INTERVIEW InterviewInterview withwith MichaelMichael AtiyahAtiyah andand IsadoreIsadore SingerSinger Interviewers: Martin Raussen and Christian Skau The interview took place in Oslo, on the 24th of May 2004, prior to the Abel prize celebrations. The Index Theorem First, we congratulate both of you for having been awarded the Abel Prize 2004. This prize has been given to you for “the discovery and the proof of the Index Theorem connecting geometry and analy- sis in a surprising way and your outstand- ing role in building new bridges between mathematics and theoretical physics”. Both of you have an impressive list of fine achievements in mathematics. Is the Index Theorem your most important from left to right: I. Singer, M. Atiyah, M. Raussen, C. Skau result and the result you are most pleased with in your entire careers? cal, but actually it happens quite different- there are generalizations of the theorem. ATIYAH First, I would like to say that I ly. One is the families index theorem using K- prefer to call it a theory, not a theorem. SINGER At the time we proved the Index theory; another is the heat equation proof Actually, we have worked on it for 25 years Theorem we saw how important it was in which makes the formulas that are topolog- and if I include all the related topics, I have mathematics, but we had no inkling that it ical, more geometric and explicit. Each probably spent 30 years of my life working would have such an effect on physics some theorem and proof has merit and has differ- on the area. So it is rather obvious that it is years down the road. That came as a com- ent applications. the best thing I have done. plete surprise to us. Perhaps it should not SINGER I too, feel that the index theorem have been a surprise because it used a lot of Collaboration was but the beginning of a high point that geometry and also quantum mechanics in a has lasted to this very day. It’s as if we way, à la Dirac. Both of you contributed to the index theo- climbed a mountain and found a plateau rem with different expertise and visions – we’ve been on ever since. You worked out at least three different and other people had a share as well, I proofs with different strategies for the suppose. Could you describe this collabo- We would like you to give us some com- Index Theorem. Why did you keep on ration and the establishment of the result ments on the history of the discovery of after the first proof? What different a little closer? the Index Theorem.1 Were there precur- insights did the proofs give? SINGER Well, I came with a background sors, conjectures in this direction already ATIYAH I think it is said that Gauss had in analysis and differential geometry, and before you started? Were there only ten different proofs for the law of quadrat- Sir Michael’s expertise was in algebraic mathematical motivations or also physical ic reciprocity. Any good theorem should geometry and topology. For the purposes ones? have several proofs, the more the better. of the Index Theorem, our areas of exper- ATIYAH Mathematics is always a contin- For two reasons: usually, different proofs tise fit together hand in glove. Moreover, uum, linked to its history, the past - nothing have different strengths and weaknesses, in a way, our personalities fit together, in comes out of zero. And certainly the Index and they generalize in different directions - that “anything goes”: Make a suggestion - Theorem is simply a continuation of work they are not just repetitions of each other. and whatever it was, we would just put it that, I would like to say, began with Abel. And that is certainly the case with the on the blackboard and work with it; we So of course there are precursors. A theo- proofs that we came up with. There are dif- would both enthusiastically explore it; if it rem is never arrived at in the way that log- ferent reasons for the proofs, they have dif- didn’t work, it didn’t work. But often ical thought would lead you to believe or ferent histories and backgrounds. Some of enough, some idea that seemed far-fetched that posterity thinks. It is usually much them are good for this application, some did work. We both had the freedom to con- more accidental, some chance discovery in are good for that application. They all shed tinue without worrying about where it answer to some kind of question. light on the area. If you cannot look at a came from or where it would lead. It was Eventually you can rationalize it and say problem from different directions, it is exciting to work with Sir Michael all these that this is how it fits. Discoveries never probably not very interesting; the more per- years. And it is as true today as it was happen as neatly as that. You can rewrite spectives, the better! when we first met in ’55 - that sense of history and make it look much more logi- SINGER There isn’t just one theorem; excitement and “anything goes” and “let’s 24 EMS September 2004 INTERVIEW see what happens”. SINGER Certainly computers have made sis. And I think the same goes for geome- ATIYAH No doubt: Singer had a strong collaboration much easier. Many mathe- try and analysis for people like Newton. expertise and background in analysis and maticians collaborate by computer instant- It is artificial to divide mathematics into differential geometry. And he knew cer- ly; it’s as if they were talking to each other. separate chunks, and then to say that you tainly more physics than I did; it turned out I am unable to do that. A sobering coun- bring them together as though this is a sur- to be very useful later on. My background terexample to this whole trend is prise. On the contrary, they are all part of was in algebraic geometry and topology, so Perelman’s results on the Poincaré conjec- the puzzle of mathematics. Sometimes you it all came together. But of course there are ture: He worked alone for ten to twelve would develop some things for their own a lot of people who contributed in the back- years, I think, before putting his preprints sake for a while e.g. if you develop group ground to the build-up of the Index on the net. theory by itself. But that is just a sort of Theorem – going back to Abel, Riemann, ATIYAH Fortunately, there are many dif- temporary convenient division of labour. much more recently Serre, who got the ferent kinds of mathematicians, they work Fundamentally, mathematics should be Abel prize last year, Hirzebruch, on different subjects, they have different used as a unity. I think the more examples Grothendieck and Bott. There was lots of approaches and different personalities – we have of people showing that you can work from the algebraic geometry side and and that is a good thing. We do not want usefully apply analysis to geometry, the from topology that prepared the ground. all mathematicians to be isomorphic, we better. And not just analysis, I think that And of course there are also a lot of people want variety: different mountains need dif- some physics came into it as well: Many of who did fundamental work in analysis and ferent kinds of techniques to climb. the ideas in geometry use physical insight the study of differential equations: SINGER I support that. Flexibility is as well – take the example of Riemann! Hörmander, Nirenberg… In my lecture I absolutely essential in our society of math- This is all part of the broad mathematical will give a long list of names2; even that ematicians. tradition, which sometimes is in danger of one will be partial. It is an example of being overlooked by modern, younger peo- international collaboration; you do not Perelman’s work on the Poincaré conjec- ple who say “we have separate divisions”. work in isolation, neither in terms of time ture seems to be another instance where We do not want to have any of that kind, nor in terms of space – especially in these analysis and geometry apparently get really. days. Mathematicians are linked so much, linked very much together. It seems that SINGER The Index Theorem was in fact people travel around much more. We two geometry is profiting a lot from analytic instrumental in breaking barriers between met at the Institute at Princeton. It was nice perspectives. Is this linkage between dif- fields. When it first appeared, many old- to go to the Arbeitstagung in Bonn every ferent disciplines a general trend – is it timers in special fields were upset that new year, which Hirzebruch organised and true, that important results rely on this techniques were entering their fields and where many of these other people came. I interrelation between different disci- achieving things they could not do in the did not realize that at the time, but looking plines? And a much more specific ques- field by old methods. A younger genera- back, I am very surprised how quickly tion: What do you know about the status tion immediately felt freed from the barri- these ideas moved… of the proof of the Poincaré conjecture? ers that we both view as artificial. SINGER To date, everything is working ATIYAH Let me tell you a little story Collaboration seems to play a bigger role out as Perelman says. So I learn from about Henry Whitehead, the topologist.