Michael
Atiyah (centre) and I.M. Singer (left) receiving the Abel Prize in 2004
courtesy Scanpix/The Norwegian Academy of Sciences and Letters
Michael
Atiyah was a distinguished mathematician with a stellar scientific
career spanning over 60 years. He passed away on January 11 aged 89. He
received the Fields Medal in 1966 and the Abel Prize in 2004, and is
best known for his work with Isadore M. Singer on the Atiyah-Singer
index theorem.
In October 2016, Atiyah spoke to C.S. Aravinda, the chief editor
of Bhāvanā, a journal of mathematics, at the former’s home in Edinburgh.
The interview is presented in full, and has been lightly edited
for clarity. Aravinda’s words are in bold and Atiyah’s are plain.
§
I am delighted to meet you. Which part of India are you from?
I am from Bangalore, which is in the southern part of India.
I have visited Bangalore several times. In fact, I have even planted a
tree in Bangalore. There was a chap there at the Raman Research
Institute who was a friend of mine, called S. [Sivaramakrishna]
Chandrasekhar – not the astronomer. He worked on liquid crystals. He
said he was also a nephew of C.V. Raman. They are all related.
Many years ago I was reading a paper of C.V. Raman. I think I was
most impressed by a statement saying something like, “It is from my
collection of diamonds, I have discovered the following facts.” [
Laughs] Do you know other scientists who say “I have diamonds”? He was a rich man and had a big diamond collection.
You also have other Indian connections. I noticed that, in
1966, the year you were awarded the Fields Medal, the Indian
mathematician Harish-Chandra gave a plenary talk at that conference.
Yes, he gave a plenary talk there. When I was a graduate student in
Cambridge, I went to Amsterdam in 1954 and he gave a plenary address
there too. I remember a very clear impression that I had, which was that
he spoke English much faster than any Englishman. In English, you have
ups and downs. But he was like a machine gun, he spoke much faster. He
spoke beautiful English. Of course I couldn’t understand anything. It
was not the English I could not understand. I was a graduate student and
he was professor or something and he was talking about advanced work. I
was just finishing my PhD.
You were finishing your PhD at that time?
I finished my Ph.D. in 1955. But I was just beginning my career, and
he was several years older than me. He was smartly dressed. Yes, I
remember very clearly that first meeting. Of course, later on we met as
colleagues. I visited Princeton, but then he was in Columbia at the
time.
I think you visited Princeton in 1955-1956. But he visited Princeton again in 1956-1957, a year after you left.
Actually, I was still there for the fall term in 1956. Of course he
became professor later on. I went to Princeton for seminars. I was
certainly there in… oh dear… [
laughs]
Atiyah (centre) receiving the Fields Medal at the 1966 ICM in Moscow. Credit: Michael Atiyah
I think you went there sometime in 1969. Because Nigel
Hitchin mentions in some place that soon after he finished his PhD, he
went with you to Princeton.
He came as my assistant. That was a good time. Anyway, later on there was another Indian who came to work with me – Patodi.
Vijay Kumar Patodi, yes. I was going to ask you about him too.
Very brilliant man. And a very sad case. He died almost at the same age as Ramanujan.
Around the same age, in his early 30s, yes. He died of renal failure.
His case was more complicated. Anyway, at that time my interests were
not very close to Harish-Chandra’s. My interests became closer much
later, possibly after he died. When did he die?
He died in 1983, at the stroke of his 60th birthday.
I know that. He was going to have an operation but he couldn’t have
it. So we didn’t really interact much. But I got interested in that kind
of mathematics probably after he died, 30 years ago.
I have read somewhere that he was going to give a talk
somewhere in England, I don’t remember where. And then he had a heart
attack and cancelled his visit.
Yes, in 1982. At that time, I was back in Oxford. There, I had a
succession of Indian visitors who came to work with me, mainly from
Bombay who were students of M.S. Narasimhan. A whole succession of them.
I got to know S. Ramanan very well, a very nice man. T.R. Ramadas came a
little later. They all came to work with me in Oxford in that period.
Yeah, it was quite a strong Indian connection. Long time ago.
Armand Borel and Harish-Chandra. Credit: Suresh Chandra
So you first heard Harish-Chandra at that Amsterdam
conference in 1954. When you were at the Institute for Advanced Study
(IAS) in Princeton later, was there any interaction?
I was at IAS as faculty from 1969 to 1972, and Harish-Chandra was
already there. We were only about eight or nine colleagues and we had to
meet to discuss things. I got to know him from close. We got to know
each other’s families, as we all lived quite close together. For
instance, their daughter Premi married Pierce, who was a fellow at
Trinity College, Cambridge.
But we didn’t really overlap much mathematically. Our mathematical
interests were a bit different. But I got to know him well. Both he and
his wife were very tall and upright. They were imposing Indians you
know, and I was short, but they were nice to us. They lived on Battle
Road. I also knew Armand Borel very well, and he was very friendly with
them. I think Harish-Chandra came from Allahabad and not from Bombay.
Originally, he was a physicist.
Correct, he did his PhD with Paul Dirac.
You know this story about Harish-Chandra and Freeman Dyson?
I do, but please tell us about it.
I think it’s a more or less true story. Harish-Chandra and Dyson were
both graduate students in Cambridge after the war. Harish-Chandra was
doing physics with Dirac and Dyson was doing number theory. While they
were walking down the streets, the story is that Harish-Chandra tells
Dyson, “Physics is in a mess and I am going into mathematics”. Dyson
said to him, “Physics is in a mess, and that’s why I am going to go into
physics.” And they switched over.
Of course, Dyson still kept an interest in mathematics.
Harish-Chandra used his knowledge of physics to direct his mathematics.
He worked on infinite-dimensional representations because of the work of
the Russian school led by Israel Gelfand. Harish-Chandra wanted to do
it more rigorously.
