Hello. I'm Martin Brighton and I'm here today to interview Nigel Hitchen.
After a long and illustrious career here at Oxford is just retiring as the civilian professor of geometry.
So, Nigel, shall we begin with John to tell us a bit about your early life and how you decided to study mathematics?
So at what point did you decide you might want to study mathematics?
Okay. So well, I grew up in a small village called Duffield near Derby.
As a matter of fact, it's the same same village where one of our colleagues, Hilary Ockendon, from her father, was a family doctor and in 1957.
And so that was when I was 11. So there was actually a brand new grammar school built in Duffield and I.
I started there along with about 73 other students.
So students, they call students nowadays pupils.
So this was like two classes. And of course, obviously with a small number like that, the school didn't have a full complement of teachers.
So the mathematics actually was initially taught by a variety of teachers.
It was a French teacher. It wasn't very good, really, but even worse was the PE teacher.
You know, he kind of got as calculating the the stagger, the staggered distance for a fool for her.
But eventually, as the school grew, after a couple of years, they hired more teachers, more specialist teachers.
And there was one teacher who actually took an interest, a serious interest in mathematics.
And in particular, you know, when when pupils responded positively to what he was telling us, he took an interest in them, too.
So he probably was was an influence on me, although, you know, there are lots of things which I quite liked at school,
some things I didn't my but but the reasonably well in chemistry for that I never liked chemistry was too much too much to learn and so can you.
What age do you think mathematics might be? What? I'll study at university, for example.
Was that clear to you when you're 15 or 16? Not really.
I mean I mean, I suppose in my younger years, I was I was quite influenced by my older brother.
I used to do all the things that I want to do or things that he so he was he was always building things with meccano.
He built a kind of an automatic transmission or something.
I could never do that, but I know he was always kind of following on what he was he was doing.
And for a long time I thought that I wanted to be an engineer, engineer.
And of course, the mixture of subjects which I chose to do it at school reflected that.
But gradually in the sixth form, I guess I realised that actually I really probably wouldn't be a very good engineer and somehow the,
the mathematics came or moved to the fore and was a geometry.
You did you sort of fall in love with geometry at some point.
Do you remember all those Greek proofs that we used to do in school?
Well, I remember them, but I don't think it was. That wasn't my favourite part of mathematics.
No, no. I think I was quite, had quite a broad appreciation of mathematics and pure and applied mathematics and ah actually the headmaster was,
he was a mathematician as well,
but he taught us applied mathematics, but it was quite an old fashioned, you know, he was going back to Cambridge trifles questions 1906 Okay.
But yeah, finally, actually what also happened was that I was quite ill for a month or so and I dropped
one of the A-level subjects that I was going to take really focussed on mathematics.
So at that point I really thought that mathematics was going to be my, my what I would do at university.
Right. And so how did how did you end up at Jesus College?
So why Oxford? My Jesus. Yeah. Okay. So then we have to talk about the headmaster, Mr. Redfearn.
This day was so he had so he he tried to create this this school.
It is it was a new school,
and he wanted to create it in this kind of I would imagine there are a lot of things which he did which were a little bit controversial.
I mean, he was kind of modelled on public schools and so on because so we weren't allowed to play football, we had to play rugby.
On the other hand, this was good for the surrounding schools because. They didn't have reputations, but they created them in a hard place.
But one in particular, he in the 1930s, he'd actually been offered a place to study mathematics at Jesus College,
but had not been able to afford to take that. And so somehow vicariously I was I was appointed an ambition actress.
He didn't have any particular links with the college, but it was just that that he wanted, you know, a mathematician to go.
I wasn't actually the first mathematician from the school to come to Oxford because I stayed
on for the extra time to take the entrance examination on the scholarship examinations,
all those things. Whereas one of my peers, she went directly, in fact, to Somerville.
Okay, that. So do you have clear memories of your interview from.
Yeah, well, I had two interviews because, I mean, they actually offered me a place in after the interview in October.
Uh huh. But then they encouraged me to take the information.
Mm hmm. Um. Yeah, so. Well, this was more or less like the interviews we do nowadays.
We, you know, pick up maybe on the topic in the entrance exam, which are not particularly interesting to me.
Mm hmm. I remember the problem was.
Was about estimating the the size of a ball of why I think you take a length and why you screw it up and you say, what is what is it?
What a good estimate of the the radius. It's extraordinary the way that these moments of life stay with you at the detail of that interview
is that's what I say you you make a crude one and then then you kind of have these approximations.
So when you arrived at Oxford, was it what you anticipated to do?
Did you find it immediately stimulating or a bit off putting?
Uh, you know, whenever you come into a new environment, you know, that, okay, maybe you were one of the best at your school in mathematics.
Of course, all the others feel the same way. You know, it's always the question of, you know, measuring yourself against the others.
And so I was a bit nervous at first about that and.
But so the teachings of Jesus at the time, whether it was Edward Thompson and Christopher Bradley,
and they had very different ways of approaching things, I think.
The Bradley was, I think, quite influential in that they encouraged us to well, to read up on something which,
you know, we could choose and give a talk a half hour and talk about it to the other students.
And so and that was that was good because, you know, we'd go to the library and look through the books and find what actually,
I think I gave a talk on a chapter in the home and Violence classical groups.
Oh, right. Kind of sheltered what it's about and the identities, I don't know.
Now, after all these years where I learned about lead groups and so forth, I still don't know what the company idea has.
