Collaboration

Both of you contributed to the index theorem with different expertise and visions - and other people had a share as well, I suppose. Could you describe this collaboration and the establishment of the result a little closer?

SINGER: Well, I came with a background in analysis and differential geometry, and Sir Michael's expertise was in algebraic geometry and topology. For the purposes of the Index Theorem, our areas of expertise fit together hand in glove. Moreover, in a way, our personalities fit together, in that "anything goes": Make a suggestion - and whatever it was, we would just put it on the blackboard and work with it; we would both enthusiastically explore it; if it didn't work, it didn't work. But often, enough some idea that seemed far-fetched did work. We both had the freedom to continue without worrying about where it came from or where it would lead. It was exciting to work with Sir Michael all these years. And it is as true today as it was when we first met in '55 - that sense of excitement and "anything goes" and "let's see, what happens".

ATIYAH: No doubt: Singer had a strong expertise and background in analysis and differential geometry. And he knew certainly more physics than I did; it turned out to be very useful later on. My background was in algebraic geometry and topology, so it all came together. But of course there are a lot of people who contributed in the background to the build-up of the Index Theorem - going back to Abel, Riemann, much more recently Serre, who got the Abel prize last year, Hirzebruch, Grothendieck and Bott. There was lots of work from the algebraic geometry side and from topology that prepared the ground. And of course there are also a lot of people who did fundamental work in analysis and the study of differential equations: Hörmander, Nirenberg... In my lecture I will give a long list of names^[2]; even that one will be partial. It is an example of international collaboration; you do not work in isolation, neither in terms of time nor in terms of space - especially in these days. Mathematicians are linked so much, people travel around much more. We two met at the Institute at Princeton. It was nice to go to the Arbeitstagung in Bonn every year, which Hirzebruch organised and where many of these other people came. I did not realize that at the time, but looking back, I am very surprised how quickly these ideas moved...

Collaboration seems to play a bigger role in mathematics than earlier. There are a lot of conferences, we see more papers that are written by two, three or even more authors - is that a necessary and commendable development or has it drawbacks as well?

ATIYAH: It is not like in physics or chemistry where you have 15 authors because they need an enormous big machine. It is not absolutely necessary or fundamental. But particularly if you are dealing with areas, which have rather mixed and interdisciplinary backgrounds, with people who have different expertise, it is much easier and faster. It is also much more interesting for the participants. To be a mathematician on your own in your office can be a little bit dull, so interaction is stimulating, both psychologically and mathematically. It has to be admitted that there are times, when you go solitary in your office, but not all the time! It can also be a social activity with lots of interaction. You need a good mix of both, you can't be talking all the time. But talking some of the time is very stimulating. Summing up, I think that it is a good development - I do not see any drawbacks.

SINGER: Certainly computers have made collaboration much easier. Many mathematicians collaborate by computer instantly; it's as if they were talking to each other. I am unable to do that. A sobering counterexample to this whole trend is Perelman's results on the Poincaré conjecture: He worked alone for ten to twelve years, I think, before putting his preprints on the net.

ATIYAH: Fortunately, there are many different kinds of mathematicians, they work on different subjects, they have different approaches and different personalities - and that is a good thing. We do not want all mathematicians to be isomorphic, we want variety: different mountains need different kinds of techniques to climb.

SINGER: I support that. Flexibility is absolutely essential in our society of mathematicians.

Perelman's work on the Poincaré conjecture seems to be another instance where analysis and geometry apparently get linked very much together. It seems that geometry is profiting a lot from analytic perspectives. Is this linkage between different disciplines a general trend - is it true, that important results rely on this interrelation between different disciplines? And a much more specific question: What do you know about the status of the proof o f the Poincaré conjecture?

SINGER: To date, everything is working out as Perelman says. So I learn from Lott's seminar at the University of Michigan and Tian's seminar at Princeton. Although no one vouches for the final details, it appears that Perelman's proof will be validated.

As to your first question: When any two subjects use each other's techniques in a new way, frequently, something special happens. In geometry, analysis is very important; for existence theorems, the more the better. It is not surprising that some new [at least to me] analysis implies something interesting about the Poincaré conjecture.

ATIYAH: I prefer to go even further - I really do not believe in the division of mathematics in specialities; already if you go back into the past, to Newton and Gauss... Although there have been times, particularly post-Hilbert, with the axiomatic approach to mathematics in the first half of the twentieth century, when people began to specialize, to divide up. The Bourbaki trend had its use for a particular time. But this is not part of the general attitude to mathematics: Abel would not have distinguished between algebra and analysis. And I think the same goes for geometry and analysis for people like Newton.

It is artificial to divide mathematics into separate chunks, and then to say that you bring them together as though this is a surprise. On the contrary, they are all part of the puzzle of mathematics. Sometimes you would develop some things for their own sake for a while e.g. if you develop group theory by itself. But that is just a sort of temporary convenient division of labour. Fundamentally, mathematics should be used as a unity. I think the more examples we have of people showing that you can usefully apply analysis to geometry, the better. And not just analysis, I think that some physics came into it as well: Many of the ideas in geometry use physical insight as well - take the example of Riemann! This is all part of the broad mathematical tradition, which sometimes is in danger of being overlooked by modern, younger people who say "we have separate divisions". We do not want to have any of that kind, really.

SINGER: The Index Theorem was in fact instrumental in breaking barriers between fields. When it first appeared, many old-timers in special fields were upset that new techniques were entering their fields and achieving things they could not do in the field by old methods. A younger generation immediately felt freed from the barriers that we both view as artificial.

ATIYAH: Let me tell you a little story about Henry Whitehead, the topologist. I remember that he told me that he enjoyed very much being a topologist: He had so many friends within topology, and it was such a great community. "It would be a tragedy if one day I would have a brilliant idea within functional analysis and would have to leave all my topology friends and to go out and work with a different group of people." He regarded it to be his duty to do so, but he would be very reluctant.

Somehow, we have been very fortunate. Things have moved in such a way that we got involved with functional analysts without losing our old friends; we could bring them all with us. Alain Connes was in functional analysis, and now we interact closely. So we have been fortunate to maintain our old links and move into new ones - it has been great fun.

Next: Mathematics and physics

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