So Harish-Chandra and Dyson had this very odd connection. I got to
know both of them. I knew Dyson very well too; he is still very much
around and must be about 90 now.
Both Harish-Chandra and Dyson were born in the same year, 1923, and so Dyson must be about 93 now.
Exactly. More recently, at the Royal Society in London, as a
tradition, if you have been a fellow for 50 years, they give you a nice
little dinner for which you could invite your friends. To succeed in
this, you have to be a mathematician or physicist, you have to be
elected very young and you have to get very old. So not many people have
qualified [
laughs].
Right!
M.S. Narasimhan. Credit: ICTS
When I was President of the Royal Society, one person did qualify,
and that was Rudolf Peierls. He was involved with the atomic bomb. I had
to preside at his dinner, and we had Hans Bethe as his guest. After I
left the Royal Society, when I was getting to my fifty years [as a
fellow], I wrote to the Royal Society, more or less saying “I hope you
haven’t forgotten that there is a tradition.” And they wrote to me
saying “Oh, we’ve forgotten all about it. For the last ten years, we
haven’t had any. Oh dear, what are we going to do?” [
laughs].
So they said, “Ah, we’ll invite everybody who were elected in that ten
year period.” Only two people came – Dyson and me. I was elected for
fifty years, and he was elected for sixty years. None of the guys in
between were able to come, so it was rather funny. I have slight
disagreements with Freeman Dyson, but he is a strong minded person with
strong views.
Harish-Chandra and Dyson were both six years older than me. So when I
went as a graduate student to Amsterdam, I was 25 and Harish-Chandra
was 31. In terms of age, the difference was only six years, but in
mathematical terms it is the difference between a post-graduate and a
senior professor.
You came back to the UK from IAS in 1972. Why did you come back?
Well, when you make a decision, first of all, you don’t know why you’re making it! [
Laughs]
There can be many different causes. Even if you can easily list the
causes rationally, the decision is usually an emotional one. And you
don’t admit it. I can talk for hours but there are a few factors. One is
we had been in America for years and my wife was not very happy. She
had a better job in the UK than she had there. Also, we used to spend
half the time there and half the time back here, which was not very good
for her because she could not get a job properly. Plus my children were
growing up – the eldest one was 16 and if we had stayed for a few more
years, he would have gone to college in America and then we would never
be able to leave. So it was now or never.
And thirdly, I was at the institute when it was an unhappy time
between the director – at that time it was Carl Kaysen – and the
faculty, particularly the mathematicians, including André Weil. I was
caught in between and I felt very unhappy. It was a difficult decision,
as I liked the institute and it suited me very well mathematically. But I
felt that I was being selfish in putting myself first. My family comes
first. A job turned out for me here and I came back. It worked out very
well and I got lots of brilliant students. It was really a toss of a
coin, and my heart made the decision. It was not rational. Sometimes
your heart makes a better decision. [
Laughs]
Yes, I think so too. As you said, when your heart makes the decision, there is no reason why.
Precisely. It is very complicated. It’s an emotional response which has many factors.
I have been to the IAS as a visitor, a young man, an older man and
I’ve known all the people and all the directors very well. It is a funny
place. It can be marvellous and vibrant for the right person at the
right time. Its most useful function is that it is a place where young
people go for postdoctoral work. The school of mathematics was a big
one, and a large number of post-graduate people came there either for
postdoctoral work or on their first sabbaticals. They were all very
young and active and some of them, having been in the army, were delayed
by five years.
[This stage in one’s research career] is a very formative period.
Everybody is full of ideas. They go to seminars at the university, at
the Institute, and the faculty on the whole have very little contact,
with a few exceptions.
The Princeton University is also close by…
Of course; the university is fine but the institute was meant to be
separate, you see. And some people were very separate – they didn’t come
in at all, they didn’t mix with students. Sometimes, each person would
select assistants who were more or less right-hand men. Otherwise, all
of them were selected by competition and some faculty would attract more
students and used to work with them – and some did not. But it depended
mainly on personalities, and in some cases the areas of interests.
As a young visitor, I hardly knew the professors. Although, through
one or two lectures, I got to know some of them, like Dean Montgomery,
quite well.
Freeman Dyson. Credit: Jacob Appelbaum/Wikimedia Commons
Was Atle Selberg also there at that time?
Yes, he was there but he didn’t interact very much. I wasn’t doing
number theory very much. We didn’t interact with all the famous people
at all. On the other hand, they were the people – the flagships – that
brought the money in that would enable the young people to come. So they
were kind of the umbrella. But the interaction between the postdocs and
the faculty was very small. Some people interacted quite a lot. For
instance, Borel interacted quite a lot because of his wide interests.
He used come to India quite often because he loved Indian music.
When I first went to IAS, he wasn’t there. He came later.
All the faculty at IAS were very clever people obviously, but
sometimes there were odd and unusual personalities who didn’t get on
with the other faculty or the students, or had some kind of problems.
There was Selberg and Arne Beurling and they worked on their own to a
great extent. They were Scandinavians – they didn’t have much contact
with others. There was Marston Morse. He was retired but he was still
active when I was there. He had a few acolytes, I would say. And Hermann
Weyl was there before I arrived. He was a marvellous guy. He was a
really great man, especially with young people.
The physicists were even younger and more active. All the
mathematicians were slightly odd – not meaning they were bad but they
were unusual. Some of them had one or two people, assistants, in little
groups who were in their area. It was a very odd situation. Everyone
wanted to go to IAS because there were famous people.
Another person who was there was Kurt Gödel. I had heard the name a
little bit. He was a very odd person, extremely odd. He eventually died
of malnutrition.
I see…
Gödel was the greatest logician of the 20th century. For a long time
at the institute he wasn’t even a full professor. He was only a sort of a
half professor. And then he had paranoid ideas. He thought that people
were out to poison him. After his wife died, he starved to death. He
just didn’t feed himself; he was really an extreme case. I knew him
before that. So if somebody had serious psychological problems, the
institute was not a good place to be. There was too much pressure.