But in those days I think I thought that I did.
So you weren't obliged to stick too closely to the syllabus during these college tutorials?
Oh, really? I mean, Edward Thompson, you would I mean,
you would go there in this cloud of pipe smoke and you'd sit down and and he would
sit there waiting for you to come up with problems which discuss had discussed.
Like, uh, Christopher Bradley was different.
So he, he actually would, you know, deal with applied mathematical problems.
But he always had a very clean approach, even though, you know, he taught us about Cartesian tendencies and using matrix methods.
It was a kind of a modern approach to, I would say, clean approach to doing applied mathematics,
which I think actually was influential and subsequent.
Right. And so who your contemporaries, did any of this go on to be?
Mathematicians. Yes. So so Lynn Thomas, who unfortunately died earlier this year.
So he was he became well, he moved from mathematics, operational research and business school,
but he became professor and in Edinburgh and in Southampton,
as Gareth Jones, who was a group theorist, and he was professor at Southampton and the others.
And I see. Well those two in particular. Yeah.
And with their proclivities for the directions they followed, did these emerge early on?
Were you obviously the John Ritter, for example, as there?
Yeah, I think I was, actually. I mean, we didn't get.
There wasn't much geometry in the course. And I that and I used to go down to the college library in the books.
And there was there was one book I remember in particular this was by Flanders.
Flanders, it was called The Differential Forms in the Physical Sciences.
And that was I was always attracted to that even though it was,
it was a non rigorous book, so it didn't actually tell you what the differential form was.
And that worried me a little bit as I was, you know, I went to the when you come to university, you I know that you starting from scratch, right?
Things have to be built up logically. Mm hmm. And he introduced these objects.
I mean, nowadays, you know, I'm more flexible if you have to be more research, more flexible in your interpretation.
But that was that was a book which I was thinking of going down on looking.
But also as a researcher, you sort of have you have a better grasp somehow of the reality of the objects you're wrestling with.
I think when you're an undergraduate, as you say, you like these clear definitions.
Yes, that's right. Yeah. I mean, I remember, you know, getting asked somebody, you know, what is the vector space?
Yeah. You know, you jump over these hurdles and everything becomes more natural.
And so so who were the dominant figures in Oxford Mathematics at that time?
And so was t influence already very strong. Yeah, well, the first year, my months year, they decided to put the top brass to give the month lectures.
So yeah. So Michael Taylor was giving the algebra course, Charles Coulson was giving the applied mathematics course.
And then they had Hammersley giving the analysis which cause some problems because his, his problem sheets were very, very difficult.
So tutors were really engaged with each other up asking, How do you do this?
Okay, now some start questions, but even the ones that I had never heard and as you might expect, McAteer went very quickly.
It was very good, very quick. So we started we sat there and by the end of the term, he was talking about Galois groups that I think.
They dropped that system, but it was clearly influenced.
I mean, I enjoyed it, but it caused it cause problems.
Yeah. But then as as your undergraduate career developed and did,
was it the lecture courses that that had a particular influence on you or you or your
peers or and how did you sort of settle on the direction for the next part of your career?
Yeah. Um, I think there were things which I, I did like and things which I didn't like.
I mean, probability and statistics I didn't like at all.
So I avoided that. I things like fluid dynamics, like playing around a bit.
But then I gave up. Eventually I focussed on more geometrical subjects and biology and there was,
there was a course on algebraic geometry with a very foundation course as well, commutative algebra, some function analysis.
They were kind of, you know, tutors gave you advice.
In my final year, Brian Steer used to give me tutorials in topology and differential geometry.
Mm hmm. Mm hmm. And so gradually I settled on these things I had always been interested in.
In popular reading about topology, sheet geometry.
I was right, you know, before I came to university, I was quite keen on that.
And then when you got points at Topology, it was a bit of a let down, right?
Yeah, right. On the other hand then when it came to doing algebraic topology,
so it was Graeme SIEGEL who gave that across there that when they started talking about,
you know, syntheses and so on, I thought, Oh, this is kind of combinatorial.
This is not a problem to that. But then when he started talking about the nerve of recovering the managing and things that I knew right.
And together much better. So it's actually the way in which these things are presented, I think.
Is this Frenchman guiding you through this?
And so later in your career, I think of your influence on physics, but also physics,
influence on the sort of geometry and the physics influence the sort of mathematics you've done.
Was that always true? Is it true? Then, for example, you sort of you found that motivating?
Oh, really? I never I never took any physics course any well, I didn't take any quantum theory courses or relativity courses.
I guess I was interested in cosmology in general relativity, but largely because of it was using the same formalism as the differential geometry.
Uh huh. But I didn't. I never got the hang of, you know, like curves and things.
I mean, it was remaining in geometry that was that.
I had a, I had a feel for that. Aha.
So, so although I had a passing interest in relativity and I used to buy one or two textbooks on it and, and I didn't take any courses on it.
So in fact, um, I basically avoided the physics related courses.
Okay. Oh, that's interesting. And what I so what you're up to versus so I 65 to 60, 68 was when I took my file.
Okay. So that's what so was the index then was what, 63.
So was that very much in the air. Did you have an appreciation of that?
Yeah. Okay. So that was when I began as a research student then.
Yeah. So I. Index then was definitely in the air.
And so brain stem is my supervisor.
Mm hmm. And in my first year, he is trying to push it towards various topics.
