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The Fields Medal


15 years of index theorem
Michael Atiyah Mathematician
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After the earlier work, including all these refinements and extensions, you know, all directions… and that might have been somehow a natural end. Then of course all the new stuff came in with the... with the Gilkey-Patodi work showing that you had these precise local formulas in the context of Riemannian geometry. So then... then that inaugurated an entirely new era which Bott and I worked on. We had… Patodi came over and worked with us at Princeton and we... we spent a long time trying to understand Patodi's proof which was very difficult to understand, the computation. We also tried to understand Gilkey's proof which was a bit messy computationally. So we ended up by producing this invariant theory approach to Gilkey's proof which we... we understood and was, we thought reasonably elegant, and that was... that was one... one story.

And then of course that led on, later on, to the work with Patodi on the… on the manifolds with boundary and that was really yet another stage. So these really are quite… I suppose the thing evolved through natural problems. The first lot of problems which are sort of, people discussed before; and then... then there's the index, the local version of index theorem understanding the Patodi-Gilkey stuff, and that being used as a kind of stepping stone to the question about manifolds with boundary and the eta invariant was yet a third stage. And each of these really required quite a bit of extensive development and... and can be refined in lots of ways.

And then, of course, after that we ran into theoretical physicists and that's the beginning of another story, and the physicists, they came in with their own improved methods and so on. And so at that stage I dropped out of the further… there were other proofs of index theory coming in using supersymmetry and so on, and that's where I certainly wasn't... wasn’t following the version; but I think there were a lot of interesting things happening. But that, as I say, I certainly was moving on to something else, but for a long time… I moved on not just because they were trying to improve the theorem, but because there were sort of different generalisations of the theorem which needed to be addressed and, you know, clearly important, so that kept me. But I think that must have taken… well, I… I remember now, when I came back from Princeton to this country which was in the beginning of ‘73, that was when I was working on the... the work on the manifold of the boundary with Patodi and Bott, and Singer rather. And I remember that in fact there was rather a critical stage because, when you… I left the institute, you know, I had a cushy job there and why should I leave the institute and come back to Oxford?

And, you know, psychology… it so happened when I got back. I spent a couple of weeks working on this problem which we'd got to a stage where it, you know, we were pretty close, but we hadn't actually got… and I spent a couple of weeks cracking the last bits that we needed to actually do the proof, you know. And I remember feeling very, very chuffed writing off to Patodi and Singer saying here… here it is now. And that was very important for my psychology, I mean, I’d come back to Oxford and do this work and things still go well. So that... that was, so anyway that was ‘73, so I... I was… so that's more than 10 years since index theorem first started and that was when we were doing manifolds with boundary and it... it carried on a bit further beyond that. So I think it must have been a good 15 years from beginning to end when it was sort of evolving in one stage or the other.

Eminent British mathematician Sir Michael Atiyah (1929-2019) broke new ground in geometry and topology with his proof of the Atiyah-Singer Index Theorem in the 1960s. This proof led to new branches of mathematics being developed, including those needed to understand emerging theories like supergravity and string theory.

Listeners: Nigel Hitchin

Professor Nigel Hitchin, FRS, is the Rouse Ball Professor of Mathematics and Fellow of Gonville and Caius College, Cambridge, since 1994, and was appointed to the Savilian Professorship of Geometry in October 1997. He was made a Fellow of the Royal Society in 1991 and from 1994 until 1996 was President of the London Mathematical Society.

His research interests are in differential and algebraic geometry and its relationship with the equations of mathematical physics. He is particularly known for his work on instantons, magnetic monopoles, and integrable systems. In addition to numerous articles in academic journals, he has published "Monopoles, Minimal Surfaces and Algebraic Curves" (Presses de l'Universite de Montreal, 1987) and "The Geometry and Dynamics of Magnetic Monopoles" (Princeton University Press, 1988, with Michael Atiyah).

Tags: Vijay Kumar Patodi, Isadore Singer

Duration: 3 minutes, 29 seconds

Date story recorded: March 1997

Date story went live: 24 January 2008