Commentary by M. Atiyah
In the Spring of 1962, my first year at Oxford,
Singer
decided to spend part of his sabbatical there. This turned out to be particularly fortunate for both of us and led to our long collaboration on the index theory of elliptic operators. This had its origins in my work on
Once we had grasped the significance of spinors and the Dirac equation, it became evident that the
Singer and I had some great advantage over the analysts investigating the problem. We were investigating a particular case, the Dirac operator, and we already “knew” the answer. Also, this case, existing in all (even) dimensions, encompassed all the global topological complications. Moreover, we had arrived at the problem starting from
With considerable help from our analytical friends, such as Louis Nirenberg, Singer and I eventually produced a proof of the general index theorem during my stay at Harvard in the Fall in 1962. This was based on the use of boundary-value problems and follows the cobordism approach of Hirzebruch’s proof of the signature theorem. We announced our results in the Bulletin note [1]. I realized at the time the significance of the index theorem and that it represented the high-point of my work, but it would have been hard to predict that the subject would continue to occupy me in various forms for the next twenty years. I would also have been extremely surprised if I had been told that this work would in due course become important in theoretical physics.
The proof of the index theorem was only briefly sketched in
[1].
The full details were presented in the seminar which I ran with
Bott
and Singer in the Fall of 1962. In due course a more comprehensive version of this appeared as the Annals of Mathematics Study, based on the Seminar at the Institute for Advanced Study run by
Borel
and
Palais.
I wrote an Appendix
[1965: The index theorem for manifolds with boundary]
for this describing the extension of the index problem for manifolds with boundary. This was joint work with Bott and is also described in
[2].
In fact understanding the role of boundary conditions for elliptic operators was by no means routine. Bott and I struggled with the problem for some time, before we saw the light. It was clear, for topological reasons, that a good boundary condition should somehow “trivialize” the symbol of the operator at the boundary, so as to define a relative
The opening of the new Courant Institute building and the International Congress at Moscow in 1966 provided opportunities for expository survey talks
[8],
[15]
on the index theorem and its relation to the topology of the linear groups (i.e., to
At the Woods Hole conference in the summer of 1964, Bott and I learnt of a conjecture of Shimura concerning a generalization of the Lefschetz fixed-point formula for holomorphic maps. After much effort we convinced ourselves that there should be a general formula of this type for maps preserving any elliptic operator (or, more generally, any elliptic complex). As a first test we applied it to elliptic curves with complex multiplication and, as we had the leading experts with us at Woods Hole (Cassells, Tate, etc.), we asked them to verify it for us. Unfortunately our formula failed the test. Fortunately we did not believe the experts: the formula seemed too beautiful to be wrong, and so it proved. We were especially convinced when one day we suddenly realized that the famous Hermann Weyl character formula was a particular case of our general formula. In due course we found the necessary proof and the result was briefly presented in [5] with full details appearing later in [6], [13].
Perhaps the most interesting application of the general Lefschetz formula concerned the action of a finite group on the middle-dimensional cohomology of a manifold. This yielded surprisingly strong theories and with
Milnor’s
help we were able in
[13]
to prove that
The first proof of the index theorem for elliptic operators, as presented in
[1],
was based on cobordism and it suffered from certain limitations. For some years Singer and I searched for a better proof modeled more on
Grothendieck’s
proof of the generalized Hirzebruch–Riemann–Roch theorem. Eventually we found such a proof, based on embedding a manifold in Euclidean space and then transferring the problem to one on the Euclidean space by a suitable “direct image” construction. This proof, given in
[11],
worked purely in a
The index theorem for families of elliptic operators [27] was an easy consequence of the new proof and could be seen as a first attempt to generalize the Grothendieck Riemann–Roch theorem. In its real form [26] it included various mod 2 index theorems, such as the computation of the dimension (mod 2) of the kernel of a real skew-adjoint elliptic operator. Despite the many interesting new proofs of the index theorem that have been produced in recent years (heat equation, supersymmetry, etc.), no other proof encompasses these refined mod 2 index theorems, and the proof in [26] remains the only one available.
Although
[27]
was analogous to Grothendieck–Riemann–Roch for a fibre map, we were still a long way from a fully fledged analogue dealing with arbitrary smooth maps. A basic reason for this was that in algebraic geometry Grothendieck had the advantage of two
In [17] Singer and I carried out a systematic investigation of mod 2 indices in the framework of Clifford algebras. Very much to our surprise we found an entirely new proof of the (real) periodicity theorem emerge, based on Kuiper’s proof of the contractibility of the unitary group of Hilbert space. The connection of this with other proofs remains to this day somewhat mysterious and so far it does not appear to have been exploited, but I understand from Quillen that it may find a natural niche in relation to Connes’ cyclic homology theory.
