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Topology and K-theory

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Working together in mathematics
Michael Atiyah Mathematician
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My collaboration with Hirzebruch – he organised the Arbeitstagungs – and I went every year for 30 years, and spent 10 days, and there were all these meetings and I was one of the early visitors there and so we got to know each other very well. And our mathematical interests were very close, but slightly complementary, so we were always, you know, in touch with what the other person was doing. And at these conferences everybody else doing interesting things would be there, and so inevitable that at various stages our paths would cross so closely we would actually collaborate. And that developed later.

With Bott I didn't see him quite so often… the early stages. Subsequently I spent visiting terms in Harvard with him, and he came and spent… he spent a year visiting in Oxford as well. Bott came from a slight… his direction was Lie groups and Morse theory, which is actually a little further from me than Hirzebruch who had worked in algebraic geometry, and so I suppose it took a little longer to establish the natural links with him. It's very difficult to tell actually.

Collaboration with another mathematician is a combination of personality and mathematical interest and expertise, and you know, you become friends with people and you collaborate with them, and one thing leads to another and they reinforce each other. It's very hard to remember what exactly what started it off, but no doubt spending a year or two in the same place when you're young people, getting to know each other well, you know, it provides a framework on which other things can follow. And you have a common… you learn… in each case I found that these are people with mathematical backgrounds different from mine – sometimes quite substantially different – but merging in a common area. And when you come to collaborate, the other person brings the other expertise in and you learn from him and he learns from you; and after many years you sort of, you know, you more or less know the same stuff, because you taught it to each other.

So that happened over a long period with all of these really – different degrees – and it's very hard in retrospect to find out exactly what triggered what. But it started off by our common stay at the institute for a long period of time, and then visits to Bonn, the Arbeitstagung, was a major annual event, and many of the other people like Bott would be regular visitors to Bonn as well. And then every… I took a sabbatical every… as soon as, whenever I could, you know, every two years I would take a term off, and I would go to Princeton or Harvard or wherever it was, and so you kept in touch by fairly extended periods, and I think that enabled you to keep these collaborations going. And sometimes you would just work in parallel, you would both be interested in the same thing and you’d find out what they were doing… they were doing but not necessarily collaborating; and then every now and again things would get sufficiently close that you actually sort of got together and made a team.

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: Mathematische Arbeitstagung, Bonn, Friedrich Hirzebruch, Raoul Bott

Duration: 2 minutes, 59 seconds

Date story recorded: March 1997

Date story went live: 24 January 2008