Sir Michael Atiyah, formerly a professor
at both Oxford and Cambridge, is one of the foremost mathematicians
of the 20th century, and still an important force in the 21st. In the
1960s Atiyah, with Isaac Singer of MIT, proved powerful and far-reaching "index theorems" making
profound connections between geometry, topology and algebra relating
to the physics of quantum operators in quantum field theory. He has
also developed a branch of algebraic geometry called K theory. These
advanced mathematical methods, as well as many others he has developed
or inspired, have had an immeasurable influence on modern theoretical
physics. Atiyah is of Lebanese and Scottish descent and was educated
in Cairo and Manchester before entering Trinity College, Cambridge.
He was elected a Fellow of the Royal Society in 1962 and awarded the
Fields Medal in 1966. His other prizes, awards and honors are too numerous
to mention but are listed here.
Sir Michael, what led you to become a mathematician?
Well, I think
I was always interested in mathematics when I was a boy, and good
at it, I enjoyed it, but there was a time when I wanted
to become a chemist. And I oscillated between mathematics and chemistry.
But one year advanced chemistry was enough for me. You had to memorize
so much stuff. In mathematics all you had to know was a few principles
and figure it out yourself. It's so much easier.
Was it organic chemistry that got to you?
No, it was
inorganic. It was how to make sulfuric acid and all that sort
of stuff. Lists of facts, just facts, you had to memorize a vast
amount of material. Organic chemistry was more interesting, there
was a bit of structure to it. But inorganic chemistry was just
a mountain of facts in books like this.
that in mathematics you don't really need an enormous memory.
You can work most things out for yourself, remember a few principles.
If you're good at that, then it comes easily. If you want to
do other things, you've got to work hard to learn a lot of facts.
There was one reason, I think. But I enjoy thinking, I'm good
at it, and will continue with it.
How do mathematicians view the history of physics? Well,
I think if you go back in the past, those in mathematics and
physics were called natural philosophers in those days, there
wasn't really any difference. Early mathematics all grew out
of practical needs and computations, you have tomato fields and
you have to do this. Newton's work in calculus was all to work
out the dynamics of motion, so there was really no difference.
All the great, well most of the great figures of the past were
mathematicians or physicists of distinguishment, many of them.
Newton, of course. Gauss, and others. Some were of course very
much more pure mathematicians by our standards, and some physicists
would of course not be very mathematical. But a large number
of them worked out the mathematics they needed and mathematics
developed out of the needs of the physics to a great extent.
Not entirely, but a large part of it, so there's been a long
history where it's both. Only in recent times is there this distinction
between what's called a mathematician and what's called a scientist.
They weren't physicists and chemists in those days, they were
natural philosophers, and natural philosophy included mathematics.
So it was much more unified.
Do you see that this specialization will continue
in the future or will they always be sort of organically knit together,
mathematics and physics? Well,
all of this always diversifies on one hand and becomes more specialized,
but there are various times when things interact again, and what's
been happening in recent knowledge with modern developments in
theoretical physics is that there's been a need to employ more
and more advanced mathematics, so mathematics independently developed
for other reasons has been brought into the fold. And so what appeared
to be diverging strands have been brought together again. There've
probably been similar specializations in different directions,
and every now and then when we've got a good period, things will
reconverge. And right at the moment we're in a reconverging period
so it's a lot of fun.
What would you say has been the impact of string
theory on mathematics?
Well that's very difficult to say. It's really to early to have any
kind of final picture. We don't even know what string theory is. But
it's had an impact on mathematics which has been really quite extraordinary.
First of all, the impact of string theory on mathematics has fairly
extensive. It covers many areas of mathematics, not just one. Geometry,
topology, and algebraic geometry and group theory, almost anything you
want, seems to be thrown into the mixture. And in a way that seems to
be very deeply connected with their central content, not just tangential
contact, but into the heart of mathematics. And at the same time, the
physics ideas have produced ways of thinking and ideas and speculations
which go back and produce quite spectacular results and conjectures
that mathematicians have been busy working on, which they had no previous
way of getting hold of.
