You listed several good alternatives to "standard" math, such as probability, cryptography, and number theory. I'll add mathematical logic, set theory, game theory, and computer programming.

The EPGY math curriculum covers logic and set theory. Set theory was an important part of the "New Math".

Financial mathematics employs probability and calculus (or at least algebra). Applying math to finance may be difficult without some grounding in algebra.


"To see what is in front of one's nose needs a constant struggle." - George Orwell