John von Neumann was an enormous personality. Both Hermann Weyl and
Einstein had died before I came. Their reputation still lives on.
Princeton was the place which had all these names – Einstein, Weyl, von
Neumann – who were great figures at the time. So you could chat with
great mathematicians and physicists.
I looked through the institute records for a while when I was there.
There was this talk by George Dyson, Freeman Dyson’s son. He had an odd
history – he ran away from school, lived in a treehouse. Then he came
back and became an author. He spent a year at the institute officially
at the invitation of the director, accessed all the archives, gave
talks. He was supposed to be writing a book. The title of the book was
‘Turing’s Cathedral’. It is about the history of computing.
A fairly recent book, yes.
It is really about von Neumann but he thought Alan Turing’s name
would attract more people. So it’s called ‘Turing’s Cathedral’.
Interesting book. I spoke to him when he was there, and he told me
stories such as, for example, that the institute tried very hard to get
Paul Dirac. They tried very hard to get Turing to stay; he had been
working with von Neumann. But he didn’t want to stay. So there were a
lot of successes and failures.
There were some people, like Carl Ludwig Siegel, who went back to
Germany after the war (1951). So they attracted people but sometimes
they lost them. But they tried very hard to get all the top people. They
paid big salaries but still some people had good reasons not to go, all
sorts of personal reasons.
As you said, sometimes it is a decision of the heart. Secondly, certain places suit certain personalities, I guess.
Of course. If you wanted peace and quiet at the institute, and you
wanted to get on with your work with one or two people to talk to, had a
good and stable enough family life, you liked the country, then it was a
marvellous place. So many of these things were good. But I had my
family worries about other problems, and eventually I left. The one
thing that I missed when I was there: I didn’t have graduate students. I
had one or two graduate students who were unofficial, a few of them who
were my assistants, and did their PhDs at the university. When I came
back to Oxford, I got a flood of good graduate students. All very
brilliant. I was very fortunate.
Yes, you have a large number of them.
Yes! Simon Donaldson was my student. Graeme Segal was my student. And
Patodi being a sort of a student. I liked having young people with me.
It worked out very well. I came back just at the time when I was well
known enough that people would come to work with me.
Since you mentioned Patodi: Perhaps you came to know about
Patodi through his paper? Or was it when he was in Princeton visiting
IAS?
I think I got to know him as a student of Narasimhan. What Patodi did
was that he gave some very clever proofs of some difficult theorems in
differential geometry. He developed his own formalism, which I couldn’t
understand. Raoul Bott couldn’t understand it. We knew the results were
interesting. So then, through Narasimhan, we invited him to come, and he
spent two years at the institute in Princeton. Bott, he and I worked
very closely together and we understood what he was doing.
Interestingly enough, subsequently, when the work we were doing got
to physicists, the physicists rediscovered what Patodi had been doing in
the language of supersymmetries. But we were there first. They gave,
subsequently, a physicists’ proofs of what Singer and I were doing, but
Patodi did it first. I don’t think they really gave him proper credit.
He did it, he found it all by his hand. He was a bit like Ramanujan.
Yes, you mention that in your preface to the collected works of Patodi.1
He had a kind of intuitive feel for formulas and he couldn’t explain
how he got them. And he was very clever. We struggled very hard to
understand them. And through that we learnt from him and we published
papers. He had very good insight.
But you see, like Ramanujan, he had extremely rigid habits of eating.
He was a Jain. He had to go wash his hands before doing anything. He
had to cook his own food. When Ramanujan came to England, he was
somewhat similar. He was very fussy about the food. He was cooking on
his own and he actually had malnutrition. He couldn’t get the vegetables
that he needed.
V.K. Patodi. Source: Author provided
And there was also the war at that time.
Of course, that didn’t help him any.
In Patodi’s case, it may have been even more rigid, because observant Jains eat before dark.
Exactly. Very rigid. He also had a lot of serious illnesses, which he
didn’t take seriously enough. He discovered he had kidney failure and
also tuberculosis.
I see. I didn’t know he had tuberculosis.
I think so. But he certainly had more than kidney failure. When I got
back to Oxford, we heard from people in India that he had kidney
failure, which is a serious illness, and he needed a kidney transplant,
which costs a lot of money. I contacted a friend of mine who was a
surgeon and an expert at [treating] kidneys and asked him if he would
consider helping. And I said, well, it costs a lot of money and I and my
friends will try to get some money together to pay for it. He said he
wanted to see his medical papers, so we arranged for it to be sent to
him.
And when he got them, he told me that it was much more serious than I
thought. He had kidney failure but also lots of other problems as well.
This meant it would be a very delicate operation. I then found out that
it wasn’t just kidney failure but that there was something more, so his
case was very difficult. Still, we were discussing whether we should
bring him to Oxford and have the operation done. But then we heard that
there was an American expert who was visiting India who would consider
doing the case. Then the expert went there but I think Patodi died while
under anaesthetic before the operation.
He died tragically before the operation could be made because his
system was so weak. He died at the age of 31. He wasn’t quite in the
same league as Ramanujan but he was very good. So there are lot of
similarities. Ramanujan went to Cambridge to work with Hardy. Patodi
came to Princeton to work with Bott and me. We wrote a lot of papers
together. I learnt a lot from him, his intuition. He learnt something
from us. So here was a very parallel story. Ramanujan was probably
ahead. But nevertheless, it was a lot of similarity.
I’m very happy to see that Patodi’s name is accorded and recognised
now. Unlike with Hardy and Ramanujan, he was a very emotional part of my
own life. I had got myself tied up because he was such a nice man. I
think he was an easier person. Ramanujan I think, was a difficult
person. Patodi was a very easy person, very nice person – a really
modest person; and you don’t find really modest people in the world –
and we didn’t have any problems. But his medical problems were more
serious and he had this very strict food regimen.
