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The L2 index theorem


Refining the eta variant
Michael Atiyah Mathematician
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So that took quite a long time to get that up, and I say, the... the last stages in that, buttoning down the exact contribution of the boundary, and some fine points, that was the last thing I did when I came back to Oxford after we'd, sort of, formulated the problem. We'd got the index of the Dirac operator with the boundary contribution, but proving the exact contribution to the boundaries, particularly the zero eigenvalues on the boundary; that was... that was the bit that had to be finished off.

But, it was, again an interesting collaboration between Patodi with his... his flair for doing explicit computations, coming up with the answer, but not telling you, you know – not in a way that other people can understand. And then Singer and I trying to extend our index formula approach to this thing and also with the… our background; we had this example of Hirzebruch which was part of the motivating procedure, motivating background, why we wanted the formula at all.

So all those fit together, and then eventually, we also had to work out the special case – which came from Hirzebruch's work – where the... the boundaries, a sort of torus bundle over a torus, initially that's just a sort of a torus... two torus bundles over the circle, more generally higher dimensional tori bundles over higher dimensional tori. And then in those... in those cases working out that the boundary contribution, as worked out, as deduced from our general theorem, could be effectively computed and related to the analytical formula coming from the Dedekind eta function, that Hirzebruch had got in the first place. And that was a very beautiful, sort of, story all by itself which took us quite a long time and which we pulled in… Donnelly also came in. Well, we first had the one dimensional case where you have the torus bundle over the circle, and that involves things like real quadratic fields; and then there's the generalisation of that.

And that was a very interesting story where I had to get involved quite lot in the number theory. I remember reading Siegel's Tata lectures on... on this in great detail, and really quite enjoying it. Something I'd never really got into that degree of detail and, you know, real quadratic fields and class numbers and… things that Hirzebruch himself was very good at, and he went on to use it in many other contexts. But I... I was, sort of, trying to connect it up with my work with Singer, and we... we succeeded, and some of it wasn't as elegant as I would have liked. I think that nowadays there may be improvements along that line.

But again it was a story that had several bits to it. The origins were a bit mixed. It wasn't completely obvious how to go about it, or even how to formulate it. By pressing… I think the secret of success here was to... to, not to be satisfied at the first stage, you know, not… we could have said when Patodi had produced this proof of the signature theorem, that's the signature theorem, here it is, here's the proof, write it up; it's a bit messy, but never mind. But we weren't satisfied, we said well this isn't really quite right, it should be more general, it should be more elegant. And we pressed and probed and eventually when we got it, the reward was there; it was a much better theorem, cleaner proof, and with more longer term implications as we've seen with subsequent work with the physics. So it was a case of not being satisfied with having the first proof that comes out and insisting on the kind of proof that you think that ought to be, that's consistent with the elegance and beauty of the problem. And I... I think that's, sort of, in some sense the style I've been trying... I’ve been trying to pursue, and you know, it... it paid off. But sometimes you... you struggle a long time to get things into the shape that you want them. They don't... they don’t come easily the first time round.

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, Friedrich Hirzebruch

Duration: 3 minutes, 31 seconds

Date story recorded: March 1997

Date story went live: 24 January 2008