Working with Bott
This is a brief memoir (written at the request of R. MacPherson) which appeared in volume 3 of Bott’s collected works.
While a graduate student at Princeton (somewhere in the early 1960s), I went to a lecture by Bott. He spoke on the Atiyah-Singer index theorem, which was quite new at the time. Here was a friendly and somewhat oversized man enthusiastically describing this marvelous index formula. He ended with a burst of great intensity, saying, “Well, I just wanted to tell you about this theory, which I find so beautiful.”
This was my first in-person contact with Bott. Earlier (as a beginning graduate student under the patient guidance of Norman Steenrod), I had studied his papers applying Morse theory to the topology of Lie groups. I had greatly admired these papers, but I was even more delighted with the author himself. His talk on the Atiyah-Singer index theorem made a deep impression on me. I was fascinated by the mixture of topology and analysis, and I was also intrigued by the love of mathematics that came through so strongly in his talk.
About four years later, I was in a very discouraged mood. I was now an assistant professor at Princeton and I had the feeling I was marching backwards. In work that everybody, including me, believed I had done, gaps and difficulties were appearing. Instead of moving ahead, I was wasting time and energy trying to repair arguments that I thought I had successfully finished one or two years before.
Into the midst of this gloom came Raoul. He gave a talk at IDA about his residue formula for holomorphic vector fields. Again, I was keenly interested in the mathematics he presented. I decided right then and there during his talk to simply abandon the problems I had been thinking about and start trying to further develop Bott’s point of view on vector fields.
So Jeff Cheeger and I got to work to see what the residue formula was for a holomorphic vector field with nondegenerate submanifolds of zeroes. It took us some months to crack this—i.e., to see a clear, simple formula coming out of Bott’s method. Once we had it, I wrote to Bott telling him our explicit answer for these residues. He wrote back informing me that he had already done this, but suggesting that Jeff and I write up the case of Killing vector fields. We did this, and the paper appeared in Topology. Eventually these ideas emerged in a somewhat changed form in work of Cheeger and Gromov.
Next, inspired by a comment of G. Washnitzer, I decided to see how Bott’s theorem for holomorphic vector fields might extend to meromorphic vector fields. On a compact, complex-analytic manifold holomorphic vector fields are really quite rare. But meromorphic vector fields are plentiful, so it seemed worthwhile to generalize Bott’s theorem to the meromorphic case. By this time Cheeger had moved on to other things, so I found myself working alone on the meromorphic vector field problem. In the late spring of 1968, I felt I had correctly generalized Bott’s method to the meromorphic situation. I wrote to Bott about this and received a letter from him saying, “Dear Paul — We certainly are working in ||!” We were thinking about exactly the same points. Shortly after this, we met in Berkeley where we were both attending the meeting on global analysis. I outlined to Bott the formula and proof that I had in mind. To my astonishment, I found that I was actually slightly ahead of him. When he realized that my proof worked, he spread his arms wide and cheerfully shouted, “It works like a charm!” The technical trick on which the proof hinged he referred to as “a slightly mysterious Kunstgriff.”
After I arrived at Brown in September 1968, there followed a period of about two years when I would sporadically visit Bott at Harvard, and we worked on a residue formula for singularities of holomorphic foliations. During this time I came to appreciate another of Bott’s qualities—his humor. Bott can be very funny. We had many good laughs together.
Strangely, almost every time I visited Bott at Harvard it would rain. When I was leaving, Bott would say, “Good-bye, Paul. See you on the next rainy day.” We made some progress on the residue question. Our work here is closely related to the differential character theory of Cheeger-Chern-Simons. We proved a rigidity theorem and we formulated a rationality conjecture that appears to be a deep and difficult problem. At the Stanford differential geometry meeting, I learned that Simons had made the opposite conjecture. He was convinced that the residues in question are not always rational. The problem remains unsolved.
One rainy day I was with Raoul in his office in the Geographic Institute discussing these matters when a tall, thin, shy graduate student came in and said, “Professor Bott, where should I apply for a job?” I said, “Apply to Brown.” The student was Bob MacPherson. His coming to Brown was one of the main events in the meteoric rise of the Brown Math Department.
Towards the end of my two-year collaboration with Bott, he was working on understanding the Haefliger classifying space for foliations and I had started working on Riemann-Roch with Bill Fulton and Bob MacPherson. I had a number of very interesting conversations with Raoul about the Haefliger classifying space. Approximately twenty years later, while working with Alain Connes, I recalled these conversations and they helped guide Alain and me towards the classifying space for proper actions that we need in our conjecture on the K-theory of group C* algebras.
What did I learn from Bott? I learned—perhaps I should say relearned—from Bott things that I already knew: that there is great beauty and wonder and fun in mathematics; that the best mathematics is not necessarily the most difficult; that there is a mainstream to our subject that talented mathematicians like Bott have an instinctive feel for; that ideas should be presented clearly and without nonsense so that other mathematicians can understand and appreciate them.
I consider myself fortunate to have benefited from the clarity and beauty of Bott’s mathematics, and from the warmth and joy of his personality.