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Further research on lacunas


Lacunas and hyperbolic equations
Michael Atiyah Mathematician
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Lars Gårding who was the sort of, you know, Swedish differential... differential equations man, and... and Hörmander's, sort of, mentor in a way. He somehow or other was visiting Oxford… I can't quite remember exactly how, you know, was passing through or he wrote to us… anyway, and Bott was... was in Oxford. And he... he came to us and said, 'I've got a problem, you chaps, you know, you should be able to help me with, and it involves some topology'. And so then he presented us with this problem which was to understand this work of Petrovsky on... on the solutions of hyperbolic differential equations governed by wave... wave equations. And... and basically the question was if you have the ordinary wave equation in space, you have the... have the light cone and you have the, sort of, regions inside the light cone, and outside the light cone where... where there's darkness so to speak; and the lacuna, the gap, the place where the solution is zero, and the question is for more general hyperbolic equations governing more complicated cones, identifying all the lacunas.

Well, Petrovsky had written this paper quite a long way back which was a very, sort of, pioneering paper, and he made a very substantial contribution to... to the problem in terms which related to the topology of algebraic surfaces, algebraic varieties which are the bases of these cones, to the lacuna problem. Because it involves… the lacuna problem involves computing the fundamental solution by some integral over certain regions and understanding which regions gave you zero for the integral. Anyway, Gårding, who, you know, made his life out of hyperbolic equations, was obviously interested in this problem of Petrovsky and wanting to redo Petrovsky's work in a modern language, because that was 20 years before. So he reckoned he needed some help on the topology side and... and the algebraic geometry that went with it.

So he... he really posed to Bott and me like a contract, you know: 'Will you, you know, here's a job, I want a... I want assistance'. So we... we… he was, I think, visiting for a short while and... and, yes, it was actually when I was at St Cat's… St Catherine's and... and I don't know whether I'd invited him or he'd come for some other scope. I can't remember quite how he came… but anyway, we worked together on this for... for a while, and, sufficiently that we could understand what the problem was about, because my knowledge of hyperbolic equations was essentially zero. You know, I knew about elliptic equations, but hyperbolic equations was a totally different game. I don't think Bott knew much more either, and Gårding didn't know very much algebraic geometry of a serious kind or just the rudiments of topology. So when we started, there was a big gap, but by sort of, you know, being together for a couple of weeks and conversing hard and exchanging, we... we got to grips with what the problem was about.

So then the collaboration began and it took quite… it evolved over quite a long number of years, partly because we weren't always in the same place, and he went back to Sweden and Bott went back to America. And there were some quite technical difficult problems. And also once we got started, Gårding wasn't satisfied with only proving – re-proving – Petrovsky's theory, he wanted to go further. And we did, we went much further. We... we generalised and proved and... and so it kept going. And it was… it turned out to be an extremely interesting, you know, application of modern algebraic geometry topology to hyperbolic equations. And for the purpose of that I had to learn some hyperbolic equations. So then I went away and read Gelfond's book, you know, the... the five volumes… I mean I didn't read all five volumes, but the... the one volume which is relevant, where he did a lot of distribution theory and constructing fundamental solutions, it's all Hedglop's [sic] formulas and so on, of hyperbolic equations. It's nice, it's a nice account, and so I... I enjoyed learning that account in Gelfond. So it helped me pick up a bit about hyperbolic equations.

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: Oxford, Cambridge, Saint Catherine's College, Sweden, Lars Hormander, Lars Gårding, Ivan Petrovsky, Raoul Bott, Alexander Gelfond

Duration: 4 minutes, 2 seconds

Date story recorded: March 1997

Date story went live: 24 January 2008