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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?

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closed as not constructive by David Z Jun 25 '12 at 3:23

As it currently stands, this question is not a good fit for our Q&A format. We expect answers to be supported by facts, references, or expertise, but this question will likely solicit debate, arguments, polling, or extended discussion. If you feel that this question can be improved and possibly reopened, visit the help center for guidance.If this question can be reworded to fit the rules in the help center, please edit the question.

This is one of those list questions that is a bad fit for the SE format, so I'm closing it accordingly. If anyone would like to argue for keeping it open, feel free to open a discussion on Physics Meta. – David Z Jun 25 '12 at 3:24
The mathematicians have not yet made rigorous the modern foundations of physics, the path integral and quantum operator algebras, so there is no such result that can be formulated in mathematical language yet. But I agree with David's close. – Ron Maimon Jun 25 '12 at 4:33

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.

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There is the lovely P vs. NP problem

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 If P=NP then efficient simulation of generic quantum statistical systems is possible... But I don't think anyone believes P=NP. :P

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