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Readership at Oxford


The real theory
Michael Atiyah Mathematician
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[Q] Were you already thinking about indices?

No, this is before that. The real theory in the context of vector bundles and periodicity theorem, because in the complex case everything can be done, there's no torsion, so everything can be done by using cohomology calculation, rational cohomology, which gives you a machinery. In the real case there are all these… mod 2 problems, which means that… to do with signs and so on… which you cannot do by cohomology calculations, so you had to do those by going back to basic homotopy theory. And in some senses, in the real case, Bott's theorems about homotopy are that more refined and there's no alternative way except going back to the fundamental principles.

And so I had to learn that.  But for example, Hirzebruch and I had made an application of the real theory to the questions about vector fields on spheres, and parallelisable spheres, so there are some nice applications you can make quite early on with the real theory, which are purely topological. They've got nothing to do with the index, that's a much later story. And at the beginning we did everything with the complex case, but the real case was one you brought along when you could, and got refinements and improvements of various kinds.

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: Friedrich Hirzebruch

Duration: 1 minute, 19 seconds

Date story recorded: March 1997

Date story went live: 24 January 2008