Interview with Michael Atiyah and Isadore Singer Martin Raussen and Christian Skau The interviewers were Martin Raussen, Aalborg University, Denmark; and Christian Skau, Norwegian University of Science and Technology, Trondheim, Norway. This interview took place in Oslo on May 24, 2004, during the Abel Prize celebrations. It originally appeared in the European Mathemati- cal Society Newsletter, September 2004, pages 24-30. The Index Theorem of zero. And certainly the Index Theorem is sim- Raussen & Skau: First, we congratulate both of you ply a continuation of work that, I would like to say, for having been awarded the Abel Prize for 2004. began with Abel. So of course there are precursors. This prize has been given to you for “the discovery A theorem is never arrived at in the way that log- and the proof of the Index Theorem connecting ical thought would lead you to believe or that pos- geometry and analysis in a surprising way and terity thinks. It is usually much more accidental, your outstanding role in building new bridges be- some chance discovery in answer to some kind of tween mathematics and theoretical physics”. Both question. Eventually you can rationalize it and say of you have an impressive list of fine achievements that this is how it fits. Discoveries never happen in mathematics. Is the Index Theorem your most im- as neatly as that. You can rewrite history and make portant result and the result you are most pleased it look much more logical, but actually it happens with in your entire careers? quite differently. Atiyah: First, I would like to say that I prefer to Singer: At the time we proved the Index Theo- call it a theory, not a theorem. Actually, we have rem we saw how important it was in mathematics, worked on it for twenty-five years, and if I include but we had no inkling that it would have such an all the related topics, I have probably spent thirty effect on physics some years down the road. That years of my life working on the area. So it is rather came as a complete surprise to us. Perhaps it obvious that it is the best thing I have done. should not have been a surprise because it used a Singer: I, too, feel that the Index Theorem was lot of geometry and also quantum mechanics in a but the beginning of a high point that has lasted way, à la Dirac. to this very day. It’s as if we climbed a mountain R & S: You worked out at least three different and found a plateau we’ve been on ever since. proofs with different strategies for the Index The- R & S: We would like you to give us some com- orem. Why did you keep on after the first proof? ments on the history of the discovery of the Index What different insights did the proofs give? Theorem.1 Were there precursors, conjectures in Atiyah: I think it is said that Gauss had ten dif- this direction already before you started? Were there ferent proofs for the law of quadratic reciprocity. only mathematical motivations or also physical Any good theorem should have several proofs, the ones? more the better. For two reasons: usually, differ- Atiyah: Mathematics is always a continuum, ent proofs have different strengths and weak- linked to its history, the past—nothing comes out nesses, and they generalize in different direc- tions—they are not just repetitions of each other. Martin Raussen is associate professor of mathematics And that is certainly the case with the proofs that at Aalborg University, Denmark. His email address is we came up with. There are different reasons for [email protected]. the proofs, they have different histories and back- Christian Skau is professor of mathematics at Norwegian grounds. Some of them are good for this applica- University of Science and Technology, Trondheim, Norway. tion, some are good for that application. They all His email address is [email protected]. shed light on the area. If you cannot look at a 1More details were given in the laureates’ lectures. problem from different directions, it is probably FEBRUARY 2005 NOTICES OF THE AMS 223 not very in- ground. And of course there are also a lot of peo- teresting; ple who did fundamental work in analysis and the the more study of differential equations: Hörmander, Niren- perspec- berg…. In my lecture I will give a long list of names2; tives, the even that one will be partial. It is an example of in- better! ternational collaboration; you do not work in iso- Singer: lation, neither in terms of time nor in terms of There isn’t space—especially in these days. Mathematicians just one are linked so much, people travel around much theorem; more. We two met at the Institute at Princeton. It there are was nice to go to the Arbeitstagung in Bonn every generaliza- year, which Hirzebruch organized and where many tions of the of these other people came. I did not realize that theorem. at the time, but looking back, I am very surprised One is the how quickly these ideas moved. Photograph by Anne Lise Flavikl. families R & S: Collaboration seems to play a bigger role Abel Prize winners Michael Atiyah (left) and index theo- in mathematics than earlier. There are a lot of con- Isadore Singer. rem using ferences, we see more papers that are written by two, K-theory; three, or even more authors—is that a necessary another is the heat equation proof that makes the and commendable development or has it drawbacks formulas that are topological more geometric and as well? explicit. Each theorem and proof has merit and Atiyah: It is not like in physics or chemistry has different applications. where you have fifteen authors because they need an enormous big machine. It is not absolutely nec- Collaboration essary or fundamental. But particularly if you are R & S: Both of you contributed to the Index Theo- dealing with areas that have rather mixed and in- rem with different expertise and visions—and other terdisciplinary backgrounds, with people who have people had a share as well, I suppose. Could you de- different expertise, it is much easier and faster. It scribe this collaboration and the establishment of is also much more interesting for the participants. the result a little more closely? To be a mathematician on your own in your office Singer: Well, I came with a background in analy- can be a little bit dull, so interaction is stimulat- sis and differential geometry, and Sir Michael’s ex- ing, both psychologically and mathematically. It has pertise was in algebraic geometry and topology. For to be admitted that there are times when you go the purposes of the Index Theorem, our areas of solitary in your office, but not all the time! It can expertise fit together hand in glove. Moreover, in also be a social activity with lots of interaction. You a way, our personalities fit together, in that “any- need a good mix of both; you can’t be talking all thing goes”: Make a suggestion—and whatever it the time. But talking some of the time is very stim- was, we would just put it on the blackboard and ulating. Summing up, I think that it is a good de- work with it; we would both enthusiastically explore velopment—I do not see any drawbacks. it; if it didn’t work, it didn’t work. But often enough, Singer: Certainly computers have made collab- some idea that seemed far-fetched did work. We oration much easier. Many mathematicians col- both had the freedom to continue without worry- laborate by computer instantly; it’s as if they were ing about where it came from or where it would talking to each other. I am unable to do that. A lead. It was exciting to work with Sir Michael all sobering counterexample to this whole trend is these years. And it is as true today as it was when Perelman’s results on the Poincaré conjecture: we first met in ’55—that sense of excitement and He worked alone for ten to twelve years, I think, “anything goes” and “let’s see what happens”. before putting his preprints on the Net. Atiyah: No doubt: Singer had a strong expertise Atiyah: Fortunately, there are many different and background in analysis and differential geom- kinds of mathematicians, they work on different etry. And he knew certainly more physics than I did; subjects, they have different approaches and dif- it turned out to be very useful later on. My back- ferent personalities—and that is a good thing. We ground was in algebraic geometry and topology, so do not want all mathematicians to be isomorphic, it all came together. But of course there are a lot of people who contributed in the background to the 2Among those: Newton, Gauss, Cauchy, Laplace, Abel, buildup of the Index Theorem—going back to Abel, Jacobi, Riemann, Weierstrass, Lie, Picard, Poincaré, Castel- Riemann, much more recently Serre, who got the nuovo, Enriques, Severi, Hilbert, Lefschetz, Hodge, Todd, Abel Prize last year, Hirzebruch, Grothendieck, and Leray, Cartan, Serre, Kodaira, Spencer, Dirac, Pontrjagin, Bott. There was lots of work from the algebraic Chern, Weil, Borel, Hirzebruch, Bott, Eilenberg, Grothen- geometry side and from topology that prepared the dieck, Hörmander, Nirenberg. 224 NOTICES OF THE AMS VOLUME 52, NUMBER 2 we want variety: different mountains need differ- Singer: The Index theorem was in fact instru- ent kinds of techniques to climb. mental in breaking barriers between fields.