One of the interesting applications of the index theorem for families of elliptic operators concerns the signature of vibrations, and this is the topic of [18]. Chern, Hirzebruch, and Serre had shown that the signature of a vibration is multiplicative (i.e., the product of signatures of base and fibre) provided the fundamental group of the base acts trivially on the cohomology of fibre. The index theorem for families clarified the role of the fundamental group and non-multiplicative examples occur naturally in algebraic geometry. As I was to realize much later from Witten these questions are related to gravitational anomalies and [18] provides the simplest examples of such anomalies.
My general interest in differential equations (both elliptic and hyperbolic) together with my background in algebraic geometry led naturally to the little note [23], where I explained how Hironaka’s theorem on the resolution of singularities gave an easy proof of the Hörmander–Lojasiewicz theorem on the division of distributions.
One of the most surprising applications of the equivariant index theorem was the result proved in my joint paper
[21]
with Hirzebruch showing that spin manifolds with non-zero
During a visit to Harvard in 1970,
Mumford
asked me whether some classical results on Riemann surfaces, concerning square roots
In
[25]
the dimension mod 2 of the space of holomorphic sections of
My colloquium lectures [56] to the AMS in 1973 (just after my permanent return to Oxford!) were never published, but they provide perhaps a useful historical survey of the Riemann–Roch story and its evolution into the index theorem. Of course, this should now be updated, since much has happened in the intervening decade, particularly in relation to theoretical physics, but that will have to await another day and possibly another pen.
The Lefschetz formula of my papers with
Bott
gave the answer as a sum of local contributions from the fixed points. For a circle action the Lefschetz number is a character of the circle, i.e., a finite Fourier series, but the local contributions have poles. In particular the Hermann Weyl character formula is of this type, and we know in that case the local contributions can be interpreted as distributional characters of infinite-dimensional induced representations. These facts led Singer and me to look for a general context in which such local contributions could be interpreted in an appropriate sense as local “distributional indices”. We had in mind possible applications to the Harish-Chandra theory of representations of non-compact semi-simple groups and the Blattner conjecture (giving the restriction of the character to a maximal compact subgroup). The ideas led us to introduce the notion of an operator which was elliptic transversally to the orbits of a
The joint paper [33] arose out of the attempts by Bott and myself to understand the remarkable results of the young Indian mathematician V. K. Patodi on the heat-equation approach to the index theorem, a topic which was to blossom later in the contact with theoretical physics. In our work on the Lefschetz formula for elliptic complexes, Bott and I had described the Zeta-function approach to the index theorem but had commented on its computational difficulty. Singer and McKean took the process a step further, in the heat-equation version, by concentrating on the Riemannian geometry. They speculated on the possibility of remarkable cancellations leading directly to the Gauss–Bonnet form for the Euler characteristic. This was proved some years later by Patodi. Even more significant was Patodi’s extension of the result to deal with the Riemann–Roch theorem on Kähler manifolds. Patodi’s approach was rather direct, and a considerable tour de force. But the algebraic manipulations were difficult to understand and it was therefore very interesting when Gilkey produced an alternative indirect approach depending on a simple characterization of the Pontryagin forms of a Riemannian manifold. On the other hand, while Gilkey’s result was beautifully simple, and easily led to the index theorem, it appeared to be enormously complicated to prove. In fact Gilkey had discovered it while performing algebraic computations on the computer.
This was the situation when Bott and Patodi joined me at the Institute for Advanced Study in 1971–72. After considerable effort we eventually realized that Gilkey’s results was a very easy consequence of the Bianchi identities, together with the main theorem on tensorial invariants of the orthogonal group. Gilkey’s proof had appeared complicated only because he had not taken this route.
Besides proving Gilkey’s result, and giving its application to the index theorem, [33] also included a leisurely account of the heat equation asymptotics based on Seeley’s approach. Moreover we gave, in the Appendix, a new proof of the theorem on orthogonal invariants. We were impelled to do this by our difficulty in understanding the notorious “Capelli identity”.