I would say the
biggest impact on mathematics has been as a whole new collection of
results and conjectures where you have to deal with, broadly speaking,
with what you might call dualities, where the same thing can appear
two different guises. In the physics framework these dualities are broadly
understood at some conceptual level, even if not technically, and in
the mathematics, the dual picture comes out to be totally different.
So it's a great challenge for the mathematicians on the ground, to see
what we got from the sky, these two things may go together. Working
out how that is going to work out in detail is going to be a big challenge
for mathematicians. It has transformed and revitalized and revolutionized
large parts of mathematics. And so you could say that mathematics in
the first half of the 20th century, large parts of it, will be devoted
to understanding the impact of string theory on physics and mathematics.
One way of looking
at it, it's not the only way, but one way of looking at it also is that
large parts of physics are concerned with things coming out of quantum
field, before we get to string theory. And quantum field theory is about
working on infinite-dimensional spaces, infinite-dimensional manifolds,
infinite-dimensional groups' function spaces, and doing it in a very
detailed way, a lot of detailed calculations and geometry and analysis.
And so you could say the mathematics of the earlier century was basically,
large parts of it, finite-dimensional. The mathematics of the 21st century
will be pretty infinite. Infinite-dimensional stuff in terms of linear
theory, Hilbert spaces and that stuff, that will have been done a long
time. But nonlinear, all the subtleties and complicated topologies and
geometry and so on, nobody did those things at all, barely. Early work
on Morse theory and sort of, closed geodesics and so on, were just to
scratch the surface. All this new stuff from physics seems to be some
overarching attempt to build a big hierarchy of things for infinite-dimensional
geometry. And so the 21st century might look like that in the future.
In the nineteenth century, all they played around with was N dimensions.
In the eighteenth they only played around with three dimensions, then
go back to two dimensions and one dimensions.
And so you can figure
in that way we're leading to a new chapter in mathematics. In its very
early days, and its hard to say what it will look like when its finished.
But if I had to make a prediction about what people will say about it
in the year 2100, then that's what I would say. That the big change
was this shift into a totally different frame. And using infinite-dimensional
ideas, you get back, of course, results in finite dimensions, that's
So it seems there's a lot of mathematical momentum
for string theory, even though the experimentalists aren't too happy
with it yet.
Why I was told just yesterday somebody was talking about the latest
experiment that seems to suggest that supersymmetry might be needed.
No I think that I agree with you. Of course it's reassuring for physicists
that what they're playing with, even if we can't measured it experimentally,
appears to have a very rich, consistent mathematical structure, which
not only is consistent but actually opens up new doors and gives new
results and so on. They're on to something, obviously. Whether that
something is what God's created for the Universe remains to be seen.
But if He didn't do it for the Universe, it must have been for something.
So it's something worthy of trying to study, there's no question about
Sir Isaac Newton, when he became a physicist,
he only had to learn first year calculus. Today's physics and math students
have to learn so much more. Where will it all end?
Just one slight correction - Isaac Newton didn't have to learn calculus,
he had to invent calculus. That's a different story altogether. And
I think the norm in mathematics in general, in the history of mathematics,
is it builds enormous structures, and we've been doing it now for thousands
of years or hundreds of years at least, so you wonder how on Earth we
can go on learning and doing more.
And the reason is
of course because mathematics has this great propensity to unify. People
find out lots of things, and then at the next stage they say, well all
these are special cases of some one simple picture. We abstract out
of them all. People object to that business abstracting, they say why
don't you stay concrete. Well the whole point about it is if you stay
concrete, you're always tied to tables and chairs, you can't see the
constantly move up a level. As they go forward, they have all these
examples of something, we'll see what's common, and give it a nice title
and unify it. And with that under our belts, we can put it in a small
textbook and move on.
