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Analysis with Singer


Russian contributions to the Dirac operator
Michael Atiyah Mathematician
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[Q] I had heard that one of the Russian formulas had a missing n-factorial or something because… is that right?

Well the Russians, you see, were only… they tended to consider the simple cases. They weren't concerned with the general cases. You just started off with low dimensional examples and they didn't worry with non-trivial tangent bundles, so they just had a formula involving the degree; which is the top formula that's showing character formula, and that wouldn't… the factorial, they would miss it. I mean it wouldn't be there in the first instant, but you don't put it in until you know the… but they might have thought there was some constant and wouldn't know what it was. They would know that the topological information would be contained in this degree, because in the simpler case they considered, that was the only topological information there was. And then they would guess that this was the answer, or there might be a factor, but they wouldn't have known the formula to expect, certainly, not in general, and they started off in dimensions two or three where the factorials aren't that big.

So the Russians formulated the problem, they hadn’t got… and they'd started working on the analysis of pseudo-differential operators, deforming differential operators through pseudo-differential operators so that you then begin to apply the continuity argument to say it had to depend on some symbol… so they got the general framework. But what they were totally missing was the machinery of the general algebraic topology side, which you had to plug into it. Now we had the… we came from the other end completely, in some sense we had to learn our analysis, but it was easier for us to learn the analysis than for them to learn all the elaborate machinery which I suppose was what Hirzebruch had contributed. So it was having the right friends at the right time that did it.

Eminent British mathematician Sir Michael Atiyah 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: Russian mathematicians, algebraic topology, continuity

Duration: 1 minute, 47 seconds

Date story recorded: March 1997

Date story went live: 24 January 2008