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. I in mathematics than earlier. There are a Lott’s seminar at the University of remember that he told me that he enjoyed lot of conferences, we see more papers Michigan and Tian’s seminar at Princeton. very much being a topologist: He had so that are written by two, three or even more Although no one vouches for the final many friends within topology, and it was authors – is that a necessary and com- details, it appears that Perelman’s proof such a great community. “It would be a mendable development or has it draw- will be validated. tragedy if one day I would have a brilliant backs as well? As to your first question: When any two idea within functional analysis and would ATIYAH It is not like in physics or chem- subjects use each other’s techniques in a have to leave all my topology friends and istry where you have 15 authors because new way, frequently, something special to go out and work with a different group they need an enormous big machine. It is happens. In geometry, analysis is very of people.” He regarded it to be his duty to not absolutely necessary or fundamental. important; for existence theorems, the do so, but he would be very reluctant. But particularly if you are dealing with more the better. It is not surprising that Somehow, we have been very fortunate. areas which have rather mixed and inter- some new [at least to me] analysis implies Things have moved in such a way that we disciplinary backgrounds, with people who something interesting about the Poincaré got involved with functional analysts with- have different expertise, it is much easier conjecture. out losing our old friends; we could bring and faster. It is also much more interesting ATIYAH I prefer to go even further – I them all with us. Alain Connes was in for the participants. To be a mathematician really do not believe in the division of functional analysis, and now we interact on your own in your office can be a little bit mathematics into specialities; already if closely. So we have been fortunate to dull, so interaction is stimulating, both psy- you go back into the past, to Newton and maintain our old links and move into new chologically and mathematically. It has to Gauss… Although there have been times, ones – it has been great fun. be admitted that there are times when you particularly post-Hilbert, with the axiomat- go solitary in your office, but not all the ic approach to mathematics in the first half Mathematics and physics time! It can also be a social activity with of the twentieth century, when people lots of interaction. You need a good mix of began to specialize, to divide up. The We would like to have your comments on both, you can’t be talking all the time. But Bourbaki trend had its use for a particular the interplay between physics and mathe- talking some of the time is very stimulat- time. But this is not part of the general atti- matics. There is Galilei’s famous dictum ing. Summing up, I think that it is a good tude to mathematics: Abel would not have from the beginning of the scientific revo- development – I do not see any drawbacks. distinguished between algebra and analy- EMS September 2004 25 INTERVIEW lution, which says that the Laws of Nature of mathematics having evolved out of the There are times when this goes out of fash- are written in the language of mathemat- physical world; it therefore is not a big sur- ion or when parts of mathematics evolve ics. Why is it that the objects of mathe- prise that it has a feedback into it. purely internally. Lots of abstract mathe- matical creation, satisfying the criteria of More fundamentally: to understand the matics does not directly relate to the out- beauty and simplicity, are precisely the outside world as a human being is an side world. ones that time and time again are found to attempt to reduce complexity to simplicity. It is one of the strengths of mathematics be essential for a correct description of the What is a theory? A lot of things are hap- that it has these two and not a single external world? Examples abound, let me pening in the outside world, and the aim of lifeblood: one external and one internal, just mention group theory and, yes, your scientific inquiry is to reduce this to as sim- one arising as response to external events, Index Theorem! ple a number of principles as possible. the other to internal reflection on what we SINGER There are several approaches in That is the way the human mind works, the are doing. answer to your questions; I will discuss way the human mind wants to see the SINGER Your statement is too strong. I two. First, some parts of mathematics were answer. agree with Michael that mathematics is created in order to describe the world If we were computers, which could tabu- blessed with both an external and internal around us. Calculus began by explaining late vast amounts of all sorts of informa- source of inspiration. In the past several the motion of planets and other moving tion, we would never develop theory – we decades, high energy theoretical physics objects. Calculus, differential equations, would say, just press the button to get the has had a marked influence on mathemat- and integral equations are a natural part of answer. We want to reduce this complexi- ics. Many mathematicians have