14/05/2018

'Gravitational force cannot be ignored,' says Atiyah.

Despite Michael Atiyah's many titles—he is a Fields and Abel Prize winner for mathematics; a former president of the Royal Society of London, the world's oldest scientific society (and former president of the Royal Society of Edinburgh); a former master of Trinity College, Cambridge; a knight and a member of the Royal Order of Merit; and, essentially, the British mathematical pope—he is, however, perhaps more aptly described as a matchmaker. He has an intuition for orchestrating just the right intellectual connections, often involving himself and his own ideas, and throughout his half-century career has bridged the gap between seemingly disparate ideas within the field of mathematics, and between mathematics and physics.

One day in the spring of 2013, for example, while sitting in the Queen's Gallery at Buckingham Palace, awaiting the annual Order of Merit luncheon with Elizabeth II, Sir Michael made a combination for his long-time friend and colleague, Sir Roger Penrose, the great mathematical physicist.

Penrose was trying to develop his "twistor" theory, a path toward quantum gravity that has been in progress for almost 50 years. "I had a way of doing it, which meant going to infinity," Penrose said, "and trying to solve a problem out there and then coming back again." He thought there had to be a simpler way. And right there Atiyah put her finger on him, suggesting that Penrose use a kind of "non-commutative algebra."

“I thought, ‘Oh my God,’” Penrose said. “Because I knew there was this non-commutative algebra that had been there all this time in twistor theory. But I hadn’t thought of using it in this particular way. Some people might have just said, ‘That’s not going to work.’ But Michael could see immediately that there was a way to make it work, and exactly the right thing to do.” Given where Atiyah made the suggestion, Penrose dubbed his improved idea “palace twistor theory.”

This is Atiyah's power. Roughly speaking, he spent the first half of his career connecting mathematics to mathematics, and the second half connecting mathematics to physics.

Atiyah is best known for the "index theorem," conceived in 1963 with Isadore Singer of the Massachusetts Institute of Technology (and aptly named the Atiyah-Singer index theorem), connecting analysis and topology—a fundamental connection that proved important in both mathematical fields, and later in physics as well. Largely for this work, Atiyah won the Fields Medal in 1966 and the Abel Prize in 2004 (with Singer).

In the 1980s, methods derived from the index theorem unexpectedly played a role in the development of string theory—an attempt to reconcile the realm of general relativity and large-scale gravity with the realm of small-scale quantum mechanics—particularly with the work of Edward Witten, a string theorist at the Institute for Advanced Study in Princeton, NJ. Witten and Atiyah began an extensive collaboration, and in 1990, Witten won the Fields Medal, the only physicist to win the award, with Atiyah as his runner-up.

Now, at 86, Atiyah hardly lowers his standards. He is still grappling with the big questions, still trying to orchestrate a union between quantum mechanics and gravitational forces. On that front, ideas are arriving fast and furiously, but, as Atiyah himself describes, they are still intuitive, imaginative, vague, and clumsy.

Still, he is enjoying this state of flowing creativity, energized by his busy schedule. In pursuit of these current lines of inquiry and contemplation, last December he gave two consecutive lectures on the same day at the University of Edinburgh, where he has been an honorary professor since 1997. He is keen to share his new ideas and, hopefully, attract followers. To that end, in November he organized a conference at the Royal Society of Edinburgh on “The Beauty of Science.” Quanta Magazine sat down with Atiyah at the Royal Society meeting and afterwards, whenever he slowed down long enough to answer questions. What follows is an edited version of those conversations.

QUANTA MAGAZINE: Where do you trace the beginnings of your interest in beauty and science?

MICHAEL ATIYAH: I was born 86 years ago. That's when my interest began. I was conceived in Florence. My parents were going to name me Michelangelo, but someone said, "That's a big name for a little boy." It would have been a disaster. I can't draw. I have no talent.

You mentioned that something “clicked” during Roger Penrose’s lecture on “The Role of Art in Mathematics” and that you now have an idea for a collaborative paper. What is this “clicking,” the process or the state—can you describe it?

It's the kind of thing that once you see it, the truth or veracity, it just stares you in the face. The truth is looking at you. You don't need to look for it. It's shining on the page.

Is this usually how your ideas come about?

This was a spectacular version. The crazy part about mathematics is when an idea pops into your head. Usually, when you're sleeping, because that's when you have the fewest inhibitions. The idea floats from the sky, you know where. It floats in the sky; you look at it and admire its colors. It's there. And then, at some point, when you try to freeze it, put it in a solid frame, or make it face reality, then it disappears, vanishes. But it's been replaced by a structure, capturing certain aspects, but it's a clumsy interpretation.

Have you always had mathematical dreams?