Raoul Bott (left) and Michael Atiyah. Credit: Bott Family
I don’t know if Patodi came with his wife.
Hang on. Was he married?
Yes. His wife’s name is Pushpa Patodi. Maybe he married after he went back to India.
He came as a young man, you see. When he came to work with me, he had
just done his PhD with Narasimhan. So he was probably only in his
middle twenties. I don’t think I remember his wife. He invited us for
dinner to his house once or twice and he did the cooking. You know, this
story of Ramanujan is somewhat similar but Ramanujan invited his guests
and walked out halfway through the dinner only because the guests
didn’t have a third helping.
When you talk about Raoul Bott, what I must definitely recall is that you wrote an obituary of his in Bulletin of the LMS,2 which I felt was very warm and had a lot of personal touch.
I put my heart into it. He died in 2005. He and I both got very
friendly with Patodi. We had lunches with him, and he used to cook us a
meal. But we learnt how fastidious he had to be about his food.
He was a vegetarian too.
Yes, that’s okay. You can be a vegetarian and eat a lot. First time
when I went to Mumbai, I was looked after by K. Chandrasekharan. He was a
vegetarian. He invited us for dinner and gave us a most enormous meal
to show us that vegetarians could eat very well and in large quantities!
[
Laughs]
So you think he would have, potentially, gone on to do very well.
Yes. Because he had a lot of insights and by the time we finished
working together, he had learnt a lot more things. He would have gone on
to be a very famous mathematician.
I was asking specially because you mentioned that later on physicists did the same thing.
Yes, he would have been able to fill that gap. There were not a lot
of people in that area. He might have been in any country in the world.
But probably because of his Indian connection, that he was a Jain, he
would have stayed in India. There are a lot of people now as it is very
popular. There is this physicist called Ashoke Sen – he is up there,
right? So there are now enough Indians who have gone back. After all, it
is 30 years now since that time; things have changed.
The advantage of living longer is that you get this panoramic view.
And you see things in different periods and in different times, the
influences of people, and you realise that what you read in the history
books is not the true story. True stories are much more complicated with
who influenced whom and how, and usually the history books are wrong [
laughs]. You know, they are always wrong. The further away you go, the more wrong they are.
It becomes speculation.
You know, it’s a bit like ordinary history. I am also interested in
history. They say that history is written by the winners. And then of
course people do research. They go back and say “Ah, this is all wrong.
He wasn’t the villain, this was the guy who was the villain!” England
history is full of it.
Mathematics is a subject where the establishment rules. People in
positions of power can influence somebody’s career. It can be for the
bad. And people don’t discover this for a long time, sometimes for
hundreds of years.
When you were growing up here, you were of an impressionable age when the war broke out in Europe.
Yes, but I wasn’t in Europe. I was in the Middle East.
Did the war affect you in any way?
Well, probably. My mother was from this country and my father was
Lebanese. He worked in Khartoum in Sudan, which is at the junction of
the White Nile and the Blue Nile. His family was in Lebanon, and my
uncle worked in Palestine at the time. We went to secondary school in
Cairo and in Alexandria.
When the war broke out, we were in England on holiday. My father had
to go back immediately, leaving my mother and children in England. We
then followed across through France, just before France collapsed, and
got to the Middle East. We spent the war entirely in the Middle East.
After the war, my father retired and came back to England.
Did it affect you academically?
Well, when I was young I went to this primary school in Khartoum,
which had only about 20 children of all ages because it was just for the
children of government officials. When I became 12 it was not good
enough, and then I was sent to the secondary school in Cairo. You see,
my father had been to a very famous school called Victoria College,
which educated many famous people. It was a British school in Egypt for
people like, for example, Omar Sharif: he was younger, a small boy, when
I was there, and the future King Hussein [bin Talal] of Jordan. So this
was the kind of people who went there [
laughs].
Statue of James Clerk Maxwell at St. Andrew’s Square, Edinburgh. Source: Author provided
It was a very good English school. My father went there before me. I
went there; I got a good education. But it affected me. English was my
native language. My mother spoke English, my father had been to Oxford.
My brother and I were both advanced for our age by the time we got into
this school, so they put us in a class which was two years older. I was
age 12 when I was with those of age 14 and finished the school when I
was 16. So I was always two years younger than everybody else. It had a
funny effect on me. If you weren’t careful you would get bullied by the
big boys.
So I usually had to have a big boy as a defender because I was helping with their homework [
laughs]. The deal was – well not really a deal – if you helped them with their homework, they would say “Don’t hit my friend!” [
Laughs].
So I got through very well there. But my brother was not so lucky. I
think he did get a bit bullied – he was younger still. When I was 16, we
left and as my father got a job in Britain, the whole family came back.
So we lived here after that.
Was there a teacher who inspired you? When did you realise that you were interested in mathematics?
The family folklore is that my father said that he knew I was going
to be a mathematician because every year when we travelled to the Middle
East, I always managed to exchange my pocket money at a good rate [
laughs]. That was his story anyway.
Good with calculations.
Yes, good with calculations. I don’t know, at school I was always
very good at mathematics. I think once I didn’t come on top in
mathematics and I was very cross. I was always expected to be top in
class in mathematics. But I liked other things, too. At one stage, I
wanted to be a chemist. But I discovered that chemistry was too much of
memorisation, you have to remember all the formulas. So mathematics was
much simpler, and I had quite a good mathematics teacher in Cairo. He
was also the sports teacher.
Do you remember his name by any chance?
Yes, his name was S.H. Griffiths. He was in charge of the cricket
team. I remember he was nicknamed Figaro. Figaro is a character in
The Marriage of Figaro.
The composition by Mozart.
