Individual work style

I heard you, Prof. Atiyah, mention that one reason for your choice of mathematics for your career was that it is not necessary to remember a lot of facts by heart. Nevertheless, a lot of threads have to be woven together when new ideas are developed. Could you tell us how you work best, how do new ideas arrive?

ATIYAH: My fundamental approach to doing research is always to ask questions. You ask "Why is this true?" when there is something mysterious or if a proof seems very complicated. I used to say - as a kind of joke - that the best ideas come to you during a bad lecture. If somebody gives a terrible lecture, it may be a beautiful result but with terrible proofs, you spend your time trying to find better ones, you do not listen to the lecture. It is all about asking questions - you simply have to have an inquisitive mind! Out of ten questions, nine will lead nowhere, and one leads to something productive. You constantly have to be inquisitive and be prepared to go in any direction. If you go in new directions, then you have to learn new material.

Usually, if you ask a question or decide to solve a problem, it has a background. If you understand where a problem comes from then it makes it easy for you to understand the tools that have to be used on it. You immediately interpret them in terms of your own context. When I was a student, I learned things by going to lectures and reading books - after that I read very few books. I would talk with people; I would learn the essence of analysis by talking to Hörmander or other people. I would be asking questions because I was interested in a particular problem. So you learn new things because you connect them and relate them to old ones, and in that way you can start to spread around.

If you come with a problem, and you need to move to a new area for its solution, then you have an introduction - you have already a point of view. Interacting with other people is of course essential: if you move into a new field, you have to learn the language, you talk with experts; they will distil the essentials out of their experience. I did not learn all the things from the bottom upwards; I went to the top and got the insight into how you think about analysis or whatever.

SINGER: I seem to have some built-in sense of how things should be in mathematics. At a lecture, or reading a paper, or during a discussion, I frequently think, "that's not the way it is suppose to be." But when I try out my ideas, I'm wrong 99% of the time. I learn from that and from studying the ideas, techniques, and procedures of successful methods. My stubbornness wastes lots of time and energy. But on the rare occasion when my internal sense of mathematics is right, I've done something different.

Both of you have passed ordinary retirement age several years ago. But you are still very active mathematicians, and you have even chosen retirement or visiting positions remote from your original work places. What are the driving forces for keeping up your work? Is it wrong that mathematics is a "young man's game" as Hardy put it?

ATIYAH: It is no doubt true that mathematics is a young man's game in the sense that you peak in your twenties or thirties in terms of intellectual concentration and in originality. But later you compensate that by experience and other factors. It is also true that if you haven't done anything significant by the time you are forty, you will not do so suddenly. But it is wrong that you have to decline, you can carry on, and if you manage to diversify in different fields this gives you a broad coverage. The kind of mathematician who has difficulty maintaining the momentum all his life is a person who decides to work in a very narrow field with great depths, who e.g. spends all his life trying to solve the Poincaré conjecture - whether you succeed or not, after 10-15 years in this field you exhaust your mind; and then, it may be too late to diversify. If you are the sort of person that chooses to make restrictions to yourself, to specialize in a field, you will find it harder and harder - because the only things that are left are harder and harder technical problems in your own area, and then the younger people are better than you.

You need a broad base, from which you can evolve. When this area dries out, then you go to that area - or when the field as a whole, internationally, changes gear, you can change too. The length of the time you can go on being active within mathematics very much depends on the width of your coverage. You might have contributions to make in terms of perspective, breadth, interactions. A broad coverage is the secret of a happy and successful long life in mathematical terms. I cannot think of any counter example.

SINGER: I became a graduate student at the University of Chicago after three years in the US army during World War II. I was older and far behind in mathematics. So I was shocked when my fellow graduate students said, "If you haven't proved the Riemann Hypothesis by age thirty, you might as well commit suicide." How infantile! Age means little to me. What keeps me going is the excitement of what I'm doing and its possibilities. I constantly check [and collaborate!] with younger colleagues to be sure that I'm not deluding myself-that what we are doing is interesting. So I'm happily active in mathematics. Another reason is, in a way, a joke. String Theory needs us! String Theory needs new ideas. Where will they come from, if not from Sir Michael and me?

ATIYAH: Well, we have some students…

SINGER: Anyway, I am very excited about the interface of geometry and physics, and delighted to be able to work at that frontier.

Next: History of the EMS

HomeNews ArchiveCalendar      Editor: Anne Marie Astad  The Norwegian Academy of Science and Letters  E-mail: dnva@online.no