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15 years of index theorem


Fredholm operators
Michael Atiyah Mathematician
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The connection between K-theory and Fredholm operators at the beginning… that was a thing which went back some time. It was the work of Jänich on the one hand, and I'd got a bit involved with the other, and with Kuiper's theorem about the contractibility of the unitary group in Hilbert space, realising that the… that K-theory was somehow the universal object for Fredholm operators, or the space for Fredholm operators was the classifying space of K-theory. That wasn't at all clear at the beginning. You know, when we first started, nothing like that was clear, and I remember partly talking to Palais on the one hand, and others, and this picture had emerged; and at the end of the day it was extremely simple and very elegant, and I included that in my K-theory notes.

So the Fredholm operators ending up as the classifying space for K-theory was a very good thing because, you know, if you're talking to analysts and they wanted to know what K-theory was, you would just explain that if you studied families of Fredholm operatives it was the only topological information all encoded in the K-theory. So for an analyst, immediately… I remember when I gave Harvard lectures once, Gleason was in the audience and came to these things, and you know, he immediately, sort of, really… he'd been working on Hilbert spaces and operators and he found this very, you know, appealing. So it was a very important link to the... to the analyst because if you tried to explain what K-theory is from the point of view to traditional topology – vector bundles and classifying spaces and such – you know, they get no idea, they're lost and think that it's a lot of abstract complicated stuff. But starting off from the index, abstract index of operators, it's an extremely obvious thing for an analyst, a functional analyst, and later on that became part of general K-theory.

So that was a kind of a route, a narrow route between the algebraic topology on the one side and the functional analysis on the other, and going through that route you got a good bridge. Well that was just general relationship between Fredholm operators and K-theory; then of course when you got into the real theory, then you got a much more subtle story because there were all these different other bits you wanted to bring in. And also there was… even in the complex case there was the relationship between Fredholm operators and self-adjoint Fredholm operators, which was the model for sort of the odd K-groups, which is very nice when you see it and that gives you a nice interpretation with things like the spectral flow and things on the circle, which are also nice from the point of view of analysis. So the… the use of Fredholm operators and self-adjoint Fredholm operators in relation to families and K-theory, was a nice thing which… and I learnt a lot of operator theory from Singer and so we worked a lot [on] this together.

And then the last stage of that really was this thing you referred to, when we did the real case, and then you have eight different cases and it gets quite delicate. And then you bring in the Clifford algebras as a systematic way of treating all… all the different classes of Fredholm operators in the different dimensions. And then you actually… well then, we did in that paper, produce another proof of the Bott periodicity theorem by a totally different route from anything else done before, which was a kind of big surprise to us because we weren't planning to do that, we were just studying it for the purpose of our applications. We suddenly realised at the end of the day, ‘Oh gosh, somehow we've got the Bott theorem and didn't use it. Where did it come from?’ And so we suddenly realised we had a proof, and so we... so we produced this other proof of the Bott periodicity theorem which incorporates all these different eight cases and things, which is, in a way, quite natural from the analytical point of view. And I think Dan Quillen has subsequently... subsequently used that for other purposes. It’s a different… really is a different approach to the theorem and has its own merits in going off in certain directions.

So that… but I think that it was that whole work on that collection of topics was a combination where Singer's background in… in operator theory especially was obviously, you know, very helpful. I was able to… I learnt enough, when you get to learn it, you didn't need that… wasn't that difficult. But you had to worry about the continuous spectrum and the singular spectrum and… and things like that. But it worked out very elegantly and I quite enjoyed that paper too, because again it's one of these things where quite tricky and quite subtle and slightly unexpected results come out.

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: Klaus Jänich

Duration: 4 minutes, 33 seconds

Date story recorded: March 1997

Date story went live: 24 January 2008