If I can offer a tip, and one that may seem rebarbative to many, but I would consider applying to the same places your referees got their PhDs from, if that's at all possible. I applied to four top 15 universities, and was accepted to the two where my referees had got their degrees from (whereas the other two I had no connection with, and was rejected from). In some respects, it may seem obvious, but I don't think I've seen it mentioned on this forum before. An admissions committee is more likely, I think, to trust the words of someone they're familiar with, and someone who successfully went through their program. Of course, this is not at all a hard and fast ruleāand may indeed play no role whatsoever in the committee's decision. It's just something to think about!