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The mod 2 index theorem


Delicacy of factor 2
Michael Atiyah Mathematician
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Among the interesting examples of using group actions are involutions, just groups of order 2, and amongst the natural operators which he uses is the Dirac operator. But because spin itself is a slightly subtle concept, you get a factor 2 turning up in there. And so studying involutions on spin manifolds is quite tricky. You've got to think rather hard not to make mistakes and get the right… and you get elements of order 4 instead of elements of order 2. So I mean, I wrote this paper which applied the involutions to spin manifolds and we got a lot of quite nice results out of it, but there was some particular point I remember, you're... you’re right, where on the one hand… we missed one factor through somewhere, and then later on by further improvement you put it back in somewhere else. There were several places where these factors of two are very delicate.

I quite enjoyed working on that paper and that thing because precisely this delicacy of powers, factors of two, and the notion of spin and the different kind of involutions on spin manifolds, the dimension of the fixed point sets are… their dimensions vary, congruence modular IV vary as you can have different cases. And it all got rather, very neat and then sort of refined. And then when you take it down in examples, it all comes out very nicely. And that was a case where I think the results weren't at all known, you know, by anybody, and it came out of this machinery and doing it very carefully you got these very precise results out.

So it was very satisfying in a way, sort of very clean as a consequence. But that was… you really needed, yes, you really needed to know the equivariant theory, but also for the subtleties you needed to know the real theory as well. So everything, you know… it was subtle in lots of different ways, and so there's plenty of scope for a few odd, odd slip-ups here and there.

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: Dirac operator, spin manifold, real theory, equivariant theory

Duration: 1 minute, 54 seconds

Date story recorded: March 1997

Date story went live: 24 January 2008