Have you ever considered working on some low-hanging fruit problem in a non-physics field?
My grad work had no connection to data/stats and I have not bumped into any low hanging fruit where Riemannian geometry might be the answer!
The central issue is understanding how changes in control parameters (for instance concentrations of catalysts in a chemical system, or local fields in a spin system) affect the evolution of the probability distribution over states. Some work has been done in close to steady state (for instance [1,2,3]) but it's far from resolved.
This has some nice applications - designing efficient protocols for microscale devices, for instance.
[1] https://arxiv.org/abs/1603.07758 [2] https://arxiv.org/abs/1507.06269v1 [3] https://arxiv.org/abs/1201.4166