If you are at all interested in mathematical physics you will want to watch Ed Witten’s recent talk on his work in knot theory that he gave at the IAS. Witten gives a general overview of how he discovered that the Jones polynomial used to classify knots turns out to be “explained” as a path integral using Cherns-Simon theory in 3D. More recently the Jones Polynomial was generalised to Khovanov homology which describes a knotted membrane in 4D and Witten wanted to find a similar explanation. He was stuck until some work he did on Geometric Langlands gave him the tools to solve (or partially solve) the riddle.
Geometric Langlands was devised as a simpler variation on the original Langlands program that is a wide-ranging set of ideas trying to unify concepts in number theory. Witten makes some interesting comments during the question time. He says that one of the main reasons that physicists (such as himself) are able to use string theory to answer questions in mathematics is that string theory is not properly understood. If it was then the mathematicians would be able to use it in this way themselves, he says. Referring to the deeper relationship between string theory and Langlands he said.
“I had in mind something a little bit more ambitious like whether physics could affect number theory at a really serious structural level like shedding light on the Langlands program. I’m only going to give you a physicists answer but personally I think it is unlikely that it is an accident that Geometric Langlands has a natural description in terms of quantum physics, and I am confident that that description is natural even though I think it mught take a long time for the math world to properly understand it. So I think there is a very large gap between these fields of maths and physics. I think if anything the gap is larger than most people appreciate and therefore I think that the pieces we actually see are only fragments of a much bigger totality.”
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