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Vector bundles


Starting to understand harmonic integrals
Michael Atiyah Mathematician
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Everybody knew there was this book which he was famous for – Theory of Harmonic Integrals [sic] – but we didn't study it, and even when I started working with him he gave… he started me off on characteristic classes sort of things, work that he was doing with Allendoerfer and Chern – Chern classes and things like that. But pretty quickly I suppose I had to pick up the harmonic integrals theory at least to know the results, know the outlines. But there was never a course on harmonic integrals, it would have been much too difficult, and it wasn't regarded as, sort of, a suitable book even for Part III students. But I must have been exposed to it in my first year of research. I pretty soon had to learn, know all about harmonic forms, and when sheaf cohomology came along which it did in my first year, I had to know. So I'm not quite sure how I picked it up, but it was known. Mind you the results are easy and the proofs are difficult, so you can learn the formalism. I did actually read through it some stage in part, but I didn't learn it directly from him and there were no courses.

So you pick these things up. My memory is a bit vague now as to exactly how, but… and later on, after a while, it was… clearly it was central to the subject and you have to relate to sheaf cohomology and so on, and so by that time you had to know more about it. And later on, I learnt a bit more of the analysis. Early stages I didn't know the analysis, but later on when you got more involved I brushed up on the analysis and learnt about it. The early stages, you just know the applications, the algebraic geometry side and things like that.

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: harmonic integrals, self-taught subject

Duration: 1 minute, 32 seconds

Date story recorded: March 1997

Date story went live: 24 January 2008