And for century
after century, mathematicians have been building this big structure
where we absorb what we've done before, put it together in a simple
pattern, and I think that not only mathematics but science as a whole,
only progresses if you can understand things. It isn't just a matter
of, you know, getting a lot of results out of the computer. If science,
all it did was to produce a string of numbers, we'd soon be terribly
lost. Its aim is to produce ideas and explain things in simple terms.
All science is like that.
And if it does that,
then all the earlier stuff that people... You know, before we understood
about the basis of chemistry, they had all sorts of ridiculous stuff.
Once they understood about atoms as a whole, it all clicked into place,
and most older stuff was forgotten. And the same is true in mathematics.
You explore and get through a lot of things, and suddenly you unify
everything. So it's this ability of mathematics to unify things what
comes before, and simplify it, so that the next generation of students
can all say, oh gosh, how easy it all looks. Calculus, I can learn that
in six months. And it took a great genius like Newton, and Leibniz,
too, their whole lifetimes struggling with it.
So we can make progress,
otherwise it would be impossible. And it's always surprising. Any one
generation.. When I was a student I only had to learn this, and now
all the students have to learn this and a lot more, how can they possibly
be... and yet you still see them coming along, cheerfully, maybe. In
no time at all, they're up to speed and on the front lines. You can
see it happening, it's always happening.
And the explanation
of why it happens is what I've given you. Mathematics and physics and
a lot of science aims to unify by simplifying and enabling the next
stage to go forward. It's easier in mathematics. Obviously with something
in the area of biology, unification isn't as straightforward. DNA unified
a lot of stuff, but there's still a lot more complication in biology.
But physics is much closer to mathematics, this end of physics, the
basic end. There are parts of physics which are also messy and complicated
where you don't really unify easily. But the parts related to mathematics
are pristine clear and simple. So I think that's always hope yet for
the next generation of students.
Back to your own work. What led you to develop
K theory, which is currently of interest in string theory?
Well I say, it was a big surprise to me to find that it was of interest
in string theory. K theory really arose out of algebraic geometry, but
basically what it's concerned with is the interrelationship of topology
and linear algebra, linear analysis. You study linear things, and you
have linear things that depend on parameters, you study how they vary,
and the topological implications of that. And K theory is the formal
outcome of that.
started with linear things. Linear things like curves. If you study
tangent spaces to manifolds, they are families of flat things, they
are families of vector spaces. So once you get from geometry, immediately
you start worrying about things like families of vector spaces, and
K theory is the outcome. So large parts of differential geometry and
topology have a natural formulation in terms of it. And so K theory
was a natural outcome. There were a bit of accidents here and there,
so it happened, but you can reasonably speculate that it was inevitable
something like this would happen. In many ways it formalizes notions
like traces in linear algebra. And of course you've gone to linear analysis,
with operators, so it gets up into index theory and so on. So I knew
that the physicists would be interested in the analysis side. That would
come up, we discovered that twenty years ago.
What I'm much more
surprised at is they're interested in the more basic topological side,
which is, although connected, in some ways independent, and elementary,
in some ways, but quite delicate. And some parts of the topology physicists
now seem to need is not just only the basic thing which I did a long
time ago, but even more refined refinements and variations on the theme
which seemed extremely recherché at the time, and some of them
so recherché that we didn't even bother to follow them up.
And now the physicists
say, we need this, we need this, we need that, and I've got to go back,
you know, and see what you can do.
But in a general
way I believe that if you work in mathematics on fundamental basic central
things like continuity and linear analysis and so on, they whatever
you do is going to be used in all sorts of places. I was just having
lunch the other day with a chap who's an electrical engineer at UCLA
and he applies K theory in control theory for robotics. He says it's
very important to study the variations of operators. So if you're doing
basic things, they are bound to turn up. But it was certainly surprising
the way they turned up now, I don't understand it. I've spent two months
trying to find out a bit more. It's fascinating. We really don't know
quite why it's necessary, why it works, but it seems to be a pragmatic
fact of life that K theory is important in string theory, and the more
people go along, the more they find out about it.
Do you have any sense or expectations of what
this program is going to result in? You did this work earlier but the
story isn't over yet.
