Millennium Prize Problems

Riemann hypothesis: a bunch of results on prime numbers, and therefore possible applications to cryptography

Of course, everything of interest has already been proved conditionally on it, and the likely "true" result will in itself not have any immediate applications.

As is often the case, the only usefulness would be possible new ideas from the proof technique, and people being more willing to prove stuff based on it without the risk of the hypothesis being false.

Yang-Mills existence and mass gap: this one has to do with findind/proving the existence of a more decent formalization of quantum field theory that does not resort to tricks like perturbation theory and effective field theory with a random cutoff value

This is important because the best theory of light and electrons (and therefore chemistry and material science) that we have today, quantum electrodynamics, is a quantum field theory.