JDG and Other Conferences

This past weekend I was in Cambridge and attended many of the talks at the JDG conference held at Harvard. The conference was nominally in honor of Shiing-Shen Chern, who died late last year, so many speakers made some connection between their work and Chern’s, especially his work on Chern classes.

Among the purely mathematical talks I attended was a very clear one by Victor Guillemin on Morse theory and convexity theorems on symplectic manifolds. The material he covered is quite beautiful, but rather old by now. His reason for covering it seemed to be that he has a new book on the topic (with Reyer Sjamaar) called “Convexity Properties of Hamiltonian Group Actions”, soon to appear from the AMS in the CRM monograph series, but also available on Sjamaar’s website.

Mike Hopkins gave an impressive talk on “Derived Schemes in Stable Homotopy Theory” which was based on very recent work by his student Jacob Lurie. This work involves defining a notion of a scheme which makes sense in the context not of the commutative rings of algebraic geometry, but instead the commutative rings of spectra in stable homotopy theory. It allows a new construction of the tmf (topological modular forms) theory of Miller and Hopkins.

Iz Singer reminisced about taking a class in geometry from Chern at Chicago in 1949, a class which he thought may have been the first one Chern taught in the US. Singer’s talk was about “Projective Dirac operators” which have an index which is a fraction. One of the main motivations for Singer’s original work with Atiyah on the Atiyah-Singer index theorem was to understand the integrality of the A-hat genus on a spin manifold as coming from the fact that it was an index. On a non-spin manifold the A-hat genus takes on fractional values, and one can use this to prove the non-existence of a spin structure. In work with Mathai and Melrose, pseudo-differential operator techniques are developed that allow one to define a sort of index in these situations where there is no spin (or even spin-c) structure.

There were several talks by physicists, or related to physics. One was by Kefeng Liu, half of which was about some new metrics on moduli space, the other half about some formulae coming out of work on topological strings. For this material, see his talk at last year’s Yamabe Conference. Vafa gave a talk on “Topological M-theory”, which he motivated by starting with the holomorphic anomaly in the topological string B-model. For quite a while it has been known that you can think of these topological string results as giving a vector in the Hilbert space one gets from quantizing H^3(M), where M is a Calabi-Yau. Topological M-theory is supposed to be something related to topological string theory in much the way the full M-theory is related to the full-string theory, so involves one-dimension higher. Thus it deals with 7-dimensional manifolds and tries to explain some of the phenomena related to topological strings on 6-d Calabi-Yaus in these terms. For more about this, there’s a talk by Andrew Neitze on-line that covers some of the same material.

Nikita Nekrasov’s talk was about “Z-theory”, which is his own name for the same ideas about topological M-theory that Vafa was talking about. He drew a version of the standard picture of the M-theory moduli space, now for Z-theory and with all sorts of mathematical objects attached to the various cusps. Nekrasov gave a similar talk in Nagoya late last year, as well as one at Strings 2004.

While a lot of interesting mathematics has come out of topological strings, the idea that that there is some grandiose unification involving thinking about 7d G2-manifolds seems to me even less promising than the idea of 11d M-theory itself, which for years now seems to have gone nowhere. Just as M-theory has led many physicists to pointless wanderings in 11-dimensions, it now seems to be leading mathematical physics away from rather rich mathematical areas into the complicated geometry of seven dimensions. Undoubtedly this will lead to some new mathematics, but it looks to me like it will be much less interesting than the mathematics emerging from string theory during earlier periods. The interaction between mathematics and physics remains dominated by the ideology of string/M-theory, and this is harming both subjects.

One aspect of the sad state of the interface between math and physics is that virtually no one from the physics department at Harvard seemed to be attending the JDG conference lectures. I’d been expecting to see at least Lubos Motl there, but he was down at Columbia attending a meeting on string cosmology. He reports on the talks here, here, and here, as usual covering very critically a talk on loop quantum gravity, quite uncritically one about the landscape and absurdly baroque constructions that try to make some contact with the standard model. I’m beginning to believe that his “leashing” did have something to do with his criticizing the landscape ideology too vigorously, since he seems to have stopped doing that.

For the latest on the landscape, see a recent talk by Lubos’s senior colleague Arkani-Hamed (whom he better not piss off too much) at the PHENO 05: World Year of Phenomenology symposium in Wisconsin, entitled The Landscape and the LHC. Arkani-Hamed’s talk begins with the usual strained historical analogy, this time a long and bizarre description of the calculation by Aristarchos of the distance to the sun by the method of parallax. The point of this is highly obscure, but seems to be that since Aristarchos was wrong to find unreasonable the huge distances to the stars implied by the lack of visible parallax, we’re wrong to find unreasonable the huge amounts of fine-tuning required by split supersymmetry.

He goes on much like Susskind for quite a while about the glories of the landscape idea, with the twist that supposedly split supersymmetry is “sharply predictive”. The only “sharp” predictions he mentions concern a relation between some coupling constants which haven’t been observed and likely never will, as well as that there may be a “long-lived” gluino. Not that he actually has a prediction for the mass or lifetime of this gluino.