I'm a pure math PhD specialized in hyperbolic topology, doing a research postdoc right now which ends in August of 2018. So I'll be starting to apply for my next job in the next couple of months. I've been thinking about looking into applied math jobs in addition to the usual math research and teaching jobs.
I realize that there are jobs one could get in industry just for holding a PhD in math, but I am interested in jobs where my particular areas of knowledge/interest have potential for important scientific applications. Barring that, I would most likely accept the best teaching or research position offer I got (work which I also enjoy).
Below are some areas of math that I have studied. I've started with some specific topics I've studied closely, then put some more general areas that you could say I work in. My question is: what are (if any) some known applications of the following disciplines from pure mathematics, to the sciences?
- Hyperbolic 3-manifolds
- Hilbert-Blumenthal surfaces or Hilbert modular varieties
- Dynamical systems using Hamilton's quaternions
- Quaternion algebras (not just Hamilton's quaternions, but generalized quaternion algebras as defined by Dickson).
- Low-dimensional topology
- Arithmetic manifolds
- Algebraic number theory, especially quadratic fields, lattices, rings of integers