My second collaboration with Patodi, this time with Singer as the third partner, concerned the signature theorem for manifolds with boundary and led to the series of papers [31], [38], [37], [43] The problem of generalizing the Hirzebruch signature theorem to manifolds with boundary had long been an intriguing question. There had been many clues, notably the work of Hirzebruch on signature defects of cusps of Hilbert modular surfaces. While Patodi was with me in Princeton I had suggested this problem to him, hoping that he could apply his virtuosity with the heat equation to the problem. In fact he succeeded on these lines, but his method was again highly computational and tied to the extensive use of differential forms. As such it did not apply to the Dirac operator and its generalizations which I felt should be included. Singer and I therefore tried to analyse the problem in its more general form. Eventually we saw that the natural formulation was that of an index problem with a “global” boundary condition. This was conceptually a major breakthrough but there were several crucial obstacles still to be overcome when I left the Institute and returned permanently to Oxford at the end of 1972. Shortly after my return I solved the outstanding technical problems. One of these involved using standard formulae from a classical textbook on heat conduction, which I found rather amusing. More important was the psychological effect of feeling that my return to Oxford had started off well, and that the difficult decision to leave the Institute for Advanced Study would not turn out to have been disastrous.
In many ways the papers on spectral asymmetry were perhaps the most satisfying ones I was involved with. The way they stretched over differential geometry, topology, and analysis with a nod in the direction of number theory appealed greatly to me. At the time these papers had only a modest impact but, a few years later when contact was made with theoretical physics, they became extremely popular. In particular
Witten’s
work on global anomalies brought our
Sadly these papers were the end of my collaboration with Patodi. He returned to the Tata Institute and we continued to correspond for a while but later his health deteriorated, and a kidney transplant became the only hope. I was involved in the medical discussions concerning this and at one stage it was planned to bring him to Oxford for the purpose. However, this was eventually deemed unnecessary and plans were made in Bombay, but sadly Patodi died of some pre-operation complications. It was a tragic loss both personal and mathematical. Patodi was a mathematician of real originality and power. He was charmingly modest, with a friendly and captivatingly simple disposition.
Although elliptic differential equations constituted the centre of my mathematical interest for many years, there was an interesting collaboration with Bott and Gårding on hyperbolic differential equations which led to the papers [58], [24], [34] Our collaboration began in an unusual way. Bott was staying at the time in Oxford and Gårding came from Lund for a few weeks. He said he had a problem for us. There was this important but obscure paper of Petrovsky which involved some homology of algebraic varieties. Bott and I were essentially contracted to understand and explain Petrovsky’s paper. We were at the time very ignorant about hyperbolic equations, but we had Lars Gårding a world expert and excellent tutor. In return we instructed him in topology, and so the collaboration began with mutual education. In due course we managed to understand Petrovsky, then to modernize and generalize, leading eventually to our rather lengthy joint paper. Altogether it was an enjoyable and instructive episode.
Another part of my education on analysis had of course been going on for some time with Singer (and at other times
Hörmander)
as tutor. In particular I learnt from Singer, who had a strong background in functional analysis, about von Neumann algebras of type II with their peculiar real-valued dimensions. We realized that
For many years I had taken a general interest in the representation theory of non-compact semi-simple groups. In fact this was such a major industry at Princeton, and it had so many ramifications, that it was impossible to ignore it. On the other hand the work of Harish-Chandra was forbiddingly technical and I constantly hoped that more geometrical methods might lead to a conceptual simplification. In particular I was attracted to the idea of realizing the discrete series representations by solutions of the Dirac equation. This seemed a natural generalization of the case of compact groups where the Borel–Weil theorem could also, as I knew, be reinterpreted in terms of the Dirac operator. I had a number of discussions on those topics with Wilfred Schmid, after which I realized that my paper [44] could be used as an existence theorem for square-integrable harmonic spinors on suitable homogeneous spaces. I wrote to Schmid, explaining this idea, and he soon saw how one could really develop much of the theory in detail from this starting point. However, to avoid relying on Harish-Chandra’s work it was necessary to find an alternative direct proof of the fundamental theorem on the local integrability of the irreducible characters. In 1975 Schmid and I both spent a term at the Institute for Advanced Study, and during that time we worked out a reasonably satisfactory proof of the local integrability based on a careful study of invariant differential operators and conjugacy classes. We planned to write this up as a joint paper, but first we decided to write up quickly the work on the discrete series. Unfortunately this took much longer than planned, ending up as a much more substantial paper [45] than originally planned. The other project, on the local integrability, got postponed indefinitely but the essential ideas were explained in a series of lectures [53] I gave at Oxford in the Spring of 1976.
Another set of lectures [40] given at a summer school in Italy in 1975, give a leisurely account of index theory. Although not containing any new material, these lecture notes remain perhaps a useful quick introduction to the subject.
The