I think so. Dreams happen during the day, they happen at night. You can call them visions or intuition. But basically they are a state of mind – without words, images, formulas or statements. It's "pre" all that. It's pre-Plato. It's a very primal feeling. And again, if you try to understand it, it always dies. So when you wake up in the morning, some vague residue remains, the ghost of an idea. You try to remember what it was and you only have half of it right, and maybe that's the best you can do.

Is imagination part of it?

Absolutely. Time travel in the imagination is cheap and easy – you don't even need to buy a ticket. People go back and imagine they're part of the Big Bang, and then ask the question of what came before.

What guides the imagination – beauty?

It's not the kind of beauty you can point to – it's beauty in a much more abstract sense.

Not long ago you published a study, with Semir Zeki, a neurobiologist at University College London, and other collaborators, entitled The Experience of Mathematical Beauty and its Neural Correlates.

This is the most read article I've ever written! It's been known for a long time that a part of the brain lights up when you listen to good music, or read good poetry, or see beautiful photos – and all these reactions happen in the same place [the "emotional brain," specifically the medial orbitofrontal cortex]. And the question was: is the appreciation of mathematical beauty the same or different? And the conclusion was: it's the same. The same part of the brain that appreciates the beauty of music, art, and poetry is also involved in the appreciation of mathematical beauty. And that was a great discovery.

You arrived at this conclusion by showing mathematicians various equations while a functional magnetic resonance imaging (fMRI) scan recorded their responses. Which equation won as the most beautiful?

Ah, the most famous of all, Euler's equation:

It involves π; the mathematical constant e [Euler's number, 2.71828…]; i , the imaginary unit; 1; and 0 – it combines all the most important things in mathematics into one formula, and that formula is really very profound. So everyone agreed that this was the most beautiful equation. I used to say it was the mathematical equivalent of Hamlet's phrase "To be or not to be" – very short, very succinct, but at the same time very profound. Euler's equation uses only five symbols, but it also encapsulates beautifully profound ideas, and brevity is an important part of beauty.

You are especially known for two supremely beautiful works, not only the index theorem, but also K-theory, developed with the German topologist Friedrich Hirzebruch. Tell me about K-theory.

The index theorem and K-theory are actually two sides of the same coin. They started out differently, but after a while they became so intertwined that you can't separate them. Both are related to physics, but in distinct ways.

K-theory is the study of flat space and flat space in motion. For example, let's take a sphere, the Earth, and let's take a large book, place it on the Earth, and move it. That's a flat piece of geometry moving over a curved piece of geometry. K-theory studies all aspects of this situation – the topology and the geometry. It has its roots in our navigation of the Earth.

The maps we use to explore Earth can also be used to explore both the large-scale universe, by going into space with rockets, and the small-scale universe, by studying atoms and molecules. What I'm doing now is trying to unify all of that, and K-theory is the natural way to do that. We've been doing this kind of mapping for hundreds of years, and we'll probably be doing it for thousands more.

Were you surprised that K-theory and the index theorem turned out to be important in physics?

Oh, yes. I did all that geometry without realizing it would be connected to physics. It was a big surprise when people said, "Well, what you're doing is connected to physics." So I learned physics quickly, talking to good physicists to find out what was going on.

How did your collaboration with Witten come about?

I met him in Boston in 1977, when I became interested in the connection between physics and mathematics. I attended a meeting, and there was this young guy with the older guys. We started talking, and after a few minutes, I realized that the younger one was much smarter than the older ones. He understood all the mathematics I was talking about, so I started paying attention to him. It was Witten. And I've kept in touch with him ever since.

What did he like to work with?

In 2001, he invited me to Caltech, where he was a visiting professor. I felt like a graduate student again. Every morning I would go into the department, see Witten, and we would talk for about an hour. He would give me my homework. I would leave and spend the next 23 hours trying to catch up. Meanwhile, he would go out and do half a dozen other things. We had a very intense collaboration. It was an incredible experience because it was like working with a brilliant supervisor. I mean, he knew all the answers before I got them. If we ever argued, he was right and I was wrong. It was embarrassing!

You said earlier that the unexpected interconnections that occasionally arise between mathematics and physics are what attract you most – you like to venture into uncharted territory.

Okay; well, you see, a lot of math is predictable. Someone shows you how to solve a problem, and you do the same thing again. Every time you take a step forward, you're following in the footsteps of the person who came before. Every now and then, someone comes along with a completely new idea and shakes everyone up. To begin with, people don't believe it, and when they do, it leads in a completely new direction. Math has uneven progress. It has continuous development, and then it has discontinuous leaps, when suddenly someone has a new idea. Those are the ideas that really matter. When you get them, they have big consequences. We're about to have another one. Einstein had a good idea a hundred years ago, and we need another one to propel us forward.

