Sure, but in no point in the pursuit of a PhD in theoretical physics did I actually use any of that machinery. Nor did it ever come up in any papers I've read,

Sure, but in no point in the pursuit of a PhD in theoretical physics did I actually use any of that machinery. Nor did it ever come up in any papers I've read, nor any talk I've seen. The point is this -- if you want to extend our model of the universe, knowing the more abstract machinery is useful; by generalising your understanding (and a hell of a lot of math is basically generalising to more abstract systems) you gain insight into how alternative models might be constructed. If you want to understand our current model, even at a quite deep level, you can get by without knowing any topology at all.