been physics because they were developed for ty to a form that the human mind can shocked at this unexpected development: physics. Other parts of mathematics are understand, to a few simple principles. new ideas from outside mathematics so also natural for physics. I remember lec- That’s the nature of scientific inquiry, and effective in mathematics. We are delighted turing in Feynman’s seminar, trying to mathematics is a part of that. Mathematics with these new inputs, but the “shock” explain anomalies. His postdocs kept is an evolution from the human brain, exaggerates their overall effect on mathe- wanting to pick coordinates in order to which is responding to outside influences, matics. compute; he stopped them saying: “The creating the machinery with which it then Laws of Physics are independent of a coor- attacks the outside world. It is our way of Newer developments dinate system. Listen to what Singer has to trying to reduce complexity into simplicity, say, because he is describing the situation beauty and elegance. It is really very fun- Can we move to newer developments with without coordinates.” Coordinate-free damental, simplicity is in the nature of sci- impact from the Atiyah-Singer Index means geometry. It is natural that geome- entific inquiry – we do not look for com- Theorem? I.e., String Theory and try appears in physics, whose laws are plicated things. Edward Witten on the one hand and on independent of a coordinate system. I tend to think that science and mathe- the other hand Non-commutative Symmetries are useful in physics for matics are ways the human mind looks and Geometry represented by Alain Connes. much the same reason they’re useful in experiences – you cannot divorce the Could you describe the approaches to mathematics. Beauty aside, symmetries human mind from it. Mathematics is part mathematical physics epitomized by these simplify equations, in physics and in math- of the human mind. The question whether two protagonists? ematics. So physics and math have in com- there is a reality independent of the human ATIYAH I tried once in a talk to describe mon geometry and group theory, creating a mind, has no meaning – at least, we cannot the different approaches to progress in close connection between parts of both answer it. physics like different religions. You have subjects. prophets, you have followers – each Secondly, there is a deeper reason if your Is it too strong to say that the mathemati- prophet and his followers think that they question is interpreted as in the title of cal problems solved and the techniques have the sole possession of the truth. If you Eugene Wigner’s essay “The that arose from physics have been the take the strict point of view that there are Unreasonable Effectiveness of lifeblood of mathematics in the past; or at several different religions, and that the Mathematics in the Natural Sciences”3. least for the last 25 years? intersection of all these theories is empty, Mathematics studies coherent systems ATIYAH I think you could turn that into then they are all talking nonsense. Or you which I will not try to define. But it stud- an even stronger statement. Almost all can take the view of the mystic, who thinks ies coherent systems, the connections mathematics originally arose from external that they are all talking of different aspects between such systems and the structure of reality, even numbers and counting. At of reality, and so all of them are correct. I such systems. We should not be too sur- some point, mathematics then turned to ask tend to take the second point of view. The prised that mathematics has coherent sys- internal questions, e.g. the theory of prime main “orthodox” view among physicists is tems applicable to physics. It remains to be numbers, which is not directly related to certainly represented by a very large group seen whether there is an already developed experience but evolved out of it. of people working with string theory like coherent system in mathematics that will There are parts of mathematics where the Edward Witten. There are a small number describe the structure of string theory. [At human mind asks internal questions just of people who have different philosophies, present, we do not even know what the out of curiosity. Originally it may be phys- one of them is Alain Connes, and the other symmetry group of string field theory is.] ical, but eventually it becomes something is Roger Penrose. Each of them has a very Witten has said that 21st century mathe- independent. There are other parts that specific point of view; each of them has matics has to develop new mathematics, relate much closer to the outside world very interesting ideas. Within the last few perhaps in conjunction with physics intu- with much more interaction backwards and years, there has been non-trivial interaction ition, to describe the structure of string the- forward. In that part of it, physics has for a between all of these. ory. long time been the lifeblood of mathemat- They may all represent different aspects ATIYAH I agree with Singer’s description ics and inspiration for mathematical work. of reality and eventually, when we under 26 EMS September 2004 INTERVIEW impact on number theory as the ideas flow across mathematics – on one extreme num- ber theory, on the other physics, and