Well, first of all, he tried to get me interested in dynamical systems, because this was the time of the snail and all this stuff, right?
Mm hmm. And I think there was a kind of preference in giving and giving research grants to, uh, in this area in SLC.
I think it was a good thing it was doing something like that. I read a few books on that, but I didn't.
You weren't drawn to it, I think, by the flavour of the subject.
But I learned a bit about smell, horse shoes and things.
So then he pointed me. Then I said, No, this wasn't pointed in another direction, which was to do with k theory of the groups.
And that was that. I didn't really get very far with that.
Mm hmm. But still, I was learning k theory, and then I got in touch with the index there, and.
And finally I started working on the Dirac operator where course there was this important application of the index there.
But the. I had just spent months. Mm hmm.
And it was at that point that that yes, I started to read read the papers on index panels, papers, etc.
And so so as you as you said, you you started with Brian Steer, but with the intention always that you would talk to Mike later as well.
Or is that as a result of this evolution of interests? No, it was because Michael had actually left for Princeton.
So he was a permanent member of the institute in Princeton. So maybe after the first year I can't remember which year it was,
but he always used to come back in the summer for the summer term because in Princeton, the finished paper, I guess.
Mm hmm. Um, so, I mean, obviously, I always felt that it would be nice to, you know, to actually work under my out there.
But then the opportunity came when Brian went on sabbatical for one term.
Mm hmm. And so. And so Michael took me on as a student for that.
That period of that summer term. Mm hmm. And then he pointed me in lots of different directions, gave me ideas and so forth.
It was really. Yeah. Which in my research. Mm hmm.
Yeah. That kind of gets to go back to when you were doing these things that did didn't resonate so much with you.
Was there a point at which you got disheartened at all and you thought, maybe maths isn't for me or so?
Sure. Um. In fact, before I came to Oxford as an undergraduate because I'd take the entrance exam and,
you know, I was admitted in, you know, I was accepted in December.
I got a job for the next nine, eight months with Rolls-Royce and Derby in that computer engineering computer department.
Mm hmm. And that was that was interesting while I was earning money 7000 shillings a week.
But there were some interesting problems there. But I went there kind of not knowing much, and I didn't even know what matrices were.
And so they were always talking about stiffness, matrices and engine design and so forth.
And so actually I picked up some mathematics there and and I worked there maybe two vacations.
And so they don't have always said that if, you know, if I wanted to there have always been a job for me.
And so, you know, it's true that halfway through my first year when I felt that I wasn't getting anywhere, I thought, well, should I go back?
And it was really only when I had the task of writing the what we now call the transfer dissertation
that I really got engaged with something which I really don't and encouraged me to stay.
Mm hmm. So jumping ahead. So. So when you've written your thesis.
So what next? What were your thoughts at that stage?
Um, okay, so I actually, I haven't quite finished my thesis by the end of the three, three years.
I mean, I hadn't written it up. I got various results.
But then, then Michael invited me to be his assistant in the institute, so he was back permanently by that stage.
Well, he was still he was still permanent. Professor at Princeton, I mean, permanent member of the.
Aha. Mm hmm. So that was great for me.
I was wonderful that, you know, I had something to go to.
You know, I haven't fully finished like this, right? Mm hmm. So, actually, it was in my first year in.
Princeton that I finished the thesis and that, okay, that was very good.
And well, the institute just being in the States was a wonderful experience, you know, because well,
you're surrounded on the one hand by postdocs are in a similar situation to you,
but then there are visitors that have the time to answer your questions as well.
And it doesn't matter if they're fairly naive questions, and then there are the big shots as well.
And then also, you know, I will get some invitations to give talks in various universities.
I found that people were interested in whatever I was doing.
So actually, yeah, I actually finished off the thesis in. Okay.
Yeah. And how long did you spend in Princeton at that time?
So that was two years. So that I the my stay there as a teacher's assistant went on for another year.
Mm hmm. And did you change direction at that time, or is it more just a flowering of things you were already interested in?
So, no, I think I was still working on that.
Well, okay, I. It expanded my horizons in various ways, so.
So, yeah. Was there one of those? Yeah. Okay.
And so we used to talk a lot about, for example, the Columbia conjecture was okay and the implications it had.
And so I was I got interested in, in particular in K3 surfaces and they had Einstein metrics.
Yeah. I suppose it was then that, that although there were still problems, issues related to the Drake equation,
then more general differential geometric problems relating to Einstein matching scalar curvature came into play and we said that.
So your first brush with algebraic geometry as well would you already know?
So, so. Right. So my thesis I started out by looking at Raymond Surfaces which so my introduction to how to break geometry
was really through studying Raymond Surfaces and then by so by having Michael materials supervisor,
he was encouraged me to look at algebraic surfaces moving up.
So but it was, you know, I, I learned stuff about algebraic geometry well, in different ways, actually.
One was by having a particular problem, namely how to interpret harmonic spinners on a came a manifold but also actually
there was a yeah and in Warwick there was a special year in algebraic geometry.
David Manfred was in Warwick and so they used to have days of seminars in Warwick and then a bunch of us graduate students used to,
we used to drive up there and back to actually not just to Brian stage used to come sometimes and we would spend the day and learn.
I learned a bit of those visits as well and so obviously being there, talking to other visitors program.
So I think so. So early on you had this this famous paper with a tier and singer.
How did that come about or what stage did that work take place?