String theory or K theory? String theory... everyone who works in this
field soon aims to understand string theory in a way not yet understood.
String theory is meant to be some approximate perturbative expansion
of some theory not yet known. People are grappling with it in the dark
and they're looking for everything they can get that gives a clue. And
I think the way K theory comes in is certainly a bit of a clue. There's
lots of places where it comes into string theory at a different level,
and since K theory is my background, I like to think about these things
and see whether they can possibly suggest what the ultimate picture
might look like. And we don't know what that picture is, but it could
be the ultimate picture will involve K theory in a some way in a rather
It's come in by
the back door, and somehow people find it's useful. Why? We don't even
quite know. It lurks around the various corners. So I think that if
there is an eventual simpler picture that emerges, then I would think
that something like K theory, or versions of it, will be an important
component. I'd like to think that, it seems rather plausible, but it's
very speculative as to what this final theory will look like and how
long it will be before it gets there. But I certainly think its happening
at the moment. It may not be that far away. In five or ten years we
may get a really big insight, some young will come along and open up
the doors. And K theory may figure quite interestingly in it.
What makes a mathematics problem fun for you?
You spoke earlier, in a talk you gave, about a fun problem. So what's
Well, there are two different things. The main thing that interests
me in mathematics always is the interconnection between different parts
of mathematics, the fact that one problem may have half a dozen different
ways of being looked at in different subjects, a bit of algebra, a bit
of geometry, a bit of topology. It's this interaction and bridges that
interest me. I'm not that keen on becoming entirely focused on a single
area where you forget about everything else and go down with a big bore
hole deep down to the middle. I prefer things that unite across the
borders. I find that exciting. Occasionally there's some fun in the
more lighthearted or frivolous sense. There are some problems which
are fun because they're elementary, but strange and difficult to solve
to all degrees, unexpected relationships. There's a problem I'm working
on at the moment, in what you refer to, is amusing probably in some
ways. I don't know what it means or why it's there. It just forces itself
on my attention as a problem that is interesting to look at and understand,
and fun in some general sense. It may turn out to be I'm peering through
a window into some new deep unknown underground treasure, I don't know
yet, or maybe that's the window into a rather small piece of scenery.
We don't know until we've drawn the veil.
But I like two things
in mathematics - unification, things that unite, unexpectedly, you know.
Many beautiful things in mathematics prove something in subject A. By
a marvelous and unexpected link a subject way in the corner over there,
if you apply this idea then presto, you get to solve those problems,
and that's the sort of thing I like. That's one form of unexpected thing.
In general, the unexpected.
If someone is plowing
along with a big machine and gradually grinding away and you chug away
at the rock face, then that's not very exciting. But if somehow there's
a breakthrough and something totally unexpected happens, occasionally.
It only happens in mathematics every decade perhaps, something like
that happens, like Donaldson's work on four-dimensional manifolds. That
was a real spectacular opening, and totally unexpected, out of the blue.
And that's really exciting. So I'm really excited by things that are
And that's why of
course when people ask you what's going to happen in mathematics, the
most interesting things are the things you can't predict, by definition.
If you can predict it... Predictable things are within your grasp, you
can get there. Unpredictable is exciting, and we hope there will be
unpredictable things for a long way into the future.
I think the way a lot of people think about
mathematics, since it's all based on logic, it can't be unpredictable.
Well, the idea that mathematics is synonymous with logic is a great
ridiculous statement that some people make. Mathematics is very difficult
to define, actually, what constitutes mathematics. Logical thinking
is a key part of mathematics, but it's by no means the only part. You've
got to have a lot of input and material from somewhere, you've got to
have ideas coming from physics, concepts from geometry. You've got to
have imagination, you're going to use intuition, guesswork, vision,
like a creative artist has. In fact, proofs are usually only the last
bit of the story, when you come to tie up the... dot the i's and cross
the T's. Sometimes the proof is needed to hold the whole thing together
like the steel structure of a building, but sometimes you've stopped
putting it together, and the proof is just the last little bit of polish
on the surface.
