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My mathematical growth


Topology and K-theory
Michael Atiyah Mathematician
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[Q] Working on the basics of what came to be topological K-theory, how did this arise?

Well that really arose, I think primarily it came from my attendance at the meetings in Bonn, the annual meetings, because the way their meetings were organised were very, very good. There were no fixed programme. Hirzebruch would just invite the people he liked, or the people he knew in the areas that he was interested in, mainly young active people. And he was very successful at doing this organisation, so lots turned up, became a very popular event and every year, more or less, you would discuss whatever was the most interesting topics going around. And more or less by general agreement you would have lots of talks on this or that, and so by attending these I was exposed to constantly, you know, changing developments and new things came along, I would get interested in them.

And so, in those early years in Bonn when I went, I suppose the late ‘50s, I was hearing of all these developments. On the one hand there was the topologists were doing Morse theory, there was Smale's work and others, classification of manifolds, Milnor's work on different aspects of topology. I mean, there was topology. I learnt a lot of topology… it was all around, topology was going through a big period. I had some topology originally, I learnt more topology in Princeton. I learnt more topology in Bonn. I didn't regard myself as a topologist but I picked it up, it was all around. And there were a lot of things happening. And Milnor was a regular attender also at Princeton. Then there were all the manifold classification, differential geometry, and all the theorems that Hirzebruch himself had proved about… which emerged from algebraic geometry, but, the theorems that various combinations of characteristic classes were integral. Which was a kind of very striking, formal consequence of the algebraic geometry, but applied through Thom's cobordism theory and others into differential manifolds. So I think it was Hirzebruch himself who really shifted the applications of these ideas involving topology in characteristic classes from the algebraic geometry side into the differential manifold side.

And so for many years that was a main theme in Princeton, I would say, so inevitably I got drawn into that and realised that manifolds and characteristic classes and topology were things you should know about and be interested in, and when the opportunity came if something happened I, sort of, got involved. And so my work in… early work in K-theory started off that way because there were these, all these ideas of characteristic numbers and divisibility theorems, my friends the algebraic topologists were all playing around with trying to prove results. Ioan James at Oxford told me about some problems that he had, and Grothendieck was giving all his lectures on his generalised Riemann-Roch theorem, and at some stage I put some of these things together and discovered that you could get some nice results by interplaying one against the other. And then the Bott periodicity theory hit the headlines about the same time, so around about a period of a year or two there was a whole range of new contributions from different people.

By just being around in the right place at the right time, and talking to and being interested in, you were inevitably sucked along certain directions. I didn't actually make any conscious move to say I was going to move from algebraic geometry into topology; it is just the subject naturally had evolved and the centre of activity was mixing these things up. And so before I knew where I was, I was having to think of myself more as a topologist. So when I started doing K-theory, I suddenly became, you know, a kind of topologist for a while. Topology is a branch of K-theory and K-theory is a branch of topology and I had to learn some more topology and K-theory became a tool for topologists to use. And so I… Henry Whitehead was at Oxford, he invited me over to give talks and was clearly impressed with what I was doing, and so for a while I sort of drifted very close to being a topologist; not really a bona fide card-carrying topologist, but sort of, you know, pseudo.

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: Mathematische Arbeitstagung, Bonn, Princeton, Oxford, Friedrich Hirzebruch, John Milnor, Henry Whitehead

Duration: 4 minutes, 14 seconds

Date story recorded: March 1997

Date story went live: 24 January 2008