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Michael Atiyah - The mod 2 index theorem (51/93)

805 views · Published 20 June 2017 · 1:30 · Indexed 21 September 2026

Channel: Web of Stories - Life Stories of Remarkable People · 2017 · Science & Technology

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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: The mod 2 index, it could have gone unnoticed because I think with the emphasis originally on real cohomology and characteristic classes, and so on, you know, you wouldn't see it and you'd have to look very hard for it. And also the original methods of proof wouldn't have allowed for it. And even subsequently if we move further on to the time when there was all this emphasis on the Riemannian geometry approach and the heat equation approach, that's also using real numbers and going even further away from this. So the mod 2 thing was a little bit on its own, could easily have been missed out at the first step, and would certainly have been, you know, trodden underfoot in the stampede later on.

So I've always had a kind of soft spot for it, you know. It was a little bit of a gem that came out. If you do things very carefully, very nicely, you have this little elegant story and it certainly wasn't something that you were going to get easily achieved by any other method. And it remains, you know, one of the sort of trickiest things to get by these... by these methods. And I tell you, you know, you had a soft spot for these kind of theorems that are on their own and are rather refined.

Obviously, many theorems you end up by… they can be attacked in half a dozen different ways, and that's very interesting. And in fact you… Gauss was said to have had 10 different proofs of the reciprocity theory; and the index theorem probably has something like getting on the same way now, and that's very, you know, significant. It means it's on the crossroads of many different routes, and that adds to its interest. But there's other ones that you can only get to by one route, the long end of a long twisting journey up the mountainside, and they have an attraction of their own too, a charm which I rather like.

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