in the middle geometry: the wind is blowing, and it will eventually reach to the farthest extremities of number theory and give us a new point of view. Many problems that are worked upon today with old-fashioned ideas will be done with new ideas. I would like to see this happen: it could be the Riemann hypothesis, it could be the Langlands program or a lot of other related things. I had an argument with Andrew Wiles where I claimed that physics will have an impact on his kind of number the- ory; he thinks this is nonsense but we had a good argument. I would also like to make another predic- tion, namely that fundamental progress on the physics/mathematics front, String Isadore Singer and Sir Michael Atiyah receive the Theory questions etc., will emerge from a Abelprize from King Harald. (Photo: Scanpix) much more thorough understanding of stand it all, we may say “Ah, yes, they are in String Theory that will give us greater classical four-dimensional geometry, of all part of the truth”. I think that that will insight into the structure of the subject, and Einstein’s Equations etc. The hard part of happen. It is difficult to say which will be along with that will come new uses of physics in some sense is the non-linearity dominant, when we finally understand the mathematics. of Einstein’s Equations. Everything that picture – we don’t know. But I tend to be Alain Connes’ program is very natural – has been done at the moment is circum- open-minded. The problem with a lot of if you want to combine geometry with venting this problem in lots of ways. They physicists is that they have a tendency to quantum mechanics, then you really want haven’t really got to grips with the hardest “follow the leader”: as soon as a new idea to quantize geometry, and that is what non- part. Big progress will come when people comes up, ten people write ten or more commutative geometry means. Non-com- by some new techniques or new ideas real- papers on it and the effect is that everything mutative Geometry has been used effec- ly settle that. Whether you call that geom- can move very fast in a technical direction. tively in various parts of String Theory etry, differential equations or physics But big progress may come from a differ- explaining what happens at certain singu- depends on what is going to happen, but it ent direction; you do need people who are larities, for example. I think it may be an could be one of the big breakthroughs. exploring different avenues. And it is very interesting way of trying to describe black These are of course just my speculations. good that we have people like Connes and holes and to explain the Big Bang. I would SINGER I will be speculative in a slightly Penrose with their own independent line encourage young physicists to understand different way, though I do agree with the from different origins. I am in favour of non-commutative geometry more deeply number theory comments that Sir Michael diversity. I prefer not to close the door or than they presently do. Physicists use only mentioned, particularly theta functions to say “they are just talking nonsense”. parts of non-commutative geometry; the entering from physics in new ways. I think SINGER String Theory is in a very special theory has much more to offer. I do not other fields of physics will affect mathe- situation at the present time. Physicists know whether it is going to lead anywhere matics - like statistical mechanics and con- have found new solutions on their land- or not. But one of my projects is to try and densed matter physics. For example, I pre- scape - so many that you cannot expect to redo some known results using non-com- dict a new subject of statistical topology. make predictions from String Theory. Its mutative geometry more fully. Rather than count the number of holes, original promise has not been fulfilled. Betti-numbers, etc., one will be more inter- Nevertheless, I am an enthusiastic support- If you should venture a guess, which ested in the distribution of such objects on er of Super String Theory, not just because mathematical areas do you think are noncompact manifolds as one goes out to of what it has done in mathematics, but also going to witness the most important devel- infinity. We already have precursors in the because as a coherent whole, it is a marvel- opments in the coming years? number of zeros and poles for holomorphic lous subject. Every few years new devel- ATIYAH One quick answer is that the functions. The theory that we have for opments in the theory give additional most exciting developments are the ones holomorphic functions will be generalized, insight. When that happens, you realize which you cannot predict. If you can pre- and insights will come from condensed how little one understood about String dict them, they are not so exciting. So, by matter physics as to what, statistically, the Theory previously. The theory of D-branes definition, your question has no answer. topology might look like as one approach- is a recent example. Often there is mathe- Ideas from physics, e.g. Quantum es infinity. matics closely associated with these new Theory, have had an enormous impact so insights. Through D-branes, K-theory far, in geometry, some parts of algebra, and Continuity of mathematics entered String Theory naturally and in