Short and simple question: as a prospective PhD student, should I consider applying to universities which aren't as good/prestigious (still top 500 worldwide, for instance) in my area, in order to study with a professor who is well-renowned, or should I try for better standing universities, but study with potentially young and less established professors?
I struggle especially because, often, there is a single (if that) professor in my area in most universities, so it's really a case of individual versus institution.
If it helps, I'm going for a PhD in pure math, but the questions seems independent of the area.
Thanks in advance!