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Mathematical microscope


My work in easier words
Michael Atiyah Mathematician
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The index theorem, which is what I spent much of my time working on in one form or another, is concerned with the linear differential equations. Many linear differential equations are… occur very often in mathematical physics… the equations of Maxwell for electromagnetism are linear equations; so linear equations aren't necessarily simple, they can be quite complicated. And in studying linear equations, but not just in a piece of flat space but perhaps on a curved space, then life gets complicated; not from the original nature of the equation, but from the space in which you’re studying it. And that… so very often you will want to study an equation not in a whole space but on some curved part, for example. And so these equations you may want to study, you may want to find their solutions. That's what... what people do; you have a problem, you want to solve it. If the questions are very complicated, then it becomes impracticable to necessarily find the exact solutions; it may be just much too difficult. And yet you want to know something about it.

So the first thing you may want to know is, how many solutions does this equation have? Has it got one solution or a large number and if so, how many? Well those kind of questions are sometimes… you can answer those questions sometimes without finding the solutions. You may be able to predict how many there are from considerations of the global picture of the whole story. And that's where the curved global aspect, the topology we study, can help you to predict how many solutions there are; without knowing them at all, individually. So it's a very powerful method and when you’re dealing with complicated problems involving many dimensions, many variables, very complicated stories… the real world, or anything approximating the real world tends to be complicated. These qualitative arguments which will predict how many solutions you may get to your equation, without finding the exact solutions, is a tremendously powerful weapon. And this occurs all over the place in parts of mathematics and physics, so that the index theorem is a use of topology, more or less, to predict how many solutions equations can have.

And it has therefore a wide range of applications because of the... the general nature of the problem and it brings together different parts of mathematics; the parts concerned with the differential calculus and differential equations, the parts concerned with the large scale behaviour of curved spaces and the topology and puts them all together in a... in a package. And the theorem itself has a rather simple… sometimes a simple form and sometimes a very complicated form, depending exactly how you look at it… but it has a very wide range of applicability. It applies in a whole range of problems and that's partly why it's so important, partly why it's interesting, partly why so many people work on it from different directions. So that's the position of the index theorem within this context.

Then the other thing you mentioned, things like monopoles and instantons; they... they arise again in contexts where... where these non-linear phenomena – these topological things which are not present in purely flat spaces – occur. There are situations in which you get complicated solutions to differential equations. For example, the... the… one of the best examples of a soliton was a thing which was observed in a canal in Scotland where a boat… barge was moving down the river and produced a big wave in the front, which then, sort of, moved off by itself and propagated down the canal as a kind of hump. And there was a very graphic description, and this was first observed by the man who saw it who galloped after it on horseback and he saw this lump of water floating down the river for a long time. That's an example of what's called a soliton. It's a, kind of… something which can occur for a certain kind of complicated differential equations where you get a lump of whatever it is – matter, electric charge, water – which is somehow confined in a particular region and moves as a sort of single entity. It behaves like some kind of particle, but actually it's not a particle because it's a bump of water within a sort of whole continuous background; and yet it behaves like a particle. And that... that soliton phenomenon is one of the most interesting discoveries to have emerged in mathematics… mathematical physics in the last 20 or 30 years, because it combines… these things only occur because of complicated non-linear effects which force things to stay together, and yet they propagate like particles. They're a mixture of particles and waves. And the magnetic monopole, that we’ve discussed before, was an example of that kind of phenomenon within the context of electromagnetic theory, in the modern... in the modern sense.

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: algebraic equations, topology, monopoles, instantons, soliton, index theorem, application of mathematics, curved space, magnetic monopole

Duration: 4 minutes, 41 seconds

Date story recorded: March 1997

Date story went live: 24 January 2008