Michael Atiyah - Mathematical microscope (93/93)

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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: So all of these are really… have a kind of common background which is that the interrelationship between geometry and differential equations involving the calculus, in the context of non-linear things, things which are complicated because they're not determined by sort of simply the flat, straight prediction. And that complication is where the topology enters. And if you understand the topology, the large scale nature of the space, the way it twists and turns, then you can make predictions about the number of solutions of an equation, or the presence of these soliton bumps, the monopoles, and what happens to them. And the context of magnetic monopoles we discussed before, was studying not just what happened to your single bump, or single lump of charge, but when you have two of them, what happens when they bounce off each other. And then very interesting things happen, they... they bounce off each other something like billiard balls, but not quite. And so there's a whole range of interesting questions which on the one hand are geometrical, on the other hand are dynamic or physical, governed by equations. But right in the background underneath it all there's underpinning of topology which is the sort of global large scale study of curved spaces or curved phenomena or complicated equations in which remarkably you can make progress without doing all the hard work. Where you can actually predict certain facts from qualitative behaviour, understanding the way in which the large scale things affect the small scale things. That's... that’s how the fascination [is] for me, and why sometimes they're important, because these qualitative topological ideas underpin so many other things. Above this superstructure you can put many different actual detailed systems.

In some sense these… when you're studying geometry, you start off by studying triangles and squares and angles and straight lines; then you study, sort of, bits of curved surfaces on them. And then... then you start to study large scale pieces, not just small pieces; and at that stage these topological notions begin to become important, determine things. And a whole large parts of mathematics have been developed in order to provide one with the tools to understand this; how you... how you can use it, put it together. And most of the… I suppose the mathematics of the last century, has been building up this kind of machinery.

And mathematicians use machinery like... like physicists used apparatus. As physics has evolved you use more and more sophisticated microscopes, telescopes, you know, X-ray machines, which enable you to penetrate harder and harder into the smaller and smaller or the bigger and bigger; and mathematicians have developed similar intellectual machines. They are things that enable them to probe much more into what happens at large scales or small scales or the link between them, in many ways very... very parallel, except these machines are intellectual machines. They're... they’re bits built up out of the past, you put them together, they're designed for particular purposes and like any good machine they can be used on a whole range of problems.

The specimen you study in a microscope can become all sorts of different possibilities. The same is true with one of these intellectual machines. They can be applied to a whole range of problems which come from different backgrounds, but some have certain things in common. So I like to think that this is what mathematicians are... are doing, is to produce intellectual machines, and the topology is... is one of the sort of key ingredients in most… many of these machines. It's somehow the backbone, in some sense, if you like, the... the large scale structure of the machine. And we've been doing that… building up these machines for a long time, and we're now putting... putting them to use.