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Refining the eta variant


Eta invariant
Michael Atiyah Mathematician
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Well we heard about Patodi because he'd produced this proof of the conjecture of Singer and McKean on the local formula for the relation with the Gauss-Bonnet theorem, and then later on with his work on the Kähler manifolds, and... and so we heard he got these fantastic algebraic cancellations. So he was obviously a very bright young Indian, and, but it was difficult to understand his work and so on, so, you know we… he was invited to come to the institute. I think, well I was responsible for inviting him, I suppose, at that stage, to come and spend a couple of years with us. So he did. And it was a very productive time. He was a very interesting young man of course, Patodi, he was slightly eccentric in some ways. He was… he came from… he was a Jain… a very strict sect, vegetarians who had to live in a certain way, and he also unfortunately disregarded his health and eventually that led to complications of which he died very young, tragically. But he had his own style, so he was a very unusual contribution. He worked his own way, he had insights of his own, and he produced these formulas.

I remember when I first worked on this problem with the manifolds with boundary, what happened was that the origin of that whole problem was again a mixed combination. On the one hand there was the input coming again from Hirzebruch's work. Hirzebruch had done this work on the cusp singularities of algebraic surfaces, and the formulas which correct the signature formula for singularities of certain types. Well the signature corrections for orbifold points, points of finite order, that we understood already from the index theorem with finite group actions. But then he had fixed-point corrections arising out of the singularities of like the cusps of Hilbert modular surface, there were a different kind of singularity. And he had corrections there which involved beautiful things like the Dedekind eta function which have nice analytical formula. And so we naturally… we'd like to understand that formula and prove it.

And so we were led to formulate the question of what happens with signature theorem for a manifold with boundary, what is the boundary contribution, and when you play around with it a bit you see what are the kind of possible constraints it might have and where it might come from. And then gradually it led us to formulate some version using heat equation methods, and we formulated a problem for Patodi to calculate. And first of all did it in the special case of the signature, and then Patodi, he went away and he calculated, and he came back and he checked it, and said, ‘It works out right, you know, the formula’.  But we couldn't understand his kind of way of doing it, Singer and I, so we tried to redo it in our own way, in particular it involved doing it for the Dirac operator and therefore for all Dirac type operators.

And so we tried to cast it in a different way, away from differential forms and in the form of a general operator. And you know, eventually we succeeded, but it took quite a long time; particularly finding this notion of these global boundary conditions formulated in that way. It didn't at all come naturally. We chased ourselves round and round [our] tail, we were also on the wrong track until we hit the right way of looking at it. So at the end of the day it all looks very simple, and that looks so obvious that's the right way to see it. But when we were working on it, by no means, that was the way it came around, and it was a very tortuous way we got there.

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: Vijay Kumar Patodi, Isadore Singer, Friedrich Hirzebruch

Duration: 3 minutes, 28 seconds

Date story recorded: March 1997

Date story went live: 24 January 2008