The Index Theorem

Michael Atiyah and Isadore Singer interviewed by Martin Raussen and Christian Skau

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 analysis in a surprising way". Both of you have an impressive list of fine achievements in mathematics. Is the Index Theorem your most important result and the result you are most pleased with in your entire careers?

ATIYAH: First, I would like to say that I prefer to call it a theory, not a theorem. Actually, we have worked on it for 25 years and if I include all the related topics, I have probably spent 30 years of my life working on the area. So it is rather obvious that it is the best thing I have done.

SINGER: I too, feel that the index theorem was but the beginning of a high point that has lasted to this very day. It's as if we climbed a mountain and found a plateau we've been on ever since.

We would like you to give us some comments on the history on the discovery of the Index Theorem.^[1] Were there precursors, conjectures in this direction already before you started? Were there only mathematical motivations or also physical ones?

ATIYAH: Mathematics is always a continuum, linked to its history, the past - nothing comes out of zero. And certainly the Index Theorem is simply a continuation of work that, I would like to say, began with Abel. So of course there are precursors. A theorem is never arrived at in the way that logical thought would lead you to believe or that posterity thinks. It is usually much more accidental, some chance discovery in answer to some kind of question. Eventually you can rationalize it and say that this is how it fits. Discoveries never happen as neatly as that. You can rewrite history and make it look much more logical, but actually it happens quite differently.

SINGER: At the time we proved the Index Theorem we saw how important it was in mathematics, but we had no inkling that it would have such an effect on physics some years down the road. That came as a complete surprise to us. Perhaps it should not have been a surprise because it used a lot of geometry and also quantum mechanics in a way, à la Dirac.

You worked out at least three different proofs with different strategies for the Index Theorem. Why did you keep on after the first proof? What different insights did the proofs give?

ATIYAH: I think it is said that Gauss had ten different proofs for the law of quadratic reciprocity. Any good theorem should have several proofs, the more the better. For two reasons: usually, different proofs have different strengths and weaknesses, and they generalize in different directions - they are not just repetitions of each other. And that is certainly the case with the proofs that we came up with. There are different reasons for the proofs, they have different histories and backgrounds. Some of them are good for this application, some are good for that application. They all shed light on the area. If you cannot look at a problem from different directions, it is probably not very interesting; the more perspectives, the better!

SINGER: There isn't just one theorem; there are generalizations of the theorem. One is the families index theorem using K-theory; another is the heat equation proof which makes the formulas that are topological, more geometric and explicit. Each theorem and proof has merit and has different applications.

Next: Collaboration

---

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