I was deeply saddened to hear this morning that Raoul Bott died a couple days ago, on December 20th, in San Diego, of cancer. Bott was one of the greatest mathematicians of the twentieth century; for some details about his life see the commemorative web-site set up by the Harvard math department. The article by Loring Tu gives an excellent outline of Bott’s mathematical career and accomplishments.

I first encountered Bott when I attended his graduate course on differential geometry at Harvard. The course was over my head since I was an undergraduate, and the fact that it met at 9am (or was it even 8:30?) didn’t help at all since I was sometimes not awake that early. But the course was extremely inspirational, giving a beautiful and revelatory view of geometry in terms of Lie groups, Lie algebras, connections and curvature. I hope someday someone who has a complete set of notes from that course will write them up. I never took Bott’s course on algebraic topology, but learned much of the subject from the book Differential Forms in Algebraic Topology, which was a write up of the course notes done with Loring Tu. Bott’s point of view on algebraic topology is a perfect one for physicists interested in the subject, starting very concretely with manifolds, deRham cohomology and Poincare duality, and only later getting to more abstract material. Later on in the 1980s when I spent another year at Harvard, one of the most rewarding experiences of that year was sitting in on an advanced course on the index theorem (especially the heat equation proof) that Bott was teaching.

During my undergraduate years I lived at Dunster House, and my last year there was livened up by Bott’s presence, when he took up the position of “Master” of Dunster House, living there with his wife and often interacting with the undergraduates. Bott was an extremely warm and friendly person, a truly wonderful human being and a pleasure to be around. He gave the outward appearance sometimes of not being all that swift, demanding that one explain things to him slowly and in as simple terms as possible. His greatest mathematical achievements came not from any ability to quickly master difficult formalisms, but from a talent for seeing deeply into a problem, getting at its very core and finding new ways of understanding what was going on in the simplest terms possible. Many of his results have given new insights into mathematics at the deepest levels that we currently are able to understand.

Some of Bott’s earliest work involved dramatic new applications of Morse theory, especially his discovery and proof of Bott periodicity, an unexpected fundamental new fact about topology that lies at the foundation of topological K-theory. Bott was intimately involved with Clifford algebras and spinors from early on, and his extremely important paper with Atiyah and Shapiro shows how crucial these are for understanding K-theory. While the proof of the general index theorem is due to Atiyah and Singer, Bott periodicity and the central role of the Dirac operator made clear by the Atiyah-Bott-Shapiro work are critical parts of the story. Bott also worked out with Atiyah an amazingly powerful fixed point theorem that also fits into the index theory story.

Despite being nominally a topologist, Bott had a big influence on representation theory, with his Borel-Weil-Bott theorem showing how to extend the Borel-Weil theorem to understand the way in which irreducible representations can occur not just as holomorphic sections of line bundles, but also in higher cohomology. He worked out the Dirac operator version of this result, an early check of the index theorem, and he was among the first to promote the idea that the notion of geometric quantization is best understood in terms of the index theorem and integration in equivariant K-theory.

His work with Atiyah on the moduli space of flat connections on a Riemann surface opened up a whole new field of mathematics, one whose implications are still not fully understood, especially the connections with quantum field theory. Over the years Bott took a great interest in physics and in communicating with physicists. He often gave lectures at physics conferences, and it was at one of these that Witten first learned about Morse theory, leading directly to his extremely influential work on supersymmetry and Morse theory. The fact that advanced age and ill-health reduced Bott’s mathematical activity in recent years has been a huge blow to the whole subject of the interaction of mathematics and physics.

Most of Bott’s papers have appeared in a four-volume set of his collected works, together with some commentary on them by him and by other mathematicians. Reading through these volumes has been a significant part of my mathematical education and I heartily recommend them to everyone interested in mathematics and physics. Bott was an exceptionally lucid thinker and thus a very clear expositor. His death marks a great loss on many levels for both mathematics and physics.

**Update:** Other blog entries about Bott can be found here, here, and here.

Pingback: Raoul Bott, 1923-2005 | Cosmic Variance

Oh, this is sad. 20-plus years ago when I was trying to be a theoretical physicist I attended some of his seminars. Your description of him fits perfectly with my memories of a large, genial, barrel-chested man, not obviously fast in movement or thought but clearly having truly great insight. Amusingly Atiyah attempted to disrupt things by playing the irritating schoolkid (“Please sir! Please sir!”). They could have been a nice comedy double act, though it wouldn’t have been worth the crushing loss to mathematics if they had changed careers.

I took a course on differential geometry with Bott when I was a grad student at MIT. It was great! He was explaining, and trying to simplify, Witten’s work on supersymmetry and Morse theory.

He always had a benign twinkle in his eye. I remember him starting one sentence with the phrase “Before I was such a bigshot…” – and the whole class cracking up in laughter.

I also remember him saying “So, if thinking doesn’t let us solve the problem, what should we do?

Superthink!” This was sometime around 1985, when mathematicians were just getting into supersymmetry, so it suited the spirit of the times.Like Dirac, Bott started out as an electrical engineer. He said that he worked his way to algebraic topology through Kirchoff’s laws, which are a nice introduction to the ideas behind deRham cohomology and Hodge theory. My friend Robert Kotiuga, who is an electrical engineer at Boston University, talked to Bott about this and got really fired up about it – he’s written a lot about electromagnetism and algebraic topology. Just another of the many people influenced by Bott….

Thanks for posting this Peter. It’s good to pass along bits of personal history. I went ahead and ordered his book (sounds like it might help improve my poor knowledge of algebraic topology) — you should consider becoming an Amazon Associate so you can get a small percentage from books you link to. Sadly, Bott won’t get his share either, but it sounds like he lead a good life.

Peter,

Beautifully written and well said.

Peter, this was inspiring.

It’s a great loss.

-drl

This is very sad news. If I can add a reminiscence, Raoul Bott showed a great deal of generosity to me and other physics students who found a warm environment where he would give freely of his time to talk about physics and math. After he expressed interest in the singularity theorems in GR, another student, he and I met regularly to work our way through the key chapters of Hawking and Ellis’s book. He required that every formula be translated into index free notation and insisted we gain an intuitive understanding of each term in each equation before moving on. I don’t know if he got anything from it but it was an education to me.

I learned from his example that one could do good science in an atmosphere that was as intellectually probing as it was warm and light hearted. At the end I felt sufficiently grateful to him that I arranged to receive my Ph.D. from him in the graduation ceremony at Dunster House, where he was master.

I met him only one time, when I attended one of his lectures. He was indeed a very warm man, very communicative. He talked about Morse theory.

Very sad news about a warm and generous man. I confess to a pang of jealousy that I was in Dunster House a couple of years too soon to experience him as house master. Instead, I met him when I was a clueless freshman taking linear algebra. I was too young and ignorant to fully appreciate my luck at being plopped into the one section taught that year by tenured faculty: Shlomo Sternberg in the first semester, and Raoul Bott in the second.

It was springtime, and my attention wandered. I tried to fake it through much of the final exam. My blue book came back with one comment written exuberantly across many of my answers: “SALAD!” Decades later, this is still the phrase I hear in my head — with its author’s accent and enthusiasm — whenever I’m editing my most awkward prose.

Pingback: Not Even Wrong » Blog Archive » Follow-Ups