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Collaborating with physicists


Euclidian version of twistor theory
Michael Atiyah Mathematician
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I remember Roger Penrose coming to me one day and saying – it was probably with one or two other of his students – saying they'd... they’d realised that in the positive definite case, things were a bit different and that you had lines through every point in the space and... and so on. And you know, could I... could I understand this? Could I help them to check it out? And was it... was it true, and so on. So that's when I went away and thought about that and I realised that it was... was this nice quaternionic fibration and it was all very classical and you could understand it very well. So I explained it to them, you know, and that was... that was fine. So I... I think… yeah, I've forgotten exactly the order, but I think that was separate, because really it was independent, it was to do with the geometry of the space, not to do with the bundles on the space. And I remember that they'd... they’d realised this themselves and they'd asked me.

And then... and then we had… on top of that we had to put the Richard Ward story together. But it was... it was an interesting case because they... they kept coming in from their end, they... they knew things in terms of their own way of calculating the computation, but somehow they – not being trained to think of it in terms of global pictures and homogeneous spaces and fibrations – they didn't have kind of the facility with... with these global techniques that people like us, trained in the other direction, had.

So it was a very fruitful dialogue. They would come along with… they'd done the work, they knew… they had the good ideas coming from the physics background and... and, I mean I think, that interesting fibration is one I think would never have occurred probably to an algebraic geometer. It's a real fibration and it seems a bit bizarre. A topologist might have used it for purely topological purposes – they do – but I don't think algebraic geometers would have really thought of it, and so it was... it was somehow being suggested from their... from their particular point of view. But once you saw it, then it was all… it all hung together very, very straightforwardly. So this was an example where Dyson's… eventually I would understand what twistors are all about. Took... took, you know, years, but it was really correct.

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: Roger Penrose, Freeman Dyson

Duration: 2 minutes, 9 seconds

Date story recorded: March 1997

Date story went live: 24 January 2008