Sir Michael Aityah has made it to ninety and the world is better for it. One of the greatest mathematicians of the 20th century, Aityah (b. 1929) has made fundamental contributions to many areas of mathematics, especially in topology, geometry and analysis. From his first major contribution (topological K-theory) to his more recent work on quantum field theory, he has worked tirelessly in explaining the insights of mathematics to physicists and vice versa, a cross-fertilization that continues to be fruitful for both sciences.
Atiyah is perhaps best known for the Atiyah-Singer Index Theorem (1963) for which he was awarded the Fields Medal in 1966 and the Abel Prize (with Isadore Singer of MIT) in 2004. The index theorem is one of the great landmarks of 20th century mathematics, profoundly touching some of the most important recent developments in topology, differential geometry and quantum field theory.
Why is the index theorem so important to science? Physicists describe the world by measuring quantities and forces that vary over time and space. The rules of nature are often expressed by formulas involving their rates of change – the so-called differential equations. Such formulas may have an “index”, the number of solutions of the formulas minus the number of restrictions which they impose on the values of the quantities being computed. The index theorem calculates this number in terms of the geometry of the surrounding space. In effect, we can find out how many solutions a system has just by knowing some simple pieces of information about the shape of the region, even if we can’t solve for the analytical solution themselves.
For a glimpse of what the index theorem precisely states, check out this summary note by Professor John Rognes (Oslo University) at: http://www.abelprize.no/c53865/binfil/download.php?tid=53804
In the 1980s, methods gleaned from the index theorem unexpectedly played a role in the development of String Theory which attempts to reconcile the macroscopic realm of gravity with the microscopic realm of quantum mechanics. Inspired by the index theorem, Edward Witten, the leading string theorist who is at the Institute for Advanced Study in Princeton, N.J., began an extended collaboration with Atiyah and also made fundamental mathematical contributions of his own (in 1990, Witten was awarded the Fields Medal, the only physicist ever to win the most prestigious prize in mathematics).
INTERVIEW WITH SIR MICHAEL ATIYAH BY QUANTA MAGAZINE (2016)
On K-theory and the index theorem:
K-theory is the study of flat space, and of flat space moving around. For example, let’s take a sphere, the Earth, and let’s take a big book and put it on the Earth and move it around. That’s a flat piece of geometry moving around on a curved piece of geometry. K-theory studies all aspects of that situation — the topology and the geometry. It has its roots in our navigation of the Earth … I did all this geometry not having any notion that it would be linked to physics. It was a big surprise when people said, “Well, what you’re doing is linked to physics.” And so I learned physics quickly, talking to good physicists to find out what was happening.
On his collaboration with Ed Witten:
“I met him in Boston in 1977, when I was getting interested in the connection between physics and mathematics. I attended a meeting, and there was this young chap with the older guys. We started talking, and after a few minutes I realized that the younger guy was much smarter than the old guys. He understood all the mathematics I was talking about, so I started paying attention to him. That was Witten. And I’ve kept in touch with him ever since.
In 2001, he invited me to Caltech, where he was a visiting professor. I felt like a graduate student again. Every morning I would walk into the department, I’d go to see Witten, and we’d talk for an hour or so. He’d give me my homework. I’d go away and spend the next 23 hours trying to catch up. Meanwhile, he’d go off 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!
On the process of doing mathematics:
People think mathematics begins when you write down 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 to put things down on paper, before you try to write a formula. You picture various things, you turn them over in your mind. You’re trying to create, just as a musician is trying to create music, or a poet. There are no rules laid down. You have to do it your own way. But at the end, just as a composer has to put it down 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 of why it works. But to understand why it works, you have to have a kind of gut reaction to the thing. You’ve got to feel it.
On mathematical thinking and the Subconscious:
“The crazy part of mathematics is when an idea appears in your head. Usually when you’re asleep, because that’s when you have the fewest inhibitions. The idea floats in from heaven knows where. It floats around in the sky; you look at it, and admire its colors. It’s just there. And then at some stage, when you try to freeze it, put it into a solid frame, or make it face reality, then it vanishes, it’s gone. But it’s been replaced by a structure, capturing certain aspects …”
Not too long ago you published a study, with Semir Zeki, a neurobiologist at University College London, and other collaborators, on The Experience of Mathematical Beauty and Its Neural Correlates.
“That’s the most-read article I’ve ever written! It’s been known for a long time that some part of the brain lights up when you listen to nice music, or read nice poetry, or look at nice pictures — and all of those 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 is it different? And the conclusion was, it is the same. The same bit of the brain that appreciates beauty in music, art and poetry is also involved in the appreciation of mathematical beauty. And that was a big discovery.”