Mathematician Dave Morrison is giving a colloquium talk tomorrow at the KITP with the provocative title How Much Mathematics Does A Theoretical Physicist Need To Know? It should soon be available for viewing on the KITP web-site, and I’m looking forward to seeing what he has to say.
I’m not at all sure myself how much mathematics a theoretical physicist needs to know, it certainly depends on what they’re trying to do. But there does seem to me to be a well-defined list of what mathematics goes into our current most fundamental physical theories, and anyone who hopes to work on extending these should start by learning these subjects, which include (besides the classical mathematical physics of PDE’s, Fourier analysis, complex analysis):
More general geometry of principal and vector bundles: connection, curvature, etc.
Lie groups and representation theory
I’m sure others have different ideas about this….
Update: Dave Morrison’s talk is now on-line here. He began his talk my noting that it had been advertised here on “Not Even Wrong”, and he put up a slide of my posting and people’s comments as an example of people’s lists of what mathematics theoretical physicists should know. He did say that that his talk wasn’t intended to provide such a list, but rather various comments about how physicists can fruitfully interact with mathematicians.
He began by giving several examples of people who had to construct new mathematics to do physics: Newton, Fourier, Heisenberg, and Gell-Mann. David Gross correctly objected that SU(3) representation theory was already known before Gell-Mann started using it, even though at first Gell-Mann wasn’t aware of this. As for more recent interactions, he mainly mentioned the connection between the index theorem and anomalies, as well as various math related to the quantum hall effect. For some reason he decided not to go into the relation of string theory and mathematics, which has been quite fruitful. He did say that he still believes there is some unknown more fundamental way of thinking about string theory that will involve now unknown mathematics. His general advice to physicists was that they should be willing to acquire mathematical tools as needed, but should be aware that if they ask a mathematician questions, they are likely to get answers of too great generality. He ended his talk early, opening the floor to a long discussion.