Yes. So that was in 19, I guess, 1977.
Yeah. So I came back from the States, so and I had a year in NYU and then I came back to the States as a postdoc working with well,
in principle working with Michael.
But at that time I was really following my own interests occasionally and doing something with the Eastern Band which related to his interest.
So at the time, the Aether Invent the heat equation approach to the index theorem was, was and in the work at Bertone it was very a current.
Um, and so I was playing around with one thing another little bit with solid times,
trying to understand the geometry behind some equations and not doing it to be honest, not doing very much nowadays.
I mean, I know I should. I should have been producing papers. Well, yeah, that's one view.
Anyway, I didn't see this impression that I have right know.
Looking back on it, I realised that, you know, I should have actually written more.
But then. Then Singer came across from M.I.T.
And he came with this problem about instant funds, which is completely new to us here.
I mean, on the one hand, it was about things that we knew about, about money, principal bundles and connections on.
And it was on the false fear, you know. On the other hand, the questions that were being asked are very different, right?
Gates Transformations Equivalence Classes Under Gates Transformation.
So this was really new at the time.
For me, it was the objects I knew about and I knew about what notion of an ice amorphous office was, but somehow studying solutions to equations.
But, you know, this notion of equivalence is a kind of new idea.
Mm hmm. Um, and anyway, that came along with this particular problem.
So the physicists constructed some solutions with a very natural differential geometric question.
And it's I mean, for me personally, of course, they linked up with Penrose's work as well.
I think with algebraic geometry, the twisted theory. There are a lot of ideas hovering around the time.
And then they kind of got directed on to this one particular problem.
Mm hmm. Uh, for me, I got involved in it initially because of some of the things that I was working on, actually starting in NYU,
which I never wrote out, which was about elliptic complexes, which were conformal invariants and non-standard elliptic complexes.
And one of these kind of came up, at least a twisted version of it in studying the deformations of these instance.
So I already had a background that studied these elliptic complexes.
I knew what the index theory gave us. I now understand the problem.
And so the first the first thing that I kind of contributed to this was the calculation of the infinitesimal deformation space of the base.
Mm hmm. Then the singer came along and said, Well, look, these modernise base are very similar to what you're doing now.
Spread geometry with modernised current issues, such as when you at what point you start saying moduli spaces.
Was this at the point in your life where you start thinking about moduli spaces?
That's right, yeah. I didn't know much about the algebraic geometry long before that.
And that's right. Mm hmm. So then. Then, of course. And with the rapid development, using all these techniques.
So. Okay, so eventually we we had this construction with the ADA construction, but the one with the lead singer was.
Was more the differential geometric background to the equation.
Mm hmm. And then how did the ADA. How did that develop?
Were you a team of four working in a concerted way?
So, no.
So the Russians, I mean, there were there's an exchange of letters, had letters in those days to a long time to get from Moscow to, uh, to the West.
And so, I mean, Michael was, was in contact with them.
And so but and I guess through other conversations with other mathematicians.
Mm hmm. But we were kind of working on it for my I mean, it was all using twisted theory and the result and some stable bundles and things like that.
But it it is true that I mean, Michael's described this in his memoirs as well,
that that we we eventually put together the vanishing theorem, which came from the differential geometry.
That was the movie with the algebraic geometry that the monads that Michael was familiar with.
We put these this thing together and then went off to lunch at St Catherine's College and then and well, actually, I was in Jensen Katz in those days.
So East Greenwich, which was I went to Wolfson College and Katz.
Uh huh. And I came back. I saw him waving a piece of paper at the end of the car.
And I just got this letter from Greenfeld and money went out there that they'd done it as well.
Okay. So it actually was, you know, we learned about it on the same day,
although obviously if you actually look at the timeline, then they probably had a little bit earlier.
Mm hmm. So we sort of skipped the bit of. So you went to the States and then you decided to come back.
Was that an obvious thing, or were you tempted at that point to stay in the States?
Or if you've been tempted at other times your life to to base yourself there?
Yeah. At that point. Yeah.
I could have stayed I could have stayed longer as a post-doc at NYU.
I think so, Mike, my husband. So those things.
But Michael was encouraging me to come back to Oxford and I've been away for three years.
I've always liked being in Oxford. And I got married, and, yeah, we decided to come back.
Mm hmm. And.
Did you look ahead and anticipate that Oxford would be sort of the centre of your life or was a de facto assumption about do you think I just gave up?
Uh, I know. I guess I never really looked ahead so much as I wanted to get on with the things that I was doing.
Mm hmm. Mm hmm. And I suppose I.
I kind of dragged my feet a little bit when it came to, um, to getting a more permanent job.
And so I after the postdoc position, I got an advanced fellowship as I saw C or maybe as SIRC.
Mm hmm. Um, so that gave you the time and freedom to think?
Yeah, that's right. So. Mm hmm. And I guess in the back of my mind, I thought, well, you know, eventually I've got to give it up and teaching, but.
And then I started applying for jobs in college, you know.
CV Uh huh. Jobs in Oxford. I wasn't successful initially, but I think I don't think I was very convincing.
And I still want I didn't give her a convincing argument that I actually wanted to engage myself in college life.
But eventually I, you know, I realised that I ought to have my ideas out there.
So yes. So I got the, the, the two, the shepherd and Katherine's going.
But later on you've enjoyed tutorial teaching, right? I know you kept some going even when you were a civilian professor, right?
