Sir Michael Atiyah and the Unity of Mathematics and Physics
Sir Michael Atiyah is here at the 4th Heidelberg Laureate Forum both as a Fields Medalist (1966) and a winner of the Abel Prize (2004). If you ask many mathematicians alive today who the greatest mathematician of their time has been, Atiyah’s is a name that’s likely to come up often.
Back in 1987, when I was a postdoc in physics, trying to figure out if going instead into mathematics was a good idea, the first math conference I traveled to attend was a symposium on the work of Hermann Weyl held at Duke University in Durham, North Carolina. There were many talks by impressive mathematicians, which I mostly didn’t understand very well. Atiyah’s talk was both the most amazing one I had ever heard, and to this day the most impressive talk I’ve heard by a mathematician or physicist. In his talk he outlined a set of new ideas about topology and geometry, rooted in very new work by Andreas Floer and Atiyah’s student Simon Donaldson. These ideas provided dramatic new insights into the topology of lower dimensional manifolds. In 3 and 4 dimensions this was Floer homology and the Donaldson polynomials based on Yang-Mills theory. The same ideas transposed from Yang-Mills connections to maps from a surface into a symplectic manifold provided new results in symplectic topology, including what are now known as Gromov-Witten invariants.
Most fascinating to me was that Atiyah related all of this to Witten’s work on quantum mechanics and Morse theory, arguing that there should be a quantum field theory version of the story. This was the birth of topological quantum field theory (TQFT), with Witten to work out a year later what the quantum field theory conjectured by Atiyah actually was. I remember that at the end of the talk, Witten came to ask Atiyah about the Jones polynomial, mentioned in the talk. Soon after the first topological TQFT, Witten was to find a quantum theory with observables the Jones polynomial, now often called the Chern-Simons-Witten theory. This was largely responsible for Witten’s 1990 Fields medal.
This staggeringly visionary talk was influential in convincing me that mathematics was the direction for me to take. Over the next few years a large part of my mathematical education came about through a project of trying to read as much as possible of Atiyah’s collected works, which appeared in 1988. In particular, his expository articles are extremely lucid, giving deep insights into a range of topics.
Atiyah has always been a visionary, and remains one to this day. His talk at the HLF was a raconteur’s very general set of stories and comments about mathematics, ending with a highly speculative remark that the Tate-Shafarevich conjecture might have something to do with the 4-dimensional smooth Poincare conjecture. Much of his time here in Heidelberg has been spent in conversation with young mathematicians. At age 87 he has more energy than many half his age, and is more forward-looking, less interested in the past than in new ideas about the future.
Listening in on some of his conversations, I don’t think I’m violating any confidentiality by reporting that he’s quite taken with the idea that if one could make sense of the “Galois group of the octonions” one would find that it lies at the heart of unification of the forces of physics. I can’t do justice to his arguments for this, but, if you can find him, I’m sure you’ll get an enthusiastic explanation. You can also get a good idea of what he’s thinking about from a recent interview by Siobhan Roberts in Quanta Magazine. Atiyah is our greatest enthusiast for a vision of the unification of mathematics and physics, arguing for a picture that will bring together such disparate subjects as number theory and physics. He has a great track record with this enthusiasm, as I first saw back in 1987, and I’m sure he’s right that there’s more to come.