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Continuing relevance of early studies


The future of mathematics
Michael Atiyah Mathematician
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It always seemed to me a miracle that as mathematics develops and becomes more complicated, each generation says, ‘Gosh, look we've worked all our life, we've mastered this stuff, we've learnt all these things – how can the next generation possibly pick up the pieces and run?’ You know, and it's the... the miracle that it keeps happening. I mean, I think, if you look back through history you'll find people constantly think that it's got to the end of the road, it's got much too difficult for anybody to come in now. But it's one of the strengths of mathematics is that it... it goes by leaps and bounds. All that came before gets beautifully synthesised, encapsulated, simplified, so that the next generation can be taught it in the one graduate course, you know.

What took mathematicians hundreds of years and thousands of pages to, sort of, get to grips with, it's stripped of its inessentials, it gets presented in a way that, you know, people can learn it… under proper guidance and so on. And so you start at a higher level. Each... each… you know, it's like climbing a big mountain and the next generation are already sort of flown in by helicopter and they land half way up the mountain. So it... it keeps happening, and if it didn't happen, you know, mathematics and most other science would have stopped long ago.

You keep wondering how long it can go on happening. And in certain areas such as the area we've been discussing where a whole lot of things have come together, and it's the interaction between quantum theory and mathematics topology and algebraic geometry, it makes all the interest. But then it is... it is… I think, it becomes more of a problem to attract graduate students into the area. There's a danger that either they come in and are too ambitious and spread themselves so thinly that they can't possibly get off the ground; or they pick up some very, very small problem and sort of beaver away in a little corner and then... then finish a rather technical point which is not of great interest. And how you... how you balance that between the two is a... is a very tricky exercise in... in guiding graduate students.

In other areas such as, I think, group theory or other parts of mathematics more narrowly defined, the graduate students have... have an easier time. There's no question about that. On the other hand I've... I’ve quite often met… told graduate students who are starting that there is this problem, they have to beware; but on the other hand, if this is where the excitement is you've got to give the intellectual challenge to the students to, you know, test their muscle against this thing. And every now and again you'll get a Donaldson who comes in and, you know, makes a big breakthrough against all the odds.

So I think you have to keep playing it, but obviously graduates have to be given good advice and, depending on their abilities, have to be started off with some modest ambition. Sort of test their... their technique, acquire a bit of general background before they can go ahead. And so it may be that there'll... there’ll be a few more years before they can really make major... major advances. But when I look and see what all the young theoretical physicists are doing, they've all mastered large amounts of physics and large amounts of complicated mathematics, and they jump around and so in a way it's... it’s still very impressive. But it may be that the age is drifting up, I'm... I’m not sure. I mean, people who seem young to me are probably not as young as they were; they're perhaps in their 30s rather than in their 20s.

So there is a problem, it's a kind of general problem that the development of science has in every field. You have new techniques, new ideas, and each generation somehow has to start off there and yet, quickly enough they can make... make a breakthrough. And it's... it’s really one of the most interesting aspects; if you look back at the history of development of, you know,  intellectual thought, how this process can keep going, and particularly since these things seem to be growing at exponential rates. And you know, what will mathematics be like in a 100 year’s time? What will it be like in a 1000 year’s time? Can... can we even begin to… the mind boggles, when you look back how much has changed in a 100 years and you look and see how many more people have come into the scene. You know, can we even think of mathematics continuing as an intellectual exercise a 1000 years down the road? It is really… I wish I was here in a 1000 year’s time to find out the answer.

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: Simon Donaldson

Duration: 3 minutes, 56 seconds

Date story recorded: March 1997

Date story went live: 24 January 2008