and getting paid for it.

508
00:59:41,980 --> 00:59:47,670
It's almost incredible. Well, thank you very much for sharing these memories and experience with us.