Exactly, so the boys would call him “Figaro! Figaro!” The boys at the
school had these funny jokes. When I was in the last year of school, we
went back to Alexandria, which is where the school was originally but
had to move out as it was given to the army as a hospital. So then I had
a different teacher. He was tall, severe and unbending. But he had a
difficult life because his wife was in a [mental health] asylum and he
couldn’t get a divorce. So it was a sad life. He was a chemist but he
was a good mathematics teacher. He helped me to get on with mathematics
quite a bit in my last year and gave me advice.
When I was at Trinity College, we had a visit from the Indian
president at that time. We had to show him the books with all the names
of people who have been there. I remember showing him three famous
Indians who had been at Trinity College at different times. One was of
course Ramanujan, the second was Jawaharlal Nehru, and the third was the
most famous Ranjit Singh
ji, the cricketer! So there were these three great figures; Ranjit Singh
ji
was way back in 1870-1930, the only Indian to play cricket for England.
Then we had Nehru, a great figure in Indian history, and Ramanujan. We
had a sportsman, a mathematician and a politician, which is pretty good.
So we had a long Indian connection.
We also discovered later on that we had Mohammad Iqbal, who was a
great poet and philosopher of Pakistan. So we had both Nehru and Iqbal
in the same college. There were many others. Subrahmanyan Chandrasekhar
was there at Trinity and we also had Ernest Rutherford and other famous
physicists.
After you came to Cambridge you met William Hodge, who was
your thesis advisor. I will now bring our discussion closer to
Edinburgh, the place associated with James Clerk Maxwell.
Edinburgh is a great place culturally in Scotland. James Clerk
Maxwell was born in Edinburgh and went to the university there. Hodge
too was born in Edinburgh and went to school and university there.
Further back, Edinburgh had the great philosophers David Hume, Adam
Smith and Walter Scott.
Many people from Edinburgh went to Cambridge; Maxwell went to
Cambridge. In fact, my wife, who did her PhD in mathematics, went to
Edinburgh University and then came to Cambridge. It’s part of a strong
tradition. Basically, in the sciences, people [from Edinburgh] went to
Trinity College.
Did Maxwell’s equations of electromagnetism provide the main ideas behind Hodge’s theory of harmonic forms?
Yes. Hodge got a good training in Edinburgh. He learnt about
Maxwell’s equations. And then in the path of his understanding, he
combined that with differential geometry and algebraic geometry. That
was a very big step which people didn’t fully understand at the time.
Hodge was doing Riemannian geometry and Maxwell was doing relativity in
Minkowski space, not Riemannian space. That means that the equations in
physics are hyperbolic equations and in mathematics they are elliptic
equations. Algebraically they were the same and Hodge didn’t mind but
people thought they were very different.
Now, in the recent developments, these things are fused together once
again. In modern theory, they go very happily from the Lorentzian world
to the Riemannian world by Wick rotation. They change the sign. It is
standard practice now in the whole of physics. What Hodge did was very
far ahead of his time. Now we understand that Hodge and Maxwell are part
of a single school, the Edinburgh school.
Somehow it seems that it is tied up with your living in Edinburgh.
It is very much so. George Street in Edinburgh ends in a square at
each end. At one end, which is called St. Andrew’s Square, you will find
a statue of James Clerk Maxwell. I put that statue up. I found the
sculptor, raised the money and I persuaded the city council to put it
up. At the time, when I was President of the Royal Society of Edinburgh,
we put it up and we opened with a conference. It is an old style
statue. If you look at its front side, you will find Maxwell’s
equations.
Maxwell’s equations of electromagnetism, engraved below his statue in Edinburgh. Source: Author provided
You said that your interest in representation theory came
after Harish-Chandra died. You also met Hermann Weyl at IAS and admired
his works, of which finite-dimensional representations forms a major
part. So… you weren’t really looking at it then?
Atiyah: Let me backtrack a bit about Hermann Weyl. You might like to read the obituary to Hermann Weyl which I wrote.
3 You
will find it in an unusual place, which is for the US National Academy
of Sciences. It was written fifty years after he died. Usually the
obituary is written shortly after you die. The US Academy finally tried
to get people to write, but everybody who knew him had died by then. So
fifty years after they elected me, they wrote to me and asked, “Would
you like to write the obituary?”
I said fine and I wrote his obituary fifty years after he died.
Usually, if somebody dies, we say, “If we look over the next fifty
years, we will find out what influence he had.” I was able to say “Look
backwards.” So I wrote it retrospectively and I think you will like it.
It is a nice and short article. It shows the influence he had on all
subjects of mathematics. He died in 1955. Until my time, everything I
did – I always found that Hermann Weyl was the guy who did it first. He
was always there first. He was the kind of man who set the tone for the
development of mathematics in the subsequent fifty years. And
Harish-Chandra grew up in that period. That is one influence.
Hermann Weyl. CreditL Shelby White and Leon Levy Archives Center, IAS Princeton
The second influence was of course, in some sense – you could say
that what Harish-Chandra was doing was extending Hermann Weyl’s work
into quantum mechanics because his work was on infinite dimensions. The
first people who worked on this theory were people like Eugene Wigner
and Valentine Bargmann, and then subsequently Russian schools like that
of Israel Gelfand.
They were consciously developing infinite-dimensional representations
for the purpose of understanding quantum theory. And they bothered
themselves to what extent they could use Hermann Weyl’s work on compact
groups but now extending them to non-compact groups. So,
Harish-Chandra’s work is naturally a combination of Hermann Weyl’s work
plus the influence of the people in physics at the Princeton school, and
then the Russian School. He built his work by combination of these two
ideas.
He also had to do a lot of analysis and he based a lot of that on the
work of Laurent Schwarz – the Schwarz distributions. That was kind of
the analytical basis. So all these were new at the time and he was using
them. He was a very technical expert who was able to prove theorems.
But he was a bit of a machine: he turned out papers exactly thirty-three
pages long.
Yeah, exactly thirty-three pages!
