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Fixed point formula


More on index theorem and K-theory
Michael Atiyah Mathematician
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I got interested in equivariant K-theory [sic]. You see when I was doing K-theory I got interested in equivariant K-theory before we did the index theorem, I think, through talking with Serre, and he'd, sort of, somehow dealt with groups and equivariant… and he, sort of, suggested to me this would be a good thing to do. So we developed equivariant K-theory quite early on. In fact when Graeme Segal was my student, one of the first things I got him to work on was really the equivariant K-theory, because that was around. I may be a bit, I may be confused now as to the exact order of these things, but certainly equivariant K-theory was a seemingly natural thing to do from the topological point of view and had various natural consequences that you can get nice results out of it, and so it was worth us doing by itself. And certainly we therefore knew that if you had equivariant index theorem there were some natural applications, and I remember particularly an application with, you know, in relation to the signature… it was quite clearly an interesting bit of topology.

So we knew there were applications of having these more general theorems which would be interesting, and so it wasn't just for its own sake. On the other hand it also seemed somehow totally natural to do, you know… forced by inner logic that it should be so… and very undesirable to have a proof that didn't work. And also, as things further developed there was a kind of feed backwards and forwards because it turned out that, to… these are further refinements in the index theorem, to prove the equivariant… deal with equivariant K-theory in the best way, you really had to have a good proof of the index theorem to go along with it. There was a kind of zigzag pattern backwards and forwards where different versions of the proof of the index theorem and bits of K-theory helped each other along; until the end they were so mixed up I didn't know whether I was going from left to right or right to left. But it's certainly true that various proofs of the index theorem later on… various proofs of the periodicity theorem for equivariant K-theory, for general non-compact groups, non-abelian groups, required partial use of the analysis, index theorems; not necessarily the full theorem but at least the notion of the index and the construction of indices, families of operators and so on.

So they all help each other along, and I think the process of going from the very first proof of the index theorem to, sort of, the latest, was a 10… 15 year process; various stages of refinement, improvement. But they were all motivated by, it seemed to me, a combination of two factors. First the inherent logic of the situation that you felt it had to go in its more general case, and it ought to do the same way; and secondly that it was pretty clear that in most cases there were interesting examples of things that you could actually compute with these new generalisations, if you wanted to do. And so it was a combination of both factors at different stages really.

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: Jean-Pierre Serre

Duration: 3 minutes, 4 seconds

Date story recorded: March 1997

Date story went live: 24 January 2008