Are there any purely mathematical, unsolved questions, whose resolution would have (great, or concrete) impact on physics? Eg. it could almost surely tell us whether particle x exist or not, assuming so and so well-accepted observational facts?
closed as not constructive by David Z♦ Jun 25 '12 at 3:23
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There is a conjecture that the zeros of the Riemann Zeta function correspond to the eigenvalues of some Hermitian operator. If this conjecture, and the Riemann Hypothesis, are true, that would be quite a remarkable connection that may or may not have a great impact on physics.
There is the lovely P vs. NP problem http://www.claymath.org/millennium/P_vs_NP/
I'm sure there are plenty of applications for this all over physics, but one interesting one relies on the proof that the fermion sign problem in quantum Monte Carlo simulation is NP-hard (see http://arxiv.org/abs/cond-mat/0408370). If P=NP then efficient simulation of generic quantum statistical systems is possible... But I don't think anyone believes P=NP. :P