topology. The impact on number theory reshaped it. We just have to wait and see has still been quite small, but there are Mathematics has become so specialized, it what will happen. I am quite confident that some examples. I would like to make a seems, that one may fear that the subject physics will come up with some new ideas rash prediction that it will have a big will break up into separate areas. Is there EMS September 2004 27 INTERVIEW a core holding things together? like biology these days where there is lots ATIYAH I like to think there is a core to be discovered. Communication of mathematics holding things together, and that the core is When I was young the job market was rather what I look at myself; but we tend to good. It was important to be at a major uni- Next topic: Communication of mathe- be rather egocentric. The traditional parts versity but you could still prosper at a matics: Hilbert, in his famous speech at of mathematics, which evolved - geometry, smaller one. I am distressed by the coer- the International Congress in 1900, in calculus and algebra - all centre on certain cive effect of today’s job market. Young order to make a point about mathemati- notions. As mathematics develops, there mathematicians should have the freedom cal communication, cited a French are new ideas, which appear to be far from of choice we had when we were young. mathematician who said: “A mathemati- the centre going off in different directions, cal theory is not to be considered com- which I perhaps do not know much about. The next question concerns the continuity plete until you have made it so clear that Sometimes they become rather important of mathematics. Rephrasing slightly a you can explain it to the first man whom for the whole nature of the mathematical question that you, Prof. Atiyah are the ori- you meet on the street”. In order to pass enterprise. It is a bit dangerous to restrict gin of, let us make the following on to new generations of mathemati- the definition to just whatever you happen gedanken experiment: If, say, Newton or cians the collective knowledge of the pre- to understand yourself or think about. For Gauss or Abel were to reappear in our vious generation, how important is it example, there are parts of mathematics midst, do you think they would under- that the results have simple and elegant that are very combinatorial. Sometimes stand the problems being tackled by the proofs? they are very closely related to the continu- present generation of mathematicians – ATIYAH The passing of mathematics on ous setting, and that is very good: we have after they had been given a short refresh- to subsequent generations is essential for interesting links between combinatorics er course? Or is present day mathematics the future, and this is only possible if and algebraic geometry and so on. They too far removed from traditional mathe- every generation of mathematicians may also be related to e.g. statistics. I think matics? understands what they are doing and dis- that mathematics is very difficult to con- ATIYAH The point that I was trying to tils it out in such a form that it is easily strain; there are also all sorts of new appli- make there was that really important understood by the next generation. Many cations in different directions. progress in mathematics is somewhat inde- complicated things get simple when you It is nice to think of mathematics having pendent of technical jargon. Important have the right point of view. The first a unity; however, you do not want it to be ideas can be explained to a really good proof of something may be very compli- a straitjacket. The centre of gravity may mathematician, like Newton or Gauss or cated, but when you understand it well, change with time. It is not necessarily a Abel, in conceptual terms. They are in fact you readdress it, and eventually you can fixed rigid object in that sense, I think it coordinate-free, more than that, technolo- present it in a way that makes it look should develop and grow. I like to think of gy-free and in a sense jargon-free. You much more understandable – and that’s mathematics having a core, but I do not don’t have to talk of ideals, modules or the way you pass it on to the next genera- want it to be rigidly defined so that it whatever – you can talk in the common tion! Without that, we could never make excludes things which might be interesting. language of scientists and mathematicians. progress - we would have all this messy You do not want to exclude somebody who The really important progress mathematics stuff. Mathematics does depend on a suf- has made a discovery saying: “You are out- has made within 200 years could easily be ficiently good grasp, on understanding of side, you are not doing mathematics, you understood by people like Gauss and the fundamentals so that we can pass it on are playing around”. You never know! Newton and Abel. Only a small refresher in as simple a way as possible to our suc- That particular discovery might be the course where they were told a few terms – cessors. That has been done remarkably mathematics of the next century; you have and then they would immediately under- successfully for centuries. Otherwise, got to be careful. Very often, when new stand. how could we possibly be where we are? ideas come in, they are regarded