[Mgccl]

You are restricted by the set of tools you already have right?

I can't believe how many times a great advancement in theory in different field comes from pure math advancement (or applying pure mathematical results). Say in economics, if Nash didn't apply topology, how would one ever know Nash equilibrium exists. Stable matching, compressed sensing and others also comes to mind.

I have a hypothesis that any field would benefit if we introduce some pure mathematicians to them.

[j2kun]

To be completely honest, the more common case is that the mathematicians follow behind physicists and engineers to verify that their applications are theoretically sound. But yes, great breakthroughs do come from pure theory.