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Delicacy of factor 2


Fixed point formula
Michael Atiyah Mathematician
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The thing I did with Bott on fixed point formulas, at one time we called… we had seminars… we even called it the Woods Hole fixed point theorem because we had this conference in Woods Hole, which is this place off, you know, Cape Cod where they have conferences and summer meetings – it's rather an attractive place. And there was this conference which Bott and I were included at, although it mainly, if I remember correctly, most of the people there were algebraic geometers. I think it was probably officially called ‘A Conference in Algebraic Geometry’ [sic]. Certainly Cassels was there and Tate was there and people like that and Shimura was there, so it was that sort of branch of algebraic geometry, number theory. And I think we were tagged on because we were friends of the people who organised it, and they thought it useful to have these chaps around.

So, and then during that meeting Shimura told us he had this conjectured formula for… on algebraic curves, for automorphisms with, sort of… so effectively what he had was a particular case of the fixed point theorem for automorphisms of curves, acting on differentials, holomorphic differentials, something like that. And he conjectured – he had this conjectured formula which I think he'd worked out in some special, some special cases or something like that – and he conjectured this formula. So then we got interested in that, and sort of, and that led us to, sort of our general fixed point formula during the course of that meeting, it was already… it worked out, and there were various interesting stages in that proof.

First of all, when we… when we started the formula, the general formula which would fit in, without knowing how to prove it, we thought we'd better check it out with some examples. So we took the simplest example we could think of, which actually was an elliptic curve with complex multiplication. And so we gave the experts, Tate and people like that, we said, ‘This formula should be true’, and so they went away and came back and calculated, they said, ‘No, it's wrong’. So you know, if we'd believed them we would have given up. But anyhow we didn't believe them, we thought this formula was too good to be, you know, to be wrong.  And so, you know, we insisted and I think, I don't know whether then or later they did it again and got the right answer, but certainly originally they told us it didn't work.

And then the other thing was, the Hermann Weyl formula for characters of representations, as in terms of the fixed point formula. Bott being expert on Lie groups and things, it was quite natural to think of that formula as a special case. We knew the work of Borel and all that, that you should study flag manifolds and things. But at first, when we looked at that, again it didn't seem… it didn’t seem to quite work, something… and I remember going, this was the case I remember going for a walk with Bott round this dusty track in Woods Hole, quite a long walk, and it was in the course of that walk that it somehow all clicked into place, and I remember explaining to Bott, you know, how it worked, I mean, and convincing him that it really did hang together as we’d sort of hoped. You know, we knew that it was a special case, it ought to work out, ought to reproduce the formula, but at the beginning we didn't understand it sufficiently well, it didn't seem to click.

So by the time we'd finished the Woods Hole meeting we, we knew that was a special case, we knew how… and then Bott wrote a paper subsequently where he kind of proved that special case by using group representation theory which you can do, so to speak. So we knew that was… so we had the conjecture coming just… just that conference, only for a matter of a few weeks. So from Shimura's initial starting point to our formulation of it, checking it out in examples, we had the conjecture by the end of that, and then it was a question of finding the proof again.

And we ran a seminar, when I visited Harvard, I think the second time probably, which was going through that, giving proof. Well what Bott and I had to work out, this approach to do with transversal intersections and the bit of analysis that goes with that. It was bit, sort of… I think we probably made a bit of heavy weather with it and subsequently people, you know, better at it, more skilled in analysis, got it a bit more cleanly. But it was, we were exploring, sort of, new territory as far as we were concerned, it wasn't pure topology, it was a bit… and, so we had to get a bit more into the analysis and operator theory and so on. So we had to learn... we had to learn a bit more analysis to do it properly and then work it out, but I don't think it took that long by the time we… from the time we started in the summer to when we’d worked it out.

But it was an interesting experience and of course subsequently… that was... that was the case of isolated fixed points, and then with maps which are not necessarily part of a compact group action.  Then subsequently with Graeme Segal we worked out the case of the compact group actions and arbitrary fixed point sets by more standard algebraic topological methods with the equivariant K-theory side. So those two... those were two extremes and I suppose we must have got the fixed point theorem with Bott first, that came first, then that suggested the general formulas, that there should be one. So probably the equivariant K-theory motivation was probably spurred by the fact that we needed to have that in order to get… to generalise the fixed point formula we'd got before. I think that must have been that order.

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: Woods Hole, Cape Cod, Raoul Bott, Graeme Segal, Goro Shimura

Duration: 5 minutes, 23 seconds

Date story recorded: March 1997

Date story went live: 24 January 2008