But the approach has to be more investigative than directive. If you try to direct science, you only lead people in the direction you told them to go. All science comes from people noticing interesting parallel paths. You need to have a very flexible approach to exploration and allow different people to experiment with different things. Which is difficult, because unless you jump on the bandwagon, you don't get a job.

Worrying about your future, you have to stay in line. That's the worst thing about modern science. Luckily, when you get to my age, you don't have to worry about that. I can say what I like.

Are you currently experimenting with some new ideas in the hopes of breaking the deadlock in physics?

Well, you see, there's atomic physics—electrons, protons, and neutrons, all the things that atoms are made of. At these very, very, very small scales, the laws of physics are the same, but there's also a force that you ignore, which is the gravitational force. Gravity is present everywhere because it comes from all the mass in the universe. It doesn't cancel out, it doesn't have a positive or negative value, it all adds up. So, however far away black holes and galaxies are, they all exert a very small force on the universe, even on an electron or proton. But physicists say, “Oh, yes, but it's so small that you can ignore it; we don't measure things that small, we get by perfectly well without it.” My starting point is that this is a mistake. If you correct this mistake, you get a theory that is much better.

Now I'm looking again at some ideas from about 100 years ago that were dismissed at the time because people couldn't understand what the ideas were trying to show. How does matter interact with gravity? Einstein's theory was that if you put some matter in, it changes the curvature of space. And when the curvature of space changes, it acts on matter. It's a very complicated feedback mechanism.

I'm going back to Einstein and [Paul] Dirac and looking at them again with fresh eyes, and I think I'm seeing things that people haven't seen. I'm filling in the holes in history, taking into account new discoveries. Archaeologists unearth things, or historians find a new manuscript, and that sheds entirely new light. So that's what I've been doing. Not going into libraries, but sitting in my room at home, thinking. If you think long enough, you get a good idea.

So you're saying that gravitational force cannot be ignored?

I think all the difficulty physicists have had is because they ignored this. You shouldn't ignore it. And the point is, I believe that mathematics becomes simpler if you include this. If you leave it out, you make things more difficult for yourself.

Most people would say you don't need to worry about gravitation when looking at atomic physics. The scale is so small that, for the kind of calculations we do, it can be ignored. In a way, if you just want answers, you're correct. But if you want to understand, you've made a mistake in that choice.

If I'm wrong, well, I made a mistake. But I don't think so. Because once you choose that idea, there are all sorts of pleasant consequences. The math fits. The physics fits. The philosophy fits.

What does Witten think of your new ideas?

Well, it's a challenge. Because when I spoke to him in the past about some of my ideas, he dismissed them as hopeless, and he gave me 10 different reasons why they were hopeless. Now I think I can stand my ground. I've spent a lot of time thinking, coming from different angles and coming back to him. And I hope I can convince him that there is merit in my new approach.

You're risking your reputation, but you think it's worth it.

My reputation is established as a mathematician. If I mess up now, people will say, "Okay, he was a good mathematician, but at the end of his life he went mad."

A friend of mine, John Polkinghorne, left physics when I was entering it; he went into the church and became a theologian. We had a discussion on my 80th birthday and he told me, “You have nothing to lose; you just keep going and think what you think.” And that’s what I’ve been doing. I have all the medals I need. What could I possibly lose? That’s why I’m prepared to make a bet that a young researcher wouldn’t be prepared to make.

Are you surprised to find yourself so overwhelmed with new ideas at this stage of your career?

One of my sons told me, “Impossible, Dad. Mathematicians do their best work until they’re 40. And you’re over 80. It’s impossible that you’d have a good idea now.”

If you're still awake and mentally alert when you're over 80, you have the advantage of having lived a long time and seen many things, and you have perspective. I'm 86 now, and it was in the last few years that I had these ideas. New ideas emerge, and you pick up pieces here and there, and the time is right now, and perhaps it wasn't right five or ten years ago.

Is there a big question that has always guided you?

I always want to try to understand why things work. I'm not interested in getting a formula without knowing what it means. I always try to dig behind the scenes, so if I get a formula, I understand why it's there. And understanding is a very difficult concept.

People think that mathematics begins when you write a theorem followed by a proof. That's not the beginning, that's the end. For me, the creative place in mathematics comes before you start putting things on paper, before you try to write a formula. You imagine various things, you turn them over in your mind. You're trying to create, just like a musician is trying to create music or a poet. There are no established rules. You have to do it your way. But in the end, just as a composer has to put it on paper, you have to write things down. But the most important stage is understanding. A proof by itself doesn't give you understanding. You can have a long proof and no idea at the end why it works. But to understand why it works, you have to have a kind of instinctive reaction to it. You need to feel it.