Which universities have the most students who apply graduate level at Princeton University successfully in Mathematics?
Probably Princeton
Actually, I’d disagree: Princeton Math is one of those departments that generally don’t take “their own” for doctoral studies. There are exceptions, of course, but nowadays they like to encourage their undergraduates to experience another mathematical culture in grad school.
Anyway, to get to your question — Harvard grads are very frequently represented among Princeton PhD ranks. As far as I can tell they seem to be the mode of the data set. But you should keep in mind that it really doesn’t matter much where you go: Great mathematicians come from everywhere, every educational landscape, every walk of life. What’s important isn’t the institution but the individual. A mathematical visionary is a mathematical visionary, whether their diploma says “Harvard” or “SUNY New Paltz.”
Best of luck to you!