as being a Actually, my pet aversion is that many In the 19th century, people said: “There is bit odd, not really central, because they mathematicians use too many technical so much mathematics, how could anyone look too abstract. terms when they write and talk. They were make any progress?” Well, we have - we SINGER Countries differ in their attitudes trained in a way that if you do not say it 100 do it by various devices, we generalize, about the degree of specialization in math- percent correctly, like lawyers, you will be we put all things together, we unify by ematics and how to treat the problem of too taken to court. Every statement has to be new ideas, we simplify lots of the con- much specialization. In the United States I fully precise and correct. When talking to structions – we are very successful in observe a trend towards early specializa- other people or scientists, I like to use mathematics and have been so for several tion driven by economic considerations. words that are common to the scientific hundred years. There is no evidence that You must show early promise to get good community, not necessarily just to mathe- this has stopped: in every new generation, letters of recommendations to get good first maticians. And that is very often possible. there are mathematicians who make enor- jobs. You can’t afford to branch out until If you explain ideas without a vast amount mous progress. How do they learn it all? you have established yourself and have a of technical jargon and formalism, I am It must be because we have been success- secure position. The realities of life force a sure it would not take Newton, Gauss and ful communicating it. narrowness in perspective that is not inher- Abel long – they were bright guys, actual- SINGER I find it disconcerting speaking ent to mathematics. We can counter too ly! to some of my young colleagues, because much specialization with new resources SINGER One of my teachers at Chicago they have absorbed, reorganized, and sim- that would give young people more free- was André Weil, and I remember his say- plified a great deal of known material into dom than they presently have, freedom to ing: “If Riemann were here, I would put a new language, much of which I don’t explore mathematics more broadly, or to him in the library for a week, and when he understand. Often I’ll finally say, “Oh; is explore connections with other subjects, came out he would tell us what to do next.” that all you meant?” Their new conceptu- 28 EMS September 2004 INTERVIEW al framework allows them to encompass out my ideas, I’m wrong 99% of the time. I succinctly considerably more than I can Individual work style learn from that and from studying the ideas, express with mine. Though impressed techniques, and procedures of successful with the progress, I must confess impa- I heard you, Prof. Atiyah, mention that one methods. My stubbornness wastes lots of tience because it takes me so long to reason for your choice of mathematics for time and energy. But on the rare occasion understand what is really being said. your career was that it is not necessary to when my internal sense of mathematics is remember a lot of facts by heart. right, I’ve done something different. Has the time passed when deep and Nevertheless, a lot of threads have to be important theorems in mathematics can woven together when new ideas are devel- Both of you have passed ordinary retire- be given short proofs? In the past, there oped. Could you tell us how you work best, ment age several years ago. But you are are many such examples, e.g., Abel’s how do new ideas arrive? still very active mathematicians, and you one-page proof of the addition theorem ATIYAH My fundamental approach to have even chosen retirement or visiting of algebraic differentials or Goursat’s doing research is always to ask questions. positions remote from your original work proof of Cauchy’s integral theorem. You ask “Why is this true?” when there is places. What are the driving forces for ATIYAH I do not think that at all! Of something mysterious or if a proof seems keeping up your work? Is it wrong that course, that depends on what foundations very complicated. I used to say – as a kind mathematics is a “young man’s game” as you are allowed to start from. If we have of joke – that the best ideas come to you dur- Hardy put it? to start from the axioms of mathematics, ing a bad lecture. If somebody gives a terri- ATIYAH It is no doubt true that mathemat- then every proof will be very long. The ble lecture, it may be a beautiful result but ics is a young man’s game in the sense that common framework at any given time is with terrible proofs, you spend your time you peak in your twenties or thirties in terms constantly advancing; we are already at a trying to find better ones, you do not listen of intellectual concentration and in original- high platform. If we are allowed to start to the lecture. It is all about asking ques- ity. But later you compensate that by expe- within that framework, then at every stage tions – you simply have to have an inquisi- rience and other factors. It is also true that there are short proofs. tive mind! Out of ten questions, nine will if you haven’t done anything significant by One example from my own life is this lead nowhere, and one leads to