Yes, that's right. I mean, you know, in retrospect, I realised that after becoming a tutor, I realised that.
You know, mixed dieting is good for you, that if you're always doing research, you can be banging your head against the wall all the time.
And that when you see students, when you give lectures to do tutorials, then it frees up part of your mind and has the cogs working again and.
And it centres is healthy healthy to mix together all these things in it apart from your own the legacy your own work in mathematics,
you've had many good students be good, good PhD students.
So how did that start as we. Yeah.
So I guess the first official assignment, Solomon was my first official student and I can't remember which year it was.
I'd. Yeah.
So he's okay? He turned 60 a couple of years ago, maybe Placer, huh?
Mm hmm. Mm hmm. Yeah.
So then I guess at the time, coming out of the work in Sometimes and South Duality, there were all sorts of questions about neonatology.
There are a lot of questions that have potential topics which which, you know, I thought should be worked on and.
And then I got students who were able to do that.
How did it work out? I mean, I think I think probably.
So I guess John James was civil in professor at that time and he was in charge of the geometry and mission.
And he he would feed me. And did you sort of have a program at that time where you would like students to work on this and
that idea of you to take in the student and then and then think about what you want them to do?
It was the latter. Really? Yeah.
I take the student explained to them what I was working on and explained things which were related to it and gradually seeing what they were happy,
happy doing. And I can't remember chronologically the but yes, but sometimes student had to just show up.
I mean, check these for example, is been very successful. Mm hmm.
He was kind of washed up on the shores of Oxford as a Rhodes scholar.
Didn't know, you know, nobody was a culminating and.
And then he became my student. Very happy to have him.
And but others were kind of trying towards me that I could change direction.
But if I sort of dragged you off the linear path.
But I remember you telling me once that you had a sort of a eureka moment about your famous paper on on Raymond Surfaces and Higgs Bundles.
Do you want to talk about that eureka moment?
Oh, yeah. And have they punctuated your life or do you think?
Yeah. Yeah. I mean, of course, the eureka moment is always built on lots of work underneath.
Yeah. I don't know whether it was the Higgs boson, but certainly in the monopoles.
Well, neither you. I remember a eureka moment.
This was when I was studying the construction of monopoles by looking at scattering along straight lines through three space.
And yeah, I mean, I was walking home across South Park and then I remember passing these bushes.
You know, stupid thing. But I remember thinking, Oh, a straight line has to end, you know?
So and that was and then in the in the 10 minutes that it took me to get home from there,
I kind of figured out how everything together in terms of algebraic geometry.
Previously, I've been looking at solutions to this equation along a straight line which decayed at one end.
And that gave me an entry point. Mm hmm. But then I've kind of forgotten about the other end.
I mean, I guess I thought that it was pure symmetry changing, complex conjugation, which gave me what was happening.
And then I realised that there was a whole morphing aspect to it, which.
So. So that was, I remember was a particular bush in South Park that let that happen.
And by the time I got home, I just wanted to write everything that was.
That was still there. Mm hmm.
So. So. So we've been taught by a very rich time in Oxford geometry and that.
So how did things develop? Did you think of that as a sort of a great time in Oxford?
And yes. You conscious? It was a special moment. Yes, it was.
Because there was there were all these I mean, there was the twist, the group,
the a lot of ideas coming from them because we'd seen that the twisted the theory was so valuable.
And then in the instance, um, picture.
But I'd also actually even earlier I'd seen that um, constructing solutions to Einstein equations and then known compact situation.
But the sort of thing that was bothering me even back in Princeton could be done through them, through Gibbon's and Hawking's work.
I mean, that showed me that you could use Twisted Theory to make these constructions.
So I think there was the the twisted theory, the algebraic geometry,
and then there was the gauge theory that was making it is, you know, telling us all the new things that were happening.
You would always in a seminar, you know, give us an account or something.
And I was in Harvard or whatever, and it was it was very exciting.
And then did Dan Quillin arrive at this time with that?
A little bit, I guess. Yeah, that was a bit later. Yes. So I remember giving let's say I gave a kind of graduate course on vector bundles on real
surfaces where towards the end I was just beginning to introduce the Higgs bundle package.
And I remember he was actually sitting in on this on this course.
I guess it must have been around then. Yeah. Mm hmm. But I didn't really interact so much with them.
Mm hmm. Mm hmm. So. So then at some point, you.
You decided to leave Oxford. And was that a difficult decision?
It seemed natural at the time. I think it's. Yeah.
So, um. So I left Oxford in 1990 to go to worry.
And so what had happened was that, uh, actually in the few years before that, so and so in like the mid-eighties, early eighties.
Mm hmm. I'd been offered a position at Stony Point where there had a very strong geometry group.
Mm hmm. And I went there for about two semesters to try it out.
So, um, I suppose my.
It wasn't so much a complaint about Oxford, but but, but, you know, c u f lectureship was a lot of teaching.
A lot of college teaching. Yeah. Mm hmm. And I really wanted to have more time for my research, so it would be 12 hours college, just that time.
Plus, I was actually pretty good. But it was to be a maybe 10 hours like that.
Mm hmm. And it wasn't so much the actual teaching. It was the way it impacted on the daily life that, you know,
if there was a seminar somewhere in Warwick or London or something I wanted to go to, there's no time to do it.
Mm hmm. Um.
