A commenter on the last posting pointed to the new video available at the IAS site of Witten’s recent public talk there on Knots and Quantum Theory. The talk is aimed at a general audience, including supporters of the IAS, so it’s rather non-technical. For the technical details behind what Witten is talking about (his recent work on Khovanov homology and QFT), see this survey for mathematicians, a survey for physicists at Strings 2011, and this paper.
For me an interesting part of Witten’s talk was how he described the evolution of his ideas about this topic, and the relationship to geometric Langlands. He also had interesting comments about number theory and the Langlands program, denying any real knowledge of the subject, but arguing that sooner or later (probably later, after his career is over), there would be some convergence of number theory Langlands and physics. He finds the coincidence of geometric Langlands showing up in QFT so remarkable as to indicate that there are deep connections there still to be explored. I suspect that he sees the likely path of information going more from physics to math, with QFT ideas giving insight into number theory. While I agree with him about the existence of deep connections, I suspect the influence might go the other way, with the powerful ideas behind the Langlands program in number theory someday providing some clues about QFT useful to physicists.
Also on the Langlands program topic, this semester we’re having a wonderful series of lectures on the topic by Dick Gross. He’s a fantastically gifted lecturer, and this series is pitched at just the right level for me, explicating many of the parts of the subject I’ve been trying to learn in recent years but have found quite confusing. It’s a beautiful, very deep, but rather intricate subject, bringing together a range of remarkable ideas about mathematics. In the end though, the Langlands program is really mostly about new ideas in representation theory, and since I’m convinced that deeper understanding of QFT will require new ideas about how to handle symmetries, which is the same thing as representation theory, perhaps finding connections between the subjects won’t have to await Witten’s retirement.
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