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Lacunas and hyperbolic equations


Bridging the gap between mathematics and physics
Michael Atiyah Mathematician
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When I was at Princeton I remember talking with Freeman Dyson, who was a colleague there at one stage, and I think I'd been consulted or I knew that there was a possibility of getting Roger Penrose back to Oxford, so… and I was going back. So I talked to Freeman Dyson about him. And I remember him saying, yes, he'd been... he’d been very impressed with some of the stuff that Roger Penrose had done about black holes and all that. But he said, ‘All this twistor stuff that he does. I don't... don’t really understand what it's about. Perhaps... perhaps you’ll... perhaps you'll understand it’. You see? I mean I didn't know anything about that either at the time, but of course that's what happened.

When I went back to Oxford, Roger Penrose turned up and we… since we'd been former students together, we knew each other and although I hadn't seen anything much of him over the last intervening 15, 20 years, I knew roughly what he'd been doing. But he'd moved off into physics and relativity, but... but, still he had the same common background – geometry. So when I came back we got, you know, it was rather natural for us to talk and for me to try to find out what all twistor theory was about, and... and you know, he talked to me about the geometry. And so that was actually a very fruitful interaction, because I remember when I first... when he first started to explain to me what he was doing. He'd come in and say that he was doing these complicated contour integrals with… and there were singularities and... and he... he had formulas which would work, but he didn't quite know the right way to... to treat them. And at some stage when he explained it to me I realised that what he was… really needed was... was sheaf cohomology, as a way, and that… once one realised that then lots of what he was doing immediately acquired a sort of a much more natural interpretation […] machinery. And so it was really quite interesting, because within a short period of time, once I'd realised this and explained it to him, and also his students, they immediately became total converts, you know, and, you know, within a few weeks they were all... all beavering away producing complicated spectral sequences and everything and... and going off, you know.

From then onwards it never stopped. So it was a, sort of, piece of good fortune in a way that we were able to converse, because the gap between mathematicians and theoretical physicists, even to those like Roger Penrose, was quite big. But because he'd had, we'd had the same background, and of course his twistor theory had… arose out of the common background of projective geometry which we'd been studying. So you know, we start out with a kind of common foundational dictionary. We knew what it was about. I had to understand what he was doing with all these complicated Feynman integrals, and then I had to explain to him about sheaf cohomology, but in a way it was, you know, it was inevitable I suppose, that... that if... if we were in the same place and we talked to each other, that... that would happen sooner or later. And... and it... it did and it was very, very productive, well both for him and for me and for a lot of other people. And so… but it was... it was a good... good, yes, good piece of good fortune and I think… I've forgotten the exact order, but certainly he was, he must have been appointed just as I was coming back… in fact I've forgotten which of us actually ended up coming back first; and it was a very, very productive relationship.

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: Princeton, Oxford, Roger Penrose, Freeman Dyson

Duration: 3 minutes, 18 seconds

Date story recorded: March 1997

Date story went live: 24 January 2008