Sir Michael Atiyah on math, physics and fun

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.
It's true 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 the miracle.

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 that..

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 bigger picture.
So mathematicians 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.
Geometry originally 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 central role.
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 fun?
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 totally unexpected.
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 so far?
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 Mozart.

So who would it be today?
In mathematics, 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.