Atiyah on Atiyah-Bott

This week I’m in Montreal attending a conference in honor of Raoul Bott (who I wrote about in some detail here a couple years ago).

On the first day of the conference, a documentary film about Bott made by his grandaughter Vanessa Scott was shown, and there was a panel discussion about the man and his influence on students and collaborators. Bott was both a wonderful mathematician and human being, and many people at the conference paid tribute to him as someone who encouraged them and taught them beautiful and deep mathematics. Quite a few commented on visits to him and his family at their summer place on Martha’s Vineyard. It adjoined a clothing-optional beach frequented by Alan Dershowitz among others, a beach where Bott was supposedly known as the “Mayor”.

Bott was very much involved with physics later in his career, and two physicists spoke at the conference: Cumrun Vafa on topological strings and Edward Witten on the 6d QFT point of view on geometric Langlands. I greatly enjoyed both talks, but I suspect they were pretty difficult for the mathematicians in the audience to follow, covering a great sweep of material bringing together new mathematical developments not understood by most mathematicians with QFT and string theory techniques far from their background.

Michael Atiyah gave a truly wonderful talk to open the conference. He described how his friendship with Bott had covered a period of 50 years, from 1955 until Bott’s death in 2005. From 1964-1984 they did some of the best work of their careers together, and Atiyah tried to summarize the high points of this, as well as point out problems their work raised that he sees as still open and worthy of investigation by a new generation. Editions of the collected works of both of them are available that include their commentaries on the papers, and these are very much worth reading. Each of them is a masterful expositor, so their joint papers are uniformly models of clarity.

Atiyah broke things up into the following main topics:

In work with Hirzebruch, Atiyah realized that Grothendieck’s construction of K-theory in algebraic geometry could be turned into a generalized cohomology theory in the topological category. To make this work uses Bott periodicity in a crucial way, to show that K(MxS²) is the tensor product of K(M) and K(S²) for any compact manifold M. Atiyah realized he didn’t know how to prove this, and got Bott to produce an appropriate proof, which appeared in a paper of Bott’s in 1959. The paper is written in French, clearly not by Bott, and Atiyah says he still doesn’t know who translated it into French for Bott.

Atiyah and Bott worked together on extending the Atiyah-Singer index theorem to the case of manifolds with boundary, where the issue of how to handle the boundary conditions so as to get a good index problem is a subtle one. As part of this, they needed a new, more “elementary” proof of Bott periodicity, finally finding one that “even MIT faculty could understand”. This is periodicity for the unitary group and crucially uses Fourier analysis. Atiyah gave as a problem deserving attention that of extending the proof to the case of the orthogonal group, where the use of Fourier series would have to be replaced by the representation theory of O(N). Multiplying Fourier series becomes taking the tensor product of representations, which is much more non-trivial to deal with.

Using the McKean-Singer formula, one can relate the computation of the index of a differential operator to the asymptotics of a related heat equation. Patodi and Gilkey had carried this through using complex and skillful algebraic calculation, but Atiyah and Bott felt they couldn’t understand these proofs, so with Patodi came up with a new proof, one that just used Weyl’s invariant theory for the orthogonal group and the Bianchi identities of Riemannian geometry.

As a problem for the future, Atiyah listed finding a better understanding of the relation of the Atiyah-Bott-Patodi argument with the supersymmetric quantum mechanics proof. More explicitly, he conjectures that one should be able to just use invariant theory, but perhaps invariant theory of the infinite dimensional group Diff(M), of a sort advocated by Gelfand.

The Atiyah-Bott fixed point formula computes the Lefschetz number one gets in the context of index theory and a mapping of the manifold to itself (coming for instance from a group action) in terms of data at the fixed points of the mapping. This has many beautiful applications, including a new proof of the Weyl character formula. The formula is sometimes known as the “Woods Hole formula”, since Atiyah and Bott conjectured it at a conference at Woods Hole, where certain experts told them it couldn’t be true, since they had computed counter-examples. Atiyah didn’t name names, but described the experts involved as now claiming to not remember this. “They deny it to a man” he said, but he remembers it distinctly while being in the frustrating position of having nothing written down to provide incontrovertible documentary evidence.

One generalization of the fixed point formula shows that for manifolds with circle action the index is zero, and Atiyah mentioned Witten’s extension of this to the case of a loop space, leading to a relation to modular forms and the subject of “elliptic cohomology”. This continues to be an active subject, with Mike Hopkins and others developing the theory of “topological modular forms”, something that has shows interesting relations to number theory. Atiyah described a “moral bet” between him and Andrew Wiles about whether QFT will ultimately influence number theory. Wiles thinks not, but Atiyah believes it will happen, and hopes to be around long enough to find out if he is right.

Morse theory and equivariant cohomology were two of Bott’s favorite tools, and he and Atiyah did some wonderful work applying these to the case of Yang-Mills theory in two dimensions. Here the Yang-Mills functional is a Morse function, the space of connections is a symplectic manifold, and the reduced space for the gauge group action is an important mathematical object that can be thought of as the moduli space of flat connections, or of stable holomorphic bundles on the 2d surface. Their main result, a calculation of the Betti numbers of this space, reproduced earlier results coming from a very different approach, that of using algebraic curves over finite fields and the Weil conjectures.

As a problem for the future, Atiyah asks if there is some infinite-dimensional version of the Weil-conjectures, and some QFT where the Feynman integral is analogous to Tamagawa measure. He says he has been thinking about this off and on for 30 years, hasn’t found anything satisfactory, but offers the problem as a gift to younger mathematicians, as long as they let him know if they solve it. For some recent work related to this, see here.

Atiyah described work of his with Bott on this topic as “performed under a subcontract” with Garding.

Finally, Atiyah commented on how his mathematical style was quite different than Bott’s with Bott always advocating an “old-fashioned” way of proceeding, involving concrete formulae, where Atiyah favored “new-fangled” abstraction, only writing down formulae when forced to by Bott (and then discovering that this gave important insight). Later he found himself in the opposite position when working with Graeme Segal, with respect to whom he was the “old-fashioned” one, resisting abstraction. He commented that Bott and Segal had written a paper together, and he was shocked to see that such a thing was possible.

He noted that he had met many fascinating people through Bott, including one of the world’s best known mathematicians: Tom Lehrer. Finally, he ended with the comment that, while Bott was no longer here, the great thing about doing math the way he did it is that you become immortal.