What interests me is the natural world. What interests me most is physics. I think theorists should be concerned with improving on the logical consistency & parsimony of the mathematical scaffolding of physical theories that is known to work --no surprise there. But to me, this is best captured by Dirac's motto, as quoted by James Gleick,

"The equation knows best."

Let me give you what I see as an example of this point that has become my overriding obsession: The Lagrangian formalism. Lagrangians are notorious for facilitating calculations and making the symmetries obvious. But their physical meaning, if any, is obscure at best. Anybody familiar with the Lagrangian formalism knows full well that it's to do with surfaces. What are those surfaces and what do they represent? Observational scope perhaps? Nature gives us the equation. It is for us to interpret what it's "saying." Nature couldn't care less about interpretation. Interpretation is for us. We are the translators.