So I was fortunate that because at the time teaching buyouts were really quite rare, but I was fortunate that in fact in my absence in Stony Brook,
certain things were put together and it was there was an arrangement that said I could have a buyout of teachers and kids.
Um, but then, uh, and that was largely on my committee know Grant.
Uh huh. So I was still working with him on various topics.
Mm hmm. So then, I guess towards the end of the 1980s.
Well, on the one hand, it was a question of, uh. Yeah, you sit back and say, am I going to do this right?
The rest. Right. Yeah. Um, and also, things were changing.
So Sir Michael went to Cambridge all my time to be a master of Trinity.
Graham SIEGEL went at the same time. And it was a time to, you know, I thought it was the time to change, right?
Because I didn't want to revert to full time advice.
Mm hmm. So then there was a chair assignment.
Christopher Zeeman and. And Warwick had moved and retired, I guess, but then subsequently became head of the college here.
And so there was a vacancy in Warwick and I decided to apply for that and.
And I. And I got in. Mm hmm.
Um, so that was. And that was change.
I enjoyed the change. I enjoyed the change of environment as a very different department is run in a different way.
It brought me face to face with departmental issues.
I college issues. Mm hmm. I had kind of. I hadn't taken.
Much part in mathematical institute business, actually.
Before that. Mm hmm. Um, so. Yeah.
So then I was. So. So work, actually. You know, when you.
When you change environment, sometimes your mathematical ideas change as well.
Yeah. That's go ask you about that. And it did. It did it to give you new interests.
Uh, so, um, yeah, it's difficult to say that the, the environment that I have because there was a natural progression.
Yeah. I mean, it's, it's true that there was a, there were visitors there, I guess, uh.
Yeah. Yeah. Different visitors pushed me in different directions on different subjects.
Actually were. It was where I first learned to use a computer.
In fact, when I got there, they said so.
Do you want a monitor or whatever it is? I have those things.
I said yes and so I said, fine. So that was on my desk and I didn't do that.
So I learned, I learnt to write papers intact and it brought me up to speed.
Have you ever used the computer as a research tool or as I use it, to search the literature?
And I've used that to do with mathematics.
Done calculations. That algebraic calculations. Yes.
Mm hmm. Though, to be honest, not in a very serious way.
I mean, you know, we always like to have if we have a closed form solution, we always like to have it eliminates the need to form as possible.
Mm hmm. And I've used, you know, algebraic manipulation.
Oh, okay. Uh huh. Mm hmm. On the other hand, I, I there was an occasion,
so when I was working on pan of equations and I got some algebraic solutions and they came out very nicely,
I thought, well, these are really nice formulas. And, and I thought, well, they must be right.
Uh huh, uh huh. But I never substituted them back in the equation using the computer.
And in fact, subsequently they became of use to Wolfgang Zeller and his collaborators in studying
metrics of positive curvature and an imbalance and to show that this formula is correct.
And then I had to go back to my paper and I realised that there were some medical errors.
Uh huh. So he actually put it in the computer and I said, Well, it's a nice for it can't be wrong that you're not totally happy.
And I don't, I don't have to.
I just skipped over the period when, when Simon Donaldson was your student, which, which was presumably an exciting time in all sorts of ways.
Oh, what was going on? And obviously, that that was a two way street, as it were.
But did that influence your subsequent research much, you think?
Uh, yeah, I guess so. So, yeah, so.
SIMON So this was another occasion when you and James told me, look,
we've got this applicant and Ray Liquorice is the best student for ten years or something like that.
And so I said, maybe we should make him only offer. And so he came and, and, and that was very rewarding time because.
You know, I would set a problem and then, you know, he would come back in a few weeks time and saying, okay, what should I do now?
Right. And gradually so but at that time,
it was interesting because I changed he kind of changed my view and I kind of put it I was after I'd done this work on instant
tons and I thought that this whole self duality kind of package was a kind of Kryptonian version of Raymond Surfaces,
and he was like kind of self-contained, a piece of geometry which replaced the complex then as by the quotidian.
But then there are these little bits of information coming from other areas.
So Talibs that showed that actually you could find self to your solutions on these non self your manifolds.
And then there's this link with stable bundles and, and so so there was clearly a kind of bigger picture emerging.
And that's kind of shook me out of my complacency about this, about where this whole area was sitting.
So so I began to give assignment problems related to this.
And he he was avidly reading the work back to the towns that I guess came here for a while as well.
And so so gradually he began working on these these broad general uses of some of the equations.
And but but he, of course, he passed on to the work with my idea.
And that was the the stage when, when he saw the.
So there was a conjecture there about the relationship between stable bundles and the existence of these emission Einstein connections.
And it was a clear conjecture. And he he noticed that actually you could reformulated in an infinite dimensional moment in that picture,
which was the kind of flavour of the month at the time there was lots and there was all the work that I tend done on convexity and moment maps.
And then the idea about picture for young males on Raymond Surfaces.
So moment maps, if you could phrase something in the language of a moment,
then you you kind of use the time somehow you'll be able to see the equations.
Mm hmm. And so that was, you know, at that time, that was when I thought, well, you know, Simon has identified this particular problem in those terms.
I benefited greatly by being there.
And so Michael and I had obviously a very personal.
Mm hmm. So can I skip it?
Skip ahead to your time in Cambridge.
So it was that was that a happy time? Was that a a new direction again or.
Yeah. So. So. Right. So.
You know, so I was approached I was approached for this chair in Cambridge, and it was clear that, you know, okay, I kind of knew that.
