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Working together in mathematics

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Mathematics at Princeton
Michael Atiyah Mathematician
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Well the topic that the people were working on at Princeton in retrospect, I suppose one could see that they were all parts of some big interactive puzzle. Bott's work on Lie groups and Morse theory has had a long subsequent history, impact on things, and I… from that I learnt a lot about Lie groups and their details. Serre's work on algebraic geometry and sheaf cohomology was being developed in a particular direction and developed subsequently in other ways.

I think the other big strand was the work of Kodaira and Spencer which was the other part of the sheaf cohomology from the analytical point of view, which they were doing kind of in parallel to Serre. And Kodaira was giving lectures at the university on the detailed applications to the poly-algebraic surfaces – currents and surfaces and the things which eventually led to his proof of the fact that Hodge manifolds are algebraic. So he was applying the sheaf cohomologies in great detail to theories of the families of curves and surfaces and so on. And then there was Hirzebruch's, and he was giving lectures that he'd sort of… the Riemann-Roch theorem and all the background to that. So those were the kind of corpus of ideas that were sort of Lie groups, sheaf cohomology, algebraic version of it, analytical version of it, and all the whole formalism of characteristic classes, which Hirzebruch put together. And of course there was the paper of Borel and Hirzebruch on characteristic classes which emerged later, which became the kind of lingua franca of everybody working in the area.

So it was really a whole collection of interrelated ideas which have all somehow come to maturity more or less around the same time, and there were all these young, very active people doing them. That really made the whole intellectual climate… I mean, there may be other things going on as well, but that's the bit that sort of impacted on me and a lot of people involved with it. And I think it was the dominating factor at the institute at that time, and many young people, other visitors took part. So it was learning about that and subsequently following them up in different directions and collaborating with them was where everything… much of my subsequent work emerged from. And it was an extremely stimulating period, and a rather unusual one in some sense in that all these different strands I think really, sort of, came together round about that time and all interacted one with the other. And of course it was a few years later, I guess, that André Weil came to the institute as well, but Borel was at the institute… I'm not sure then, maybe a little later. So, my memory of  time is a bit confused, and I went to Princeton many subsequent occasions as well, but that first time, these were the kind of general… this was the general level of mathematics that was going on.

There were other people involved, of course there was Deane Montgomery who represented the sort of topology of the more hard, classical type and people who proved the Hilbert's sixth problem and that sort of thing. And of course, there were people like… there were the hard analysts like Beurling and Selberg doing the number theory, but I had less to do with those, and I don't think they, to be fair, had the same kind of big school.

In fact the interesting thing about the institute in those days was that the people who had the biggest effect weren't the main professors, they were these visitors, I mean the young chaps, and they were the people who… plus a few of the people at the university who were giving courses. We used to go backwards and forwards between the institute and the university, constantly attending seminars there and coming back. I remember this car load of people, Bott and Singer… to go to these, three days a week we'd go to hear Kodaira, Hirzebruch and so on. It was a very, very stimulating period altogether, and I think quite a remarkable period in the development of mathematics in the post-war era. There was the big development in France, in Paris, with the Cartan [...], but Princeton was where much of it was transported to and added to by the people coming from other directions.

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, Jean-Pierre Serre, Armand Borel, Kunihiko Kodaira, Doanld Spencer, Friedrich Hirzebruch, Deane Montgomery

Duration: 4 minutes, 12 seconds

Date story recorded: March 1997

Date story went live: 24 January 2008