Are there any open problems in pure mathematics whose resolutions would resolve a well-formulated open physics question, whose solution could (at least in principle) explain an experimentally measurable phenomenon?
For example, I've heard that a proof of the Riemann hypothesis could settle some important open physics problems, but I don't know the details.
This question is broad, so let me try to clarify what I'm looking for. By an "open question in math" I mean a question that is considered to have purely mathematical interest apart from any possible physical applications, and so is under active research by pure mathematicians rather than physicists. (For example, the question of whether the Hubbard model on the square lattice has a d-wave superconducting phase is certain a purely mathematical question which would have lots of important physics implications, but I don't think any mathematicians are working on it, as it's not terribly interesting from a pure math standpoint.)
Two obvious candidates are the Millenium Prize Problems on the Yang–Mills existence and mass gap and on Navier–Stokes existence and smoothness. But my understanding is that a proof of Yang-Mills existence and mass gap probably wouldn't be of much physical interest to working theoretical (as opposed to mathematical) physicists. Unless the proof had a surprising physical consequence as a corollary or led to new calculational techniques, it probably wouldn't really change the day-to-day practice of most working quantum field theorists. (That is, QFT practitioners are pretty confident that there's some mathematical structure which is described by their QFT's, and if that structure turns out not to satisfy the Wightman axioms required by the Millenium Prize problem, that wouldn't be a big deal practically.) Similarly, a proof of Navier-Stokes existence and smoothness (which didn't lead to new calculational techniques) wouldn't necessarily change the study of fluid mechanics in the sense of making novel predictions that could be tested by experimentalists.
Ideally, I'd like an open problem worked on by real mathematicians which would conclusively answer some question with both theoretical and experimental physics importance (at least within some widely-accepted theoretical framework). I know that's a bit vague and subjective, but hopefully not too much.