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Three manifold invariants


The lack of a background in physics
Michael Atiyah Mathematician
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[Q] Singer once told me that he regretted that he'd not spent a dedicated period of time learning quantum field theory, and that he was then too old. And he told me that I was still young enough to do it, but I have to say I didn't follow his advice. Do you in any way regret not having been through this process that the physicists all have been through?

I don't think so really, because if you go... go back and see what physicists were doing, my contemporaries for example had been doing this, they had lots of false starts and going off in different directions. There were different times when different things were thought to be the great solution: current algebras and S-matrix theory. All of which eventually perhaps had their little place in the background, but they spent an awful lot of time chasing, sort of, unnecessary, you know, hares. And by the time we got to be… hear about it, they'd homed in on gauge theory as being the right thing for them to do; in turn gauge theory was very easy to understand for geometers, we were... we were saved an awful lot of messing around with unnecessary theories.

Of course we didn't have the background of the quantum field theory machinery, we had to pick that up, but I think, you know, we picked it up by talking to the physicists like Ed Witten and others on a very regular basis over a period of many years. At a stage when we were in a position to appreciate it, because having sort of got the, sort of, lock with the index theorem on the one had and... and gauge theories and the full Dirac operators, we were sort of, we were all, sort of well prepared, well adjusted to understand the significance of various bits and pieces. So then by that time when... when the, sort of, extra layers of all the quantum theory and path integrals were thrown at us, we... we were quite receptive and sort of could see how they fitted together.

If you had to learn them yourself from the beginning, without the kind of motivation of the end point, it's much harder. I'm sure it would have taken, you know… had to spend an awful lot of time, and it would have been probably not sufficiently interesting to keep one’s attention. So, I think, Singer, sort of, in theory might have been right, but in practice I don't think it would necessarily have worked like that. I think it worked out quite well as it was.

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: Isadore Singer, Ed Witten

Duration: 2 minutes, 3 seconds

Date story recorded: March 1997

Date story went live: 24 January 2008