He would say, “I assume you have read all of my six previous papers,”
and carried on. Which meant that only the very very devoted followers
would read it. And so his papers had these drawbacks. He was very, very
mechanical, he was very thorough, very detailed, coming up with very
good results. He had a big influence on people using bits of it. But I
think it was too formal a way to do it. I myself recently got interested
in this and worked with Wilfried Schmid. He was at Harvard, and we
redid some of Harish-Chandra’s work and I have recently come to
understand the physics better. So I think Harish-Chandra’s work will be
re-done from a modern point of view.
Do you mean to say that doing mathematics is more intuitive,
but when one writes papers, that’s not exactly the way one should
present it?
Well, you shouldn’t do it. In the old days, they did write like that.
Then it became more formalised. If you go back and read mathematics
written over a hundred and twenty years ago, it was not so formalised.
People gave you their ideas and then they give you some details. But in
more modern periods, this was not very formally correct. But they’re
wrong: it was correct at the time. I think the fundamental insights of
mathematics are by intuition and insight, and with examples making sure
you are right. The details on proof – they are a function of time. As
years went by, the nature of proofs changed, they evolved to become more
formalised. Too much so. Now it is a bit too formalised. That was part
of the influence of Nicolas Bourbaki.
Exactly! I was going to come to that. I remember reading
somewhere that Harish-Chandra was considered for the Fields Medal in
1958 but a forceful member in the committee thought that
Harish-Chandra’s style was Bourbaki. I think it was Rene Thom who got
the Fields Medal that year, in 1958.
In Edinburgh. I was here at the time. Hodge presided over that
congress. I was young and had to help people move rooms and that sort of
thing. Now, the story about Bourbaki is that Bourbaki was sort of made
up of a reforming group of French mathematicians in the post
First-World-War period. They were trying to restructure French
mathematics.
They started around… well, structures. They wanted to emphasise the
importance of structure. All the words we use today – like isomorphism –
come from Bourbaki. But at the end of the day, the abstract notions and
ideas, they were the good parts. The bad parts were that they then
wanted to make proofs, because the previous era had been full of the
Italian algebraic geometers who had no foundations. Bourbaki wanted to
have very rigorous foundations. In this direction, they went too far. So
they were a good influence and a bad influence.
They worked together because the original guys who founded Bourbaki,
they knew what they were doing, and they could do both together. The
next generation, who weren’t the founders, they only inherited the
formal part. So they went off in a totally wrong direction.
The tragedy of mathematics with Bourbaki was that they were very
successful in their fundamentals but they were over-enthusiastic, or
their supporters were over-enthusiastic, and that led to the bad
reputation of Bourbaki. And so, in that sense, Harish-Chandra was not
formally a member of Bourbaki, but he grew up under the influence of
Bourbaki and the rigorous style of writing. And so, I’m sure his work
will be taken apart and redone in a totally different way with an
emphasis on ideas.
I’m now trying to understand things. I think I’m understanding a lot
of things [now] that I didn’t understand when I was younger. I
understood partially [then]. You know, understanding is a slow process.
You evolve. As you get nearer and nearer to dying, you understand the
world better and better. And finally when you get out there you go “Ah!
It’s all obvious.” [
Laughs]
So would you say that earlier, mathematics was in some sense
driven by application and in the 20th century, mathematics developed for
its own sake and not necessarily for its application?
That was always like that, for its own sake. When the Greeks did mathematics, it was always done for its own sake.
But perhaps the sophistication became more pronounced in 20th century, in some sense.
Well, up and down. In the medieval period, people did it for the
glory of god. You built cathedrals for the glory of god. You built
mathematical theorems for the glory of god. Whatever you call it, you do
it for its own sake. You weren’t building cathedrals as houses for the
people. Somebody else built houses for the people. You built cathedrals.
Mathematicians basically build cathedrals. Cathedrals are big works of
intellectual architecture with theories. And they were built because
they were beautiful, because you like beauty. Beauty was motivating. You
try to build a building without an idea of beauty, you will build a
concrete tower. It may be very good for standing a fort or a castle but
it is not inspiring.
The Greeks – Euclid, Pythagoras – did it for its own sake. The Arab
world did it for the glory of Mohammad. The Indians and Chinese all did
it for its own sake in their own language. They would refer to their own
gods or their culture or their beauty.
Bertrand Russell made a very provocative remark about Pythagoras.
Bertrand Russell was, interestingly enough, the only mathematician to
get the Nobel Prize for literature. He said that Pythagoras was the most
important person who ever lived. Why? Because he made two fundamental
discoveries. One was that numbers were at the heart of music – the tonal
scales.
//You strive to truth but beauty is your guide. Beauty is the torch that lights the path//
The mathematical basis of musical verses were known before that, but
Pythagoras was the first person who realised the special significance
that numbers have. Secondly, he realised the real of importance of the
Pythagoras Theorem. Everybody knew Pythagoras’ theorem well before
Pythagoras but what Pythagoras emphasised was that you could have
Pythagorean right-angled triangles with rational sides.
So the two most important things in the world – music, a part of our
culture and geometry, a part of our agriculture and the utilitarian
world – both of these things depend on numbers. So this means that the
world is rational because it is built on numbers. We don’t need to have
mysterious gods, we don’t need to pray to the gods for it to rain. We
can appeal to rationality. That was the beginning of modern science.
In other words, what Pythagoras said, according to Russell, was that
there are laws of nature. It is our job to find out the laws of nature.
That’s what scientists do. The laws of nature… well, we don’t know who
created them. But we believe there are laws of nature. And that’s a
faith. But then you can work in this field but still be motivated by the
beauty of nature. And that is what all of them believed: they believed
they worked to enhance the glory of god. The beauty was the driving
force; it always has been.
So mathematics has always been developed two ways. First, not for
utilitarian purposes but for beauty. Secondly, for utilitarian purposes
because you had to do engineering. So it was always these two strands.
