This is not an answer as much as a request for answers. Here are some topics for which I don't know an example of this sort, but would really like to see one (and nobody seems to have done them yet). Feel free to edit this post if good examples appear for some of these topics, or if you have a topic to add to the list. They are in no particular order,

• Symplectic geometry (I'd really like to know...)

• Algebraic geometry (there should be a ton of stuff here, it's so diverse)

• Category theory

• Homological algebra (is there an example simpler than some monstrous calculation?)

• Group theory (so far, all that comes to my mind is that it applies to Galois theory, which in turn solves equations. But I think there should be a ton of simpler and nicer applications, shouldn't there?)