something the time you are forty, you will not do so famous problem about vector fields on productive. You constantly have to be suddenly. But it is wrong that you have to spheres solved by Frank Adams where the inquisitive and be prepared to go in any decline, you can carry on, and if you man- proof took many hundreds of pages. One direction. If you go in new directions, then age to diversify in different fields this gives day I discovered how to write a proof on you have to learn new material. you a broad coverage. The kind of mathe- a postcard. I sent it over to Frank Adams Usually, if you ask a question or decide to matician who has difficulty maintaining the and we wrote a little paper which then solve a problem, it has a background. If you momentum all his life is a person who would fit on a bigger postcard. But of understand where a problem comes from decides to work in a very narrow field with course that used some K-theory; not that then it makes it easy for you to understand great depths, who e.g. spends all his life try- complicated in itself. You are always the tools that have to be used on it. You ing to solve the Poincaré conjecture – building on a higher platform; you have immediately interpret them in terms of your whether you succeed or not, after 10-15 always got more tools at your disposal own context. When I was a student, I years in this field you exhaust your mind; that are part of the lingua franca which learned things by going to lectures and read- and then, it may be too late to diversify. If you can use. In the old days you had a ing books – after that I read very few books. you are the sort of person that chooses to smaller base: If you make a simple proof I would talk with people; I would learn the make restrictions to yourself, to specialize in nowadays, then you are allowed to essence of analysis by talking to Hörmander a field, you will find it harder and harder – assume that people know what group the- or other people. I would be asking ques- because the only things that are left are hard- ory is, you are allowed to talk about tions because I was interested in a particular er and harder technical problems in your Hilbert space. Hilbert space took a long problem. So you learn new things because own area, and then the younger people are time to develop, so we have got a much you connect them and relate them to old better than you. bigger vocabulary, and with that we can ones, and in that way you can start to spread You need a broad base, from which you write more poetry. around. can evolve. When this area dries out, then SINGER Often enough one can distil the If you come with a problem, and you need you go to that area – or when the field as a ideas in a complicated proof and make to move to a new area for its solution, then whole, internationally, changes gear, you that part of a new language. The new you have an introduction – you have already can change too. The length of the time you proof becomes simpler and more illumi- a point of view. Interacting with other peo- can go on being active within mathematics nating. For clarity and logic, parts of the ple is of course essential: if you move into a very much depends on the width of your original proof have been set aside and dis- new field, you have to learn the language, coverage. You might have contributions to cussed separately. you talk with experts; they will distil the make in terms of perspective, breadth, inter- ATIYAH Take your example of Abel’s essentials out of their experience. I did not actions. A broad coverage is the secret of a Paris memoir: His contemporaries did not learn all the things from the bottom happy and successful long life in mathemat- find it at all easy. It laid the foundation of upwards; I went to the top and got the ical terms. I cannot think of any counter the theory. Only later on, in the light of insight into how you think about analysis or example. that theory, we can all say: “Ah, what a whatever. SINGER I became a graduate student at the beautifully simple proof!” At the time, all SINGER I seem to have some built-in sense University of Chicago after three years in the ideas had to be developed, and they of how things should be in mathematics. At the US army during World War II. I was were hidden, and most people could not a lecture, or reading a paper, or during a dis- older and far behind in mathematics. So I read that paper. It was very, very far from cussion, I frequently think, “that’s not the was shocked when my fellow graduate stu- appearing easy for his contemporaries. way it is supposed to be.” But when I try dents said, “If you haven’t proved the EMS September 2004 29 INTERVIEW to make sure that Georgia is part of Europe. Now, the politicians have to decide where the borders of Europe are. It is good that the EMS exists; but you should think rather broadly about how it is evolving as Europe evolves, as the world evolves, as mathematics evolves. What should its function be? How should it relate to national societies? How should it relate to the AMS? How should it relate to the governmental bodies? It is an opportunity! It has a role to play!

Apart from mathematics…

