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.