I don't know whether this is true today, but 30 years ago I had the impression that there were Math Ph. D. programs in the United States that would accept anybody with a pulse, just so they would have warm bodies to put in front of their Calculus classes. Naturally, we're not talking about Princeton here, but there were (and are) a lot of Ph. D. programs in the US, and they couldn't all be real choozy. Where there's a will, there's a way.