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My work in easier words


Simple explanation of my work
Michael Atiyah Mathematician
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[Q] Your major results have all involved concepts like vector bundles, indices of differential operators, instantons and monopoles. Do you think you could explain to a layperson what these terms mean?

Well it's quite a... quite a task, but let me try and start. I mean, most... most of these questions have to do with the… what happens when you go from the... the easy world of...of the flat plane, straight lines, where everything is predictable by just carrying on the same way. And what happens when you get to a more complicated world where things start to curve and twist and you come back to where you started and so on. It's that kind of interplay between the, sort of, the predictable straight line flat world, or flat phenomena, and the... the really complicated things which most things are where they have… they curve and they twist and they bend. And the... the… studying what happens in this way, on a larger scale, what happened if you go round and whether you come back like Christopher Columbus going round the world and coming… wasn't flat and coming back. Those are what topologists study and that's what the mathematicians concerned with these questions examine.

Now, the word vector bundle, the first example, lets explain what that means. If you're studying flat things then you study flat spaces, and then they're given straight lines and directions and... and so that's what the word vector has to do with, just studying directions in... in flat spaces. If you have a curved surface for example, then at every point you can approximately think of it as flat and you have little directions on it, and that's a little flat piece of world. But as you move round the curved surface, then you follow this… these tangent directions round, then you have a moving family of... of straight lines. At every given stage you can approximate it by straight lines, but the whole thing is curved. So one way to, kind of, get a grip of this big curved animal is study the continuously varying family of straight lines that form these directions. And that's really what a vector bundle is. Of course the… you can generalise it and they aren't always tangent directions; they might be directions into other... other space, they might be something more complicated. But fundamentally the vector bundle is the intermediate notion, how to get hold of a curved animal by studying very small flat regions of it, and then trying to piece them together. That's what you're doing when you study vector bundles.

And again then when you go beyond that, and you study the differential equations, the same sort of thing underlies that. You'll study a differential equation, it's like if you study ordinary algebra; you have algebraic equations which give you problems where you look for solutions. If the equations just involve something like 2x+3y and so on, is equations where… which are linear… where the answer's given by straight lines. If they have x2 in it and y2 they become quadratic, they become curved or higher degree. And so algebraic equations are straight linear equations which are easy, and then you have more complicated equations, non-linear, which are difficult. And the same thing when you go to the differential calculus. Differential calculus studies differentiation and you try to find solutions… curves which satisfy differential equations which try to predict what happens in the large from what happens in a very small region by following the solution of differential equations. And again the differential equations come in... in the simple form where things are, sort of, flat and straight so they just solve this equation then you get it, and equations which are non-linear.

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: Scotland, Christopher Columbus

Duration: 3 minutes, 29 seconds

Date story recorded: March 1997

Date story went live: 24 January 2008