As I put it in some respects, I know that it would have been nice to come back to Oxford and, you know, in the chair.
Yeah, but that wasn't I wasn't available.
Available to her and so. But, you know, to be to be a, you know, office, they have a chair in Cambridge is a terrific thing.
And so I was very happy to go there. And I thought, this is going to be one of, you know, the rest of my life.
Uh huh. Mm hmm. Chair And one of the best departments in the UK.
And that was. And, of course, that meant kind of, hey, was the change, you know, going back to bwari to an environment which was more life.
Right. So it's familiar in some respects. Mm hmm. And I had a college attachment there, a in college, which is very nice.
I had other responsibilities there. And it was it was good.
In fact, there I had more contact with a physicist, actually know people like Gary Gibbons.
And we used to hold the geology seminar in the endowment so that Stephen Hawking could can.
Graham SIEGEL was there. We yeah, I, I used to.
Yeah. I followed different courses of.
Well, I follow different pieces of mathematics and research projects.
So I was I was happy that we had a nice house. But then then I got this phone call that because I'm in the civilian chair, it was held briefly.
And then I came back and again and then I got this, this phone call from Oxford suggesting I, I applied for that.
And that was you know, that was something I you know, I somehow had always liked Oxford better than.
I think it all goes back to where they whenever you know. I agree that.
Yeah. It was your heart. Yes. Then you got to know every street corner and every street corner means something to you.
And Cambridge is a wonderful place. But but still somehow also I mean, our children had grown up in Oxford and we we made we have many friends.
And so so it was coming back with something which I was happy to do and no reflection on.
So yeah. Yeah. So, so you mentioned you had some of the physicists in Cambridge, but so throughout your career,
I guess the sort of stimulus of physics asking questions of mathematics and the mathematics that's produces it,
but was that that particular time when that came more into focus,
or was it really something that was always that is I think the first time it happened was,
uh, in the seventies when, when I learned about this, this work of.
Hawking and his collaborators on Euclidean gravity.
So this was where. So I'm trying to do quantum gravity where they shifted from the Laurentian signature to positive definite signature.
Um, so this meant that, that they had these constructions and the rents in case which they adapted to the Romanian case,
and suddenly they were producing examples of these solutions to Einstein's equations.
Ritchie Flat manifolds, which we mathematicians though when I was in Princeton,
there were mathematicians there trying to even look at local solutions and getting any lower.
And yet here with the physicists, with their intuition and their methods actually producing examples of these,
I think that was the first time that I realised that,
that, that, that, that their intuition, when it was focussed on something which was relevant to the real mathematics,
that was really something valuable to hold onto. And I guess then and also the work of Penrose and his and his collaborators.
So the twist, the theory that again departed from the Lorenz signature because they were usually working over the complex numbers.
So for Penrose, he would say that the complex numbers are fundamental for quantum theory.
If we're going to have a quantum theory of gravity, which should building the complex numbers right at the beginning.
Mm hmm. So, again, what was Lorenz and became complex and complex and what was you know,
if you complexity by something which is positive definite then it's it's you know, it's the same with the same setting.
So those two pieces of that kind of change of point of view even meant that the
intuition in the background of the physicist could actually be relevant to positive,
definite signatures, changes that remain in geometry. So, so that was that.
Those are the first two instances where I really thought, okay, I can take physicists intuition seriously.
And then along came the instantaneous stuff, which is where the physicists had a problem, which was perfectly accessible and resilient.
Suddenly the flow of information was perhaps the other way there,
that that their intuition enabled us to make certain constructions, but we had no general point of view to to go back.
But I think it all started in the seventies with the whole given as well.
Mm hmm. And then the whole sort of symmetry story, is that influenced you a lot more recently?
Uh, so, yes, definitely. Uh, I've tried to.
Okay. So that was one of the things that I guess I started trying to learn about when I was in Cambridge,
because at that time there was this strange Ja Zaslow interpretation of symmetry, which was very geometrical.
And it was so trying to put that into effect made me try and look at what special advantages and enhance and fast the
various things that and subsequently they the Higgs bundle bunch of nice base which fitted into this s y z scheme very,
very nicely. I've been, you know, following up the implications of mirror symmetry and now largely by listening to physicists, people like Gourcuff,
Witten and Capus and others who tell me that, okay, you know, according to us,
something should happen, this correspondence between this and that, then checking with it.
And generally speaking, you can see you see how it happens in a particular instance.
That's, you know, that's that's been valuable in recent years. I pick up on that.
So and there's lots of interesting mathematical objects that bear your name, the hitchin component, the hitchin vibration and so on.
But then people take these up and run with them and you try to keep track of how people are using your name, as it were.
So, I mean, the mathematics that flows from it, do you feel a particular attachment to it as well?
That you know that it's true? There are conferences on the hitchin this and I know that maybe it's gone far beyond.
I have you know, I'm just in the work of go you know right over again is far removed from her that I work on.
And equally, these hitchin components, I mean, I'm very interested in the the work of people like Guichard and me not gnabry on these components,
but I know that that techniques are very far removed from the ones that I use.
And so following them means, yeah, understanding what they're saying about it.
But then in the back of my mind, I'm thinking, you know,
is it possible that I can actually use the techniques I'm familiar with to do some of these rights?
Um. But it's. But it's.
Yeah, it's. It's true that. That these things can be.
