Michael Atiyah - Vector bundles (20/93)

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British mathematician Michael Atiyah (1929-2019) studied in Cambridge where he became a Fellow of Trinity College and later held professorships at Princeton and Oxford. He is best known for his work on the K-theory and the Atiyah-Singer Index Theorem. [Listener: Nigel Hitchin; date recorded:1997]

TRANSCRIPT: I think what Hodge put me on to was the vector bundles and characteristic classes. I read Chern's papers on Chern classes, and there was some work that Allendoerfer and Weil had done. And so… he gave me that direction and that got me started on the characteristic classes and vector bundles.

There was then, secondly, we had a visiting American, a chap called Newton Hawley, who spent a sabbatical in my first year, and I got to know him quite well, very friendly with him, and he was just beginning to work on vector bundles, [...] vector bundles. And… but it was rather ironical actually, it was rather delicate, because what happened was that he was giving – he'd written a paper which had been published in the Proceedings of the National Academy of Sciences on [...] vector bundles. And he explained this paper to me and we discussed it, and then I started working on vector bundles and algebraic geometry, and I soon realised that actually his paper was all wrong – fundamental error – it was absolutely totally fundamental. He'd claimed that the topological classification worked and everything okay, so that all the jumps that we know about didn't happen. So my first paper was a bit embarrassing. I had to sort of have a little appendix pointing out this thing, and Hodge told me I had to tone it down a little bit, you know. So it was… I got friendly with him and he liked me and we got on very well, but my introduction was slightly bizarre that he’d introduced me a bit to the subject and I discovered he'd made this fundamental error and that helped me get started. That was one side.

Then of course the second side direction – that was the vector bundle side – then all the stuff to do with sheaf cohomology, well that was being published at that time, all in the Comptes Rendus notes in Paris, and I don't quite know how, but, I mean, I got on to that. And I used to go into the library every week and there'd be another note by Cartan and Serre and all these people, explaining all this sheaf cohomology, which I sort of, you know, picked up. And I discussed it with Hodge. He was interested too, but he didn't really know… understand it that well, I think, and the topologists didn't really know about it here. But I wrote my… the Smith's Prize Essays that you submit in your first year… well my Smith's Prize Essay which was done in the first year really, was about the application of sheaf cohomology to the classification of Ruled Surfaces. So I obviously had an… I'd seen at that stage a letter that Serre had written about the Riemann–Roch theorem which was going round informally.

So in that first year I must have picked up all about sheaf cohomology and the Kodaira and Spencer work was all coming in pretty fast, and of course by the time of the international congress in Amsterdam in ‘54, which was the end of my second term as research student, by that time, you know, Serre got the Fields Medal and Kodaira also. And Hodge had been over to Princeton a little before that and met Kodaira and Spencer, so I pretty rapidly fully integrated into the sort of information network about these things. But I think it was just, you know, it was good timing. The time I started these things were just taking off and with a bit of help from Hodge and bit of good fortune in the library, I sort of found out this was the exciting thing to learn about and then subsequently I met Hirzebruch and others. And Hawley helped a bit.