I was out in Stony Brook for the past couple days, to attend festivities surrounding the inauguration of the new building there which will house the Simons Center for Geometry and Physics. The Center is funded by Jim Simons, whose Renaissance Technologies is one of the world’s most successful hedge funds. The plan is to ultimately have six permanent members (half in physics, half in math) and a director, as well as quite a few postdocs and visitors. The founding director of the center is John Morgan, who until recently was my colleague here in the math department at Columbia. Mike Douglas is a permanent member in physics, current and recent senior visitors include Nikita Nekrasov and Graeme Segal.
The building is quite attractive, with the two lower floors forming a public area that includes two auditoriums, an atrium and a dining area which will serve lunch and bring together the local math and physics communities. The three upper floors have offices and seminar rooms. Construction is nearly finished, but not quite, with part of the ground floor still walled off for some last-minute work. The atrium features a large mural containing a selection of historically important equations, the choice of which evidently was a major undertaking. Guests noted one typo, but luckily the current mural is a temporary printed one, with the plan to cut the equations in stone not yet implemented.
Tuesday night featured a gala opening event, with music provided by the Emerson quartet, and short speeches from various dignitaries. Simons was presented with an original edition of one of the works of Isaac Newton, with the comment that he had shown much greater success than Newton at turning things into gold. He gave a wonderful talk describing the early stages of interaction between math and physics that he was involved in as chair of the math department at Stony Brook in the early seventies. Evidently soon after his arrival he was invited to meet with Frank Yang, who had just started up an institute for theoretical physics there. Yang described his current research on gauge theory, which Simons claims to not have understood a word of. The same thing happened again a year later. However, the third time this happened, Simons all of a sudden realized that what Yang was talking about was something that geometers knew very well: a connection in a principal bundle. This led to a series of lunch-time lectures by Simons to the physicists, to visitor Is Singer getting interested and taking what he learned back to MIT and Atiyah at Oxford, and finally to the modern period of interaction between math and physics that began in the mid-seventies centered around questions related to instantons.
On Wednesday, the Simons Center hosted a day-long inaugural conference (videos and slides of talks should appear on their web-site at some point). Appropriately, the first talk was an inspirational one from Michael Atiyah, going over a wide variety of different mathematical ideas. One theme was the quaternions, with Atiyah pointing out that Hamilton had written down a square-root of the Laplacian many decades before this trick was re-discovered by Dirac in writing down the Dirac equation. After recalling the relation between the division algebras (real and complex numbers, quaternions, octonions) and the Hopf invariant one problem, Atiyah suggested that the Freudenthal magic square has a similar relationship to the Kervaire invariant one problem, with the recent complicated proof by Hopkins et al. an analog of the original Adams proof in the Hopf case, with the analog of Atiyah’s “postcard proof” still to be discovered. He ended with some comments about ideas of Alain Connes about non-commutative geometry and the Riemann hypothesis, and suggested that the conjectured self-adjoint operator that could explain the Riemann hypothesis might be the Hamiltonian of quantum gravity. I noticed that Atiyah was supposed to be giving a talk at the IAS today with the impressive title of “Quantum Gravity and the Riemann Hypothesis”, but it appears to have been canceled.
The second morning talk was a rather rambling and elementary one by Polyakov about topics related to Wick rotation, ending with the claim that in the gravitational case the usual ideas about Wick rotation fail and this has something to do with explaining the cosmological constant. The afternoon began with a talk by my colleague Andrei Okounkov using symplectic resolutions to study quantum cohomology, followed by Witten whose topic was “A New Look at Khovanov Homology”. This talk covered similar material to the ones described here, involving a really beautiful story about various 3 and 4 dimensional topological quantum field theories. As in the earlier talks though, at the end there’s a transition to 5 and 6 dimensional theories where I got just as lost as before. I had been hoping that the Simons Center talk would explain the ideas I was missing, but I fear that there’s no way around digging into the details in Witten’s recent paper about this. The last talk was by Cumrun Vafa, who gave a very nice and elementary discussion of the “wall-crossing” phenomena in certain 2d qfts, as motivation for recent work on wall-crossing in 4d gauge theories.
The talks were uniformly of very high quality, it’s wonderful to see that the Simons Center is off to a great start.
Update: I just did take another look at Witten’s recent paper, and realized that the part involving 5 and 6 dimensional theories is not there, but in a paper in progress on “Five-branes and Knots”. So, that story will just have to wait for now…