And have you been drawn. Specific. Have you been drawn into geometric langlands through in this sort of way?
But for the game, I restrict my so I rest restrict myself to to seeing in what way langlands duality reflects their symmetry.
So actually, you know,
formulations of the geometric langlands correspondence in terms of their I have categories that equivalence of different categories
is is something which I know I would never be able to formulate that is in the language which I only partially understand,
but it's the specifics which which intrigued me. The prediction usually I think of them as predictions from the physicists,
but of course the seeing them realise in algebraic terms is very, very satisfying.
Mm hmm. I think so. And talking satisfying.
So. So. So you just won the show prize this year and was that very satisfying?
Obviously is very nice in many ways. But and and it's not that you were short of international recognition before.
And and but it's did it bring a sense of as a warm glow of appreciation or is it.
Yeah, I think it's um and it is. I mean, I did appreciate it as international recognition and I mean, I know, okay,
maybe this recognition because I get asked to give me lectures around, you know, old and so forth.
But this, this was you a tangible. A tangible appreciation.
I was very honoured to get it. Obviously it was a great surprise and but but I've been fortunate to win various prizes from the society,
but this was an international profile of the prize.
I mean, was what really appreciated both. If you had to.
So as we've discussed, your work has wide influence and you and you have great students who perpetuate your influence in many ways.
But what are you most proud of yourself?
Which part of your work I am most proud of or most fond of?
I guess a tough question, I guess.
You know, the I think the citation for the show prize talked about corners of geometry, which I lay dormant or something like that.
I mean, I always like it when there's something that I've been working on which has some historical, you know, the quaternions, for example.
Yeah, that's, uh, uh. Um, and even, you know, these 19th century mathematicians knew what they were doing,
but of course, what they were doing was framed in a different way nowadays.
And so we must actually dismiss them. But no.
So what was giving me? Most pleasure?
Uh. Well, you know, I think. I think this.
You know, the Higgs. I mean, I wrote this paper which started the Higgs bubble.
What? And that was a paper which I just came in day by day and kept discovering new things about.
I mean, I started it in a in a simple fashion.
I, I knew that there was some infinite dimension, dimensional hyperscale quotient,
which was going to give me a margin I space related to bundles under the surface.
That was my starting point, just HyperCard and geometry.
But then the more I looked at these these this particular object, the more I found.
And then it was over a period of a few weeks, actually, that somehow every feature seemed to generate a new new result, new aspect.
On the one hand, it was a new proof of uniform ization. Now, I could also imitate the loss theory approach to it.
But I can do this. I can do that. It was that was very, very satisfying because it I mean, the paper took much longer to write,
but still, ultimately, the the pieces, the components of it, they all fell together very quickly.
Mm hmm. So you you knew. You knew pretty quickly.
This was a really rich vein of ideas. Yeah, yeah, yeah.
I didn't realise that it is so influential, but. But it was pretty clear that there was.
There was something genuinely new there. So.
So as we're chatting this, that and certain influences on you are obvious.
But are there any influences we haven't touched on in people or sets of ideas that are less obviously influences on you?
Uh. They're obvious individuals, you know?
I mean, you know, people like my friend Tia is singer.
I mean, the ones who facilitated the connection between physics and mathematics.
Edward Whitten example. I mean, I'm not saying we haven't written any papers together,
but the people have had the patience to understand that there are two languages and you need to explain in order to really make the relationship work.
You have to explain things in the right formulas, even if you know, even if the original way of thinking about them is very different.
Mm hmm. And so. So people like that. Who.
Who make this interface between physics and mathematics work requires requires patience and understanding of both sides of the road.
And those three are of three examples.
People have rarely done that in my life. So. So you're retiring a civilian professor.
You're not retiring as a mathematician? I imagine so. It's a bit.
What? What do you want to do the next part of your life?
What really interests you still? What? What mathematically and unknown mathematically did you hope to do next?
Uh, I'm still I still got the momentum of what I was doing, you know, for the previous years.
So I have been coming in virtually every day to the office.
I have is I think that's going to change very much over the next year or so.
Hmm. Um, there is, there are projects which I'm, I'm working on.
I mean, this, this kind of infinite dimensional version of timeless space, which is intriguing me very much.
And which I mean,
Francois Barea is another kind of version of it and it's not at all clear about whether his is exactly the same as the one that I'm looking at.
Um, so, and I'm not sure. So I, now I have one research student left and he will finish the end of.
Of this academic year. Mm hmm. Um, I'm not quite sure how things will go.
I don't. I mean, for the sake of the research students, I don't really want to have anymore, because it will be a divided supervision.
And in any case, they need me to absorb new ideas.
Mm hmm. So I'll. I'll carry on with. With the many of the projects that I have and hopefully learn some more.
I mean, I go to conferences, that's, uh, a conference and.
Well, actually, to be honest, uh, part of the work that I.
So next month I'm giving some lectures in Brandeis and named lectures.
So I have to think about what I'm talking about. This is, this is way off right now from I think, Oh, I've got to give some lectures.
Let me think a little bit more about this topic. And then gradually maybe a paper comes out.
And then then, okay. There are now all sorts of 60th birthday conferences from her, from a student.
Right. I know that these are all coming out her these these are things which keep keep me going because I give a talk.
You know, I think I need to think of something new to talk. And it's always something in the back that which I can I can work on.
That was great. Nigel, thanks very much for talking to us today. My pleasure.