So the most time
mathematicians are working, they're concerned with much more than proofs,
they're concerned with ideas, understanding why this is true, what leads
where, possible links. You play around in your mind with a whole host
of ill-defined things.
And I think that's
one thing the field can get wrong when they're being taught to students.
They can see a very formal proof, and they can see, this is what mathematics
is. My story I can tell. When I was a student I went to some lectures
on analysis where people gave some very formal proofs about this being
less than epsilon and this is bigger than that. Then I had private supervision
from a Russian mathematician called Bessikovich, a good analyst, and
he'd draw a little picture and say, this -- this is small, this -- this
is very small. Now that's the way an analyst thinks. None of this nonsense
about precision. Small, very small. You get an idea what is going on.
And then you can work it out afterwards. And people can be misled, if
you read books, textbooks or go to lectures, and you see this very formal
approach and you think, gosh that's the way I gotta think, and they
can be turned off by that because that's not an interesting thing, mathematics,
you see. You aren't thinking at that point imaginatively.
But you mustn't get
carried away by the other extreme. You mustn't go all the time with
airy-faery ideas that you can't actually write and solve a problem.
That's a danger. But you've got to have a balance between being able
to be disciplined and solve problems and apply logical thinking when
necessary. And at other times you've got to be able to freely float
in the atmosphere like a poet and imagine the whole universe of possibilities,
and hope that eventually you come down to Earth somewhere else. So it's
very exciting to be a practicing mathematician or a physicist. Physicists
in principle have to tie themselves down to Earth more than mathematicians,
and one day look at experimental data. Well mathematicians have to tie
themselves down in other ways too. A proof is one of the things that
instantly ties them down. But it's a mistake to think that mathematics
and logic are the same. They overlap in important ways, but it's a big
mistake. I'm not very good at logic.
What is the most fun problem you've worked on
Well I suppose the problem I like most of the things I did was a problem
I attacked which concerned the thing called an index, the formula for
a manifold with a boundary, which ended up by being a formula that connected
three different terms, one of which was a topological invariant, one
of which was an analytical invariant in terms of eigenvalues of operators,
and the third of it was an integral expression involving curvature.
So topology, differential geometry and analysis, all written into one
simple formula, with a rather nice geometrical interpretation. And I
think that was the thing I really most enjoyed doing, traveling across
three different borders. It's one that's now used by physicists, that
have different meanings, it also has applications in some bits of number
theory. It's a very nice example of straddling right across the three
different subjects like that.
Do you have a favorite mathematician from before
the 20th century?
Well it's a bit like asking who's your favorite musician. Depending
how you feel, one day you can say Beethoven, another day you can say
So who would it be today?
obviously you have to begin with what the word favorite means. If you're
just trying to say who were the greatest mathematicians of all times,
you can talk about Newton or you can talk about Gauss, and so on. But
if you're trying to use the word favorite in a slightly more personal
sense, the person I think I like, well, there are two people that come
to mind. One is Riemann. He was a person, first of all, his collected
works occupy one volume, unlike Euler, where we talk about thirteen
volumes. And in that one volume, he put forward the foundations of
modern differential geometry, the Riemann zeta function, an important
problem in fluid mechanics. He had a whole range of things which he
initiated. And in fact although this was the 19th century, it turned
out to determine a large part of the work in the 20th century. He was
very far in advance of his times, very deep, original. And being nice
and compact in one volume has a kind of appeal, a quality.
The other persona
I have a lot of admiration for in a personal way was Walter Hamilton,
Walter Rowan Hamilton, who was a mathematical physicist. Hamiltonian
mechanics, Hamiltonians, is specific to physics, but he also invented
quaternions, which is a great part of mathematics, which I'm very fond
of as well. He was an original mathematician in many ways, a slightly
difficult character as a person. But I like the unusual. You said before
the 20th century, so that rules out the people who overlap the 18th,
19th and 20th century like Poincaré and so on. For the 19th century
I think it has to be Riemann and to a lesser extent, Hamilton.