And Pythagoras realised that. Individual mathematicians are motivated by
the beauty of mathematics, but they also have to make sure that the
foundations were correct. If a bridge is collapsed, you build a new
bridge. So you have to balance between beautiful theory and a theory
that works. But a theory that has no beauty will never work.<>
That’s a beautiful quote!
Hermann Weyl has a beautiful quotation that I like to use a lot. He
said “All my life, I pursued two objectives: truth and beauty. But, when
in doubt, I have always chosen beauty.” I like this quotation because
if you are a mathematician, you think surely truth is more important.
But Weyl’s dictum can be explained in the following way. Beauty is
something you see here. I see something – say, that tree is beautiful.
You can’t deny me my view because I feel it here, emotionally. If I say
something is true, that is, I apply my logical reasoning, I’m never
quite sure I’m right because the nature of proof changes with time. What
is accepted now, what you may have, is a perception of truth. Truth is
something you never reach. Truth is the ultimate god. You strive to
truth but beauty is your guide. Beauty is the torch that lights the
path.
So you have both truth and beauty but truth you never reach. Beauty
you have here, now. So if you are in doubt, you see a mountain, and you
ask “Should I go this way or that way?” Beauty would tell me to
go that way. And usually, it is a correct guess. But you’ve got a choice
to make. Do you choose the formalistic engineering approach or do you
choose the beautiful approach? And most of the time our heart says
“beautiful” and the heart will be right. So it’s a very subtle thing.
Hermann Weyl said it very explicitly several times and I believe in that
strongly.
I personally admire Hermann Weyl because he defined what a manifold is as we know today.
Of course, the idea was around that time. But he was the first person to formally define it.
Because even Élie Cartan himself says at one point that it is difficult to define what a manifold is.
Hermann Weyl was a master of ideas. I’m an enormous admirer.
Interestingly enough, when I wrote that obituary, I said I am going to
focus on the influence he had but I never focussed on the parts of
mathematics that I don’t understand. The two parts I said nothing about
were his very great concern with the foundational questions of
mathematics – big arguments with David Hilbert and Gödel, which he was
very much involved with. I never paid much attention to logical
foundations so it’s not for me.
And the other thing that I said I wouldn’t do anything about was
this: he had some very interesting questions in bits of number theory
that I didn’t know much about. I mean, I did know some number theory,
but he obviously went into much greater depth, such as with Diophantine
approximations. In the last ten years, I got interested in both these
things of Hermann Weyl. I realised what he was trying to do about the
logical foundations of mathematics and in number theory.
I can write now another interesting addition to that obituary. That
would be another excuse for bringing it up to date. I always thought I
would follow in Hermann Weyl’s footsteps and I was following them only
in the areas I knew, but now I’m following them in the areas I didn’t
know. And I’m still following Hermann Weyl. He is my spiritual beacon.
And I’m still trying slowly to catch up with him. My only regret is that
I didn’t meet him. He was sixty years ahead.
But I heard him talk at the Amsterdam Congress [ICM]. He was the
president of the Fields Medal committee that year and he had to give big
speeches about the Fields medallists. The two Fields medallists at the
time were people I knew very well: Jean-Pierre Serre and Kunihiko
Kodaira. They were different characters, although for a while their
interests overlapped. And Hermann Weyl’s talk was a very magnificent
speech. First he talked about Kodaira because Kodaira was working in the
things that Hermann Weyl studied at great length: differential
equations and tensor harmonic forms. Weyl brought Kodaira from Japan,
where he was working alone. Kodaira was his Weyl’s protégé. Serre he
didn’t know. He poetically concluded his speech with the following
remarks:
Here ends my report. If I omitted essential
parts or misrepresented others, I ask for your pardon, Dr. Serre and
Dr. Kodaira; it is not easy for an older man to follow your striding
paces. Dear Kodaira: Your work has more than one connection with what I
tried to do in my younger years; but you reached heights of which I
never dreamt. Since you came to Princeton in 1949 it has been one of the
greatest joys of my life to watch your mathematical development. I have
no such close personal relation to you, Dr. Serre, and your research;
but let me say this that never before have I witnessed such a brilliant
ascension of a star in the mathematical sky as yours. The mathematical
community is proud of the work you both have done. It shows that the old
gnarled tree of mathematics is still full of sap and life. Carry on as
you began!
He wrote this book about the classical groups.
4 It
is a marvellous book written in response to an attack. In the old days,
people did invariant theories and when Hermann Weyl wrote this book,
people said “Oh, a Hermann Weyl book. He says all this formal nonsense.
He doesn’t understand any invariant theory. We old guys, we do invariant
theory.”
Hermann Weyl was so cross. He did understand invariant theory much
better than anybody else. So he wrote this book as a response just to
show that he did understand it. And he wrote a beautiful book; it’s a
book you read many, many times. There are so many things that it isn’t a
single-track book. As opposed to the book written around the same time
by Claude Chevalley on Lie groups. Single track – you follow the axioms
and you end. By the time you finish, you know all of it and you throw
the book away and never look at it again. Hermann Weyl’s book was the
contrary – you read it and you reread it. Every time you get some new
insights. I’ve done that myself. So it’s a fantastic book.
That’s very interesting, what you said. Again it brings me
back to Harish-Chandra. I think he was quite influenced by Chevalley in
Columbia. He heard his lectures and there is this wonderful recollection
by George Mostow. Apparently, Chevalley’s lectures were very planned,
very well laid out and everything. And at one place, Chevalley kind of
stopped and sort of went to a corner of the board, wrote something and
covering that part of the board…
Yeah, I’ve heard that story too. He drew a picture.
[
Laughs]
And then he said “My assertion is certainly
correct, but I don’t see at the moment how to prove it.” Apparently,
Harish-Chandra later remarked to Mostow: “How can one know a
mathematical statement is true without knowing how to prove it?”