The Abel lectures were given by I. Singer, E. Witten, M. Atiyah and J.- M. Bismut Could you tell us in a few words about your Riemann Hypothesis by age thirty, you countries; on the other hand, mathemati- main interests besides mathematics? might as well commit suicide.” How infan- cians are much more argumentative. When SINGER I love to play tennis, and I try to tile! Age means little to me. What keeps me it comes to the fine details of a constitution, do so 2-3 times a week. That refreshes me going is the excitement of what I’m doing then they are terrible; they are worse than and I think that it has helped me work hard and its possibilities. I constantly check [and lawyers. But eventually – in principle – the in mathematics all these years. collaborate!] with younger colleagues to be good will was there for collaboration. ATIYAH Well, I do not have his energy! I sure that I’m not deluding myself – that Fortunately, the timing was right. In the like to walk in the hills, the Scottish hills – I what we are doing is interesting. So I’m meantime, Europe had solved some of its have retired partly to Scotland. In happily active in mathematics. Another rea- other problems: the Berlin Wall had come Cambridge, where I was before, the highest son is, in a way, a joke. String Theory needs down – so suddenly there was a new Europe hill was about this (gesture) big. Of course us! String Theory needs new ideas. Where to be involved in the EMS. This very fact you have got even bigger ones in Norway. I will they come from, if not from Sir Michael made it possible to get a lot more people spent a lot of my time outdoors and I like to and me? interested in it. It gave an opportunity for a plant trees, I like nature. I believe that if you ATIYAH Well, we have some students… broader base of the EMS with more oppor- do mathematics, you need a good relaxation SINGER Anyway, I am very excited about tunities and also relations to the European which is not intellectual – being outside in the interface of geometry and physics, and Commission and so on. the open air, climbing a mountain, working delighted to be able to work at that frontier. Having been involved with the set-up, I in your garden. But you actually do mathe- withdrew and left it to others to carry on. I matics meanwhile. While you go for a long History of the EMS have not followed in detail what has been walk in the hills or you work in your garden happening except that it seems to be active. – the ideas can still carry on. My wife com- You, Prof. Atiyah, have been very much I get my Newsletter, and I see what is going plains, because when I walk she knows I am involved in the establishment of the on. thinking of mathematics. European Mathematical Society around Roughly at the same time as the collapse SINGER I can assure you, tennis does not 1990. Are you satisfied with its develop- of the Berlin Wall, mathematicians in gen- allow that! ment since then? eral – both in Europe and in the USA – ATIYAH Let me just comment a little on began to be more aware of their need to be Thank you very much on behalf of the my involvement. It started an awful long socially involved and that mathematics had Norwegian, the Danish, and the European time ago, probably about 30 years ago. an important role to play in society. Instead Mathematical Societies! When I started trying to get people interest- of being shut up in their universities doing ed in forming a European Mathematical just their mathematics, they felt that there The interviewers were Martin Raussen, Society in the same spirit as the European was some pressure to get out and get Aalborg University, Denmark, and Physical Society, I thought it would be easy. involved in education, etc. The EMS took Christian Skau, Norwegian University of I got mathematicians from different coun- on this role at a European level, and the Science and Technology, Trondheim, tries together and it was like a mini-UN: the EMS congresses – I was involved in the one Norway. French and the Germans wouldn’t agree; we in Barcelona – definitely made an attempt to spent years arguing about differences, and – interact with the public. I think that these 1 More details were given in the laureates’ unlike in the real UN – where eventually at are additional opportunities over and above lectures. the end of the day you are dealing with real the old-fashioned role of learned societies. 2 Among those: Newton, Gauss, Cauchy, problems of the world and you have to come There are a lot of opportunities both in terms Laplace, Abel, Jacobi, Riemann, to an agreement sometime; in mathematics, of the geography of Europe and in terms of Weierstrass, Lie, Picard, Poincaré, it was not absolutely essential. We went on the broader reach. Castelnuovo, Enriques, Severi, Hilbert, for probably 15 years, before we founded Europe is getting ever larger: when we Lefschetz, Hodge, Todd, Leray, Cartan, the EMS. started we had discussions about where Serre, Kodaira, Spencer, Dirac, On the one hand, mathematicians have were the borders of Europe. We met people Pontrjagin, Chern, Weil, Borel, much more in common than politicians, we from Georgia, who told us very clearly, that Hirzebruch, Bott, Eilenberg, are international in our mathematical life, it the boundary of Europe is this river on the Grothendieck, Hörmander, Nirenberg. is easy to talk to colleagues from other other side of Georgia; they were very keen 3 Comm. Pure App. Math. 13(1), 1960. 30 EMS September 2004