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The L2 index theorem
Michael Atiyah Mathematician
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The L2 index theorem, the version I reported on at the… I think it was the meeting in Paris – it was the Schwarz birthday conference or something like that – that was a version which got me involved with von Neumann algebras. Well, I'd been… because Singer knew about functional… I learnt about von Neumann algebras from him, and there was a time when we were all quite interested in... in doing K-theory for type II von Neumann algebras. And there was this chap called Breuer who was from Germany who was specialised, and we... we talked with him, and you can do a lot of things for type II von Neumann algebras, the real numbers replacing the integers, and you can do index theory, and... and it's... it’s quite amusing.

But the question is: What's the pay-off? Are there any particularly interesting applications? And so, this example with... with infinite coverings of manifolds, discrete groups, seemed to me... me a sort of natural application where the discrete group gave von Neumann algebra and the general theory suggested that... that you should have a theorem. So I focused from this lecture presenting it on...on just that problem by itself. I didn't use von Neumann algebras in a very heavy way, but I had to use the basic analysis of Hilbert spaces, operators commuting with the group actions. And I remember at that stage reading... reading Dixmier's book, understanding in some detail about von Neumann algebras, so I could really get to grips with this and combine the... the sort of elliptic differential operator theory of parametrices and so on with the von Neumann algebras; bridging that gap. So I did spend a bit of time teaching myself a bit about... about von Neumann algebras for that purpose, but I got rid of most of it in the actual written version. So that was… I quite liked that because it was, sort of, a nice concrete application, as opposed to churning out the general theorems which you could do, but it wasn't quite clear what... what the point of them was.

So that was one thing, and... and the… that’s L2 index theorem and then... then of course, got interested quite a few other people… and people followed it up in various contexts, also Dodziuk got interested in the combinatorial version, and then I think subsequently Gromov and people, they worked with L2 putting numbers in combinatorial contexts. But the stuff on... on the discrete groups, that was partly motivated, or had applications for the work I did with Schmidt. The point was the L2 index theorem with this… in this infinite covering space showed that you could deduce non-trivial theories about the existence of things like harmonic forms, or L2 square-integrable cohomology by passing down to a quotient which was compact, and then using the topological formulas, and then if the index was positive that meant some space, it wasn't zero and upstairs it wasn't zero. So we... we could use it to prove existence theorems of the kind that you needed to prove that certain representation space that you constructed for non-compact semi-simple groups was not trivial, because there was something there. And so the purpose of that application was to push it in the direction of that.

And then I got involved working with Schmidt on… following that up, although actually that turned out to be rather a, sort of, odd story. I... I had this idea, I wrote to Schmidt, I said, ‘Well I think’… I knew... I knew him… ‘maybe we can try to re-prove Harish-Chandra's theorems more elegantly’. I mean, Harish-Chandra, you know, my colleague at the institute, and he wrote these enormous papers which were chopped off at 30 pages at a time, you know, he... he was a kind of factory, produced the most… when he got to the end of 30 pages, turn it up, published, and you kept... kept going. And you kept the same notation all the way through, so it was impenetrable… except for the real experts. And yet the results were very, you know, interesting; beautiful in some ways. So I was very keen to understand enough to see whether we could prove some of this stuff, you know, in a way that… it clearly related to index theorems in general, and so that... that was my attempt.

And so I talked with Schmidt who was knowledgeable about the Harish-Chandra theory and he... he agreed that something like this could be done. So we agreed we would write a joint paper to... to… which would actually set up the… re-do the main Harish-Chandra theorems, starting with this index theory as a base. But before we could start on that, we had to do something else – I’ve forgotten what it was – we had to write another paper which was meant to be the foundational paper. Yes, well we'll polish this one off quickly and then we'll get back and do the other thing. Well, we spent about a year and a half writing the first paper, and the other paper, second paper never… never appeared. But I learnt quite a bit about representation theory groups with him, and the version that… well no, maybe I've forgotten which way round it was. The other bit had to do with… the other part of the Harish-Chandra theory had to do with the… showing that the characters of irreducible representations are locally squared… locally integrable functions, and that... that gets into difficulties when you have very singular orbits, conjugacy classes, and... and you have to treat that by... by some rather direct analysis and differential equations. And... and Schmidt worked out how it should be done, but we never actually put all that together in proper papers, but I gave these lectures in Oxford which George Wilson wrote up which incorporate some of that.

So sometimes collaborations never... never finish in the sense that you... you aim to do something, you plan a big scheme, chapter I, chapter II, chapter III, and you... and you… chapter I grows and grows, and you never get on to chapter II. That... that was an example where we never achieved our full ambitions, and then by that time you… you're tired and you've moved on to something else and you're working with somebody, a different person and… so our collaboration never... never totally achieved what it did. But it was... it was quite interesting as far as it went and it got me involved a bit more into... into the Harish-Chandra sort of theory.

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: Paris, Oxford, Harish-Chandra, Wolfgang Schmidt, Isadore Singer

Duration: 5 minutes, 49 seconds

Date story recorded: March 1997

Date story went live: 24 January 2008