Both of them were true. I think the point is that Chevalley, when he
wrote books about algebraic geometry, people said to him – this is
another story – “When you think of an algebraic curve, what do you think
about?” He wrote in the corner “
f(
x,
y) = 0″ [
laughs].
Then he rubbed it out you see. To hide the fact that… I mean, these
stories were all aspects of the truth. I think Harish-Chandra is more
like Chevalley than Hermann Weyl by a long shot. He was more fastidious
about proofs but Chevalley probably had the proofs – maybe he wrote a
bit too fast. Chevalley was a bit older.
Michael
Atiyah (centre) and I.M. Singer (left) receiving the Abel Prize in 2004
courtesy Scanpix/The Norwegian Academy of Sciences and Letters
How is a physicist’s approach to mathematical truth different from that of a pure mathematician?
That’s a very interesting question. I can’t answer it in two
sentences because my ideas are quite subtle on that. I’ll give you a
quick summary.
First of all, there is no one class of physicists. Physicists are a big spectrum – appropriate terminology [
laughs].
So one end you get physicists who are only interested in having some
kind of a crude model which will fit with experiment up to the accuracy
they’re concerned with.
They are not like engineers, but they’re like an engineer with some
understanding. They understand the mathematics… well, some of them
bother. Some are just engineers who want a formula which works. Some
will say that the formalism or the mapping breaks down and I don’t know
what to do. An engineer whom the formula comes from will give it to a
physicist, who is a little bit of a mathematician who understands it.
But then it keeps shifting.
So you move further along this line, and eventually, you will end up
at the far end with the physicist who says, “I want to understand
everything. All of it. I want to understand gravitation,
electromagnetism. I want a theory that is completely universally
perfect.” They’re at the extreme wing. By the time you get to the
extreme wing, you have outflanked the mathematicians. The mathematicians
are somewhere in the middle.
Now, “everything” means physics at all scales – big, small,
classical, quantum. I think mathematicians are in between with a shorter
range. They are influenced by applied mathematics and engineering. They
know they have to find technical tools, so they have something in
common with engineers. The engineers and the applied mathematicians are
much the same. Then the mathematicians develop mathematics but more for
the sake of following the beauty. So mathematics is much more diverse
and it also has many more applications.
Physicists are focussed on, well, they call it the theory of
everything, but they’re really meaning fundamental physics. The
mathematicians are more diverse and there are more schools of thought.
Then, as you go further up, mathematicians are interested in more
advanced physics, quantum theory and so on. They have to follow the
physicists, keeping track with the physicists. Their mathematical models
have to fit the physical theory upto some point.
But then the question is how far on the scale do you want the
physical theory to work. I think in this way you get towards the extreme
end of the realm of the mathematicians who are logicians, the
fundamentalists like Gödel, Turing and Hermann Weyl, all who are
concerned with the logical foundations of mathematics. The questions
they are asking seem to be demanding answers of a fundamental nature
that in the real world doesn’t arise.
The physicists are interested in the foundations in their own
language. They don’t really want to worry about the logical foundations
like the mathematicians. They want to worry about the formal
mathematical foundations for their own purposes. So they are kind of
weaker fundamentalists; they are fundamentalists of a different kind
still. They are really separated from the logicians. There are a few
people trying to cross the border, but very few of them.
You can differentiate different physicists based on the scale they
operate. Are you only concerned with the nucleus and the atom? Are you
interested in molecules? Are you interested in galaxies? Are you
concerned with the universe? What happened before the Big Bang? You can
keep increasing the scale, and you get into deeper and deeper waters.
Mathematicians down here are also struggling.
There are some people trying to bridge the gap. Gödel was interested
in physics, and you’ve heard of John Conway. But they are rather scarce
in number. No division is clear cut but broadly speaking I see a tree
which starts with engineering and goes along with physics and
mathematics in parallel, with a few ups and downs, but later on going
off into… like a river delta valley. It’s rather strong… a big
mainstream here and there, with a few rivers in between. That’s not a
bad analogy.
I am somewhere… I think I climbed a little hill and am looking down
on this lot trying to get a good view. If you are too high up, you can
only see a single line. If you are too low down, you see too much
detail. You’ve got to be just right. And probably best if you are in a
balloon going up and down. So you change your view depending on your
perspective. If you are higher up, you see different scales. If you want
to see a mouse, you come down closer to the earth.
Could one describe this as two sides of a coin? For example, in mathematics, the two sides can be algebra and analysis.
I don’t think I accept your distinction. My distinction would be
algebra and geometry. Analysis is more across the border. In analysis,
you worry about the continuum in geometry. Analysis in algebra is when
you go, for example, from polynomials to analytic functions. Analysis is
a bridge that connects algebra and geometry. And algebra has old
traditions in India and so on, and geometry has an old tradition with
Pythagoras and the Greeks.
Analysis is common ground. And of course, the physicists grow out of
this. Algebra and geometry connected by analysis is a good bridge. Then
you follow that through to physics. Algebra leads to some formal
computation, geometry leads to insights – Dirac lives here and [Abdus]
Salam lives there. Of course, which is up and which is down is… [
laughs] I would put them on a plane so that we are not distinguishing.
Algebra leads to analysis immediately once you start, for example, to
consider infinitely many things. Finite algebra is finite. Finite
geometry is very finite. But things only get interesting when you allow
infinite things. Infinite series, decimal approximations of numbers,
continuum limits, differential calculus – how do you define the
derivative? As soon as it becomes infinite – and of course, it can be
infinite in different degrees – how big is your infinity? And in this
whole scale, how big is infinity? What is infinity? There is more than
one infinity. You can write twenty five books about infinity [
laughs]. But that is a story for another time.
It has been a great honour and a pleasure talking to you.
I always like talking. It is one of my weaknesses.
It was really delightful.
This interview was conducted by C.S. Aravinda. It was originally published by Bhāvanā
magazine and has been reproduced here with written permission. Read the original here.
