The n-Category Café

He is most famous for his contribution to the Atiyah–Singer index theorem, proved in 1963, so let me say a word about that.

Briefly put, the Atiyah–Singer index theorem gives a topological formula for the dimension of the space of solutions of a differential equation on a manifold. As such, it’s a great way to use topology to learn something about the solutions of differential equations without actually solving them. And conversely, it’s a great way to use differential equations to learn things about the topology of manifolds!

More precisely, the Atiyah–Singer index theorem is a formula for the “index” of a differential equation, where the index is the dimension of the space of solutions (the “kernel”) minus something else (the dimension of the “cokernel”). So, you have to be clever to use it to get precise information about the dimension of the space of solutions. But it easily gives a lower bound.

Furthermore, it only works when the manifold is compact and the differential equation is “elliptic” (like the Laplace equation, not like the wave equation or heat equation).

On the bright side, it works for differential equations where the solutions are not just functions, but vector fields, spinor fields, or more general sections of vector bundles. A great example is the elliptic version of the Dirac equation — the equation that describes the behavior of an electron. In fact the Dirac equation turns out to be the key to the general case! Because this equation involves spinors and Clifford algebras, the Atiyah–Singer index theorem highlights the role of spinors and Clifford algebras in topology.

I spent a lot of time as an undergrad reading Seminar on the Atiyah-Singer Theorem, by Richard Palais. It had great explanations of the analysis prerequisites, like pseudodifferential operators on vector bundles, Sobolev spaces, and chain complexes of Hilbert spaces. Somehow all that seems second nature to me by now. But I still don’t feel I have a great intuitive feel for the topology: that is, why the formula for the index in terms of characteristic classes is what it is.

When I went to grad school at MIT, Singer was one of the stars there: he ran a seminar on mathematics connected to quantum field theory, and a huge crowd attended, including Raoul Bott and many other bigshots. When Singer’s friends like Witten or Atiyah came to town, he’d tell us about their talks, and we’d all go. There was electricity in the air.

At that time — say, 1982-1986 — it seemed that everyone wanted to learn the Atiyah–Singer index theorem and generalize the heck out of it. My friend Varghese Mathai and I attended a seminar where Dan Quillen was trying to find a really simple proof of the Atiyah–Singer theorem. It was very exciting, because Quillen was working in real time. Each class he would start flawlessly, and then continue until he got stuck on something. In the end he was scooped by Ezra Getzler, who however used some fancy analysis that Quillen would have wanted to avoid. But still, one fun thing about this seminar is that given all his work on homotopy theory, you would not expect Quillen to be messing around with differential equations. And the work wasn’t wasted: the Mathai–Quillen formalism was born in this seminar.

So, the importance of the Atiyah–Singer index theorem was in part sociological. It built a bridge between two fields — partial differential equations and topology — which had not existed before. And because the differential equations it applies to are important in physics, it connected physics more firmly to both these fields. So the theorem, very much like Singer himself, brought people together and sparked new ideas.