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Technical problems in K-theory

Michael Atiyah
Mathematician

Views | Duration | ||
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31. Working with my boss | 903 | 03:20 | |

32. Mathematics at Princeton | 1117 | 04:12 | |

33. Working together in mathematics | 969 | 02:59 | |

34. Topology and K-theory | 1032 | 04:13 | |

35. My mathematical growth | 1051 | 02:10 | |

36. And topological K-theory was born | 808 | 02:59 | |

37. Technical problems in K-theory | 749 | 02:00 | |

38. The real theory | 730 | 01:19 | |

39. Readership at Oxford | 658 | 02:18 | |

40. Differences between Oxford and Cambridge | 1439 | 01:39 |

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Well there was an interesting technical barrier at one stage when I was playing around with these things, putting them together. I somehow… everything seemed to work so beautifully, so smoothly, I sort of charged along and it wasn't until Bott wrote to me at one stage, or I think maybe it was Hirzebruch, he was giving a seminar at Paris to explain our results, and he wrote and I'd more or less… we'd got the whole thing worked out. He wrote to me, ‘How do you… why is this true?’ And I said it back to myself, and said, ‘God, I didn't really know’. It just seems it had to be true, it's so beautiful, you know, I'd assumed it had to be because that's the way everything fitted together, and I suddenly realised, gosh, I had no idea how to prove this. And then I sat down and actually I managed to get a quite elegant proof.

But it would have been embarrassing because he was giving the seminar in Paris, you know, in a few weeks time and the results were being announced. And this great machine, it all had to do with the compatibility of the periodicity theorem and tensor products. And partly because of the origin, the relation, the way it arose, in algebraic geometry with Grothendieck's work… Grothendieck's work and the Bott's periodicity theorems, and when… if you put the two together it seemed obvious they had to match up. And I suppose I somehow assumed that happened, and when presented with the question I realised that I'd never actually consciously even realised that there was a problem.

So then I quickly had to produce a proof and fortunately got one quite quickly and then I wrote to Bott afterwards, and he actually then produced better proofs and in fact subsequently we had to do the corresponding theory for the real vector bundles which is much subtler. And he, we got… but it was a case where I'd been carried away if you like by the whole beauty and simplicity and elegance of the theory, I assumed it had to be true, and I was right, but it could have been embarrassing, you know, if it had been wrong: the whole thing would have collapsed like a pack of cards. But, I remember that one, but it didn't last long fortunately. We quickly came out.

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.

**Title: **Technical problems in K-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:**
Paris

**Duration:**
2 minutes

**Date story recorded:**
March 1997

**Date story went live:**
24 January 2008