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Cambridge in 1990


Geometry, physics and the future of mathematics
Michael Atiyah Mathematician
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[Q] In the last 20 years we've seen this enormous interplay between geometry and physics. Where do you see it going in the next 20 years?

Well that's an interesting question. I mean when it first happened I remember people like Jackiw, who I met at MIT, saying, ‘Well now, we don't... don’t know whether this is just a, sort of, a.. a brief interlude where the paths cross and then you go off in different directions; or whether it's going to be a long term association’. And I think the physicists on the whole, most of them thought probably it would have been a passing phase. But as we've seen over the last 20 years, it's been anything... anything but that. Every... every year going forwards and something new happens and the thing gets even more thoroughly intertwined. So now having developed these strands together over 20 years it's a pretty tough rope. I mean, I don't see it, sort of, you know, coming apart and there's... there’s no evidence that it's breaking up.

So I think it's going to keep going and in some ways its impact on mathematics has been quite spectacular. More and more areas of mathematics have been brought in, been shown to be affected by ideas from physics and quantum physics; we have quantum groups, we have symplectic geometry, algebraic geometry, topology… a whole range of things which… number theory, modular forms, you almost say there's a kind of quantum mathematics evolving. In fact there's a nice phrase I saw somebody refer to – high energy mathematics. It's not quite clear what it is, except that it all emerges and is related to things from high energy physics one way or the other… all the mathematics you need.

In some ways, I... I… my picture of it is something you might say like this; I mean, there's a... there’s the kind of classical mathematics which goes across the board, and now we are being forced to develop in various ways a whole extra layer of... of quantum mathematics which sort of rests above... above this. And you first build a bit of a tower here and it starts and then you find that you can do it here, and... and it's gradually spreading further and further. And so in some sense we – in the next century or as we enter the next century – we're going to start off with this whole additional super-layer of mathematics at the quantum level. It's... it’s difficult to define what it means, it's difficult to predict in advance what you want or what you need. It's just pragmatic and you discover that there are marvellous things taking place at this level which justify the development of the mathematical ideas.

If you sat back and said, ‘Well, I want to combine mathematics here and quantum theory there and produce’… you’d produce rubbish. It had to emerge sort of naturally.  But as we see it now it's a, sort of, very rich wide-spread structure and I would forecast that, you know, in the 20th [21st] century this will become more and more firmly embedded and the words quantum mathematics or whatever, will become, you know, part of the standard jargon. And you might say that, you know, if you start off at the beginning, logically, you might say that mathematics starts off with discrete things – algebra and combinatorics and counting and – then you get into calculus and you get into the infinities and... and you tame the infinities by doing differential calculus and integration. And then you do the whole of geometry and all... all at a finite dimensional level; and then the next step up is to go to infinite dimensional spaces, infinite dimensional – double infinities – infinities of infinities, and how you do that and... and what the right way is, is of course far, far from clear. And physics... physics shows what you need to do […] how to develop it. And so you could see it... see it as a natural evolution from discrete, continuous, to sort of doubly continuous, and in that sense, you know, it, sort of, it may seem perfectly logical to our successors, they might say, ‘Well, what were you mathematicians all, you know, fussed about? It was inevitable, you know, how we're going to get there’.

So I'd... I’d forecast fairly firmly that that's the kind of view that will gradually emerge, you know, in the next... next 20 years and it'll seem that we have a whole new way of treating these more sophisticated higher dimensional, infinite dimensional things, which will cover all sorts of ranges which will have different aspects – algebra, geometry, statistics, physics – and will gradually make a better and better picture.

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: MIT, Roman Jackiw

Duration: 4 minutes, 9 seconds

Date story recorded: March 1997

Date story went live: 24 January 2008