Assume that someone has great grades in math classes from a top-tier college, has taken a lot of advanced math classes (including grad classes), does well on the GRE math subject test, and shows passion for and dedication to math. Then it seems to be the consensus that that person can get into a top math grad program. Will having a great letter of recommendation from a very famous professor be a big enough bump to make admission into Harvard or Princeton a somewhat high probability for that person?
closed as unclear what you're asking by RoboKaren, scaaahu, user3209815, David Richerby, gman Oct 11 '16 at 10:49
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I think your question is coming from a (natural, understandable) urge to extract guarantees/promises about the future. The thing about the future is that in many cases, to know what will happen you have to wait for it.
As you say, having excellent grades in challenging courses including some graduate courses, strong GRE scores, great recommendations from faculty whom the committee can trust to know what kind of student is likely to succeed in a top PhD program, and showing an enthusiasm for mathematics and the mathematical profession is the recipe for success for admission into a top graduate program in mathematics.
But even a recipe for success doesn't guarantee success. Harvard and Princeton can pick from and will mostly fill their entering classes with students with this profile. An application with the profile you describe will get very serious consideration. Whether you get admitted to the end depends upon how you stack up against a relatively small number of other candidates with similar profiles, including on considerations of balance and diversity: e.g. they will probably hesitate to admit more than two students from the same school. Past a certain point you just can't know, so much so that it's not worth worrying about: that kind of worry is for Harvard and Princeton, not you. The way to leverage against uncertainty is not to foresee it all in advance, but rather to apply to more than two programs.
When I applied to math grad school (almost 20 years ago, but since then I have done graduate admissions and think things have not changed much), I had a similar profile to yours. I applied to four of the top five programs and another in the top ten. And sure, I sweated it a bit before eventually getting admitted at all of them. Now that I think about it, I did have one thing going for me that you didn't mention: I went to a top undergraduate program (Chicago) and shared with two others a prize for the best undergraduate math major. The prize itself is not so significant, but it showed that no one else from my program was going to get recommended ahead of me. I think the top ten programs in the US believe in each other's excellence enough so that the top graduate from one school probably has a place in one of the others.
I think you'll do well. Try to be patient about it.
Added: As suggested, let me address the "famous professor" aspect more directly. As I said above, the letters need to come from faculty whom the committee can trust to know what kind of student is likely to succeed in a top PhD program. Such faculty definitely need a PhD of their own, and it is better if they have some direct experience with PhD students and faculty in top programs (either having been one or the other, having collaborated with them, and so forth). Beyond that, I don't think the "fame" of the professor matters so much. For instance, a tenured faculty member at a strong department who has placed other students in top ten departments over the years is probably getting close to maximum credit for their "fame". One thing I didn't appreciate when I first read the question, but picked up on after reading @GEdgar's comment, is that the OP doesn't say whether she took classes from or did long-term research with the famous professor. If not, then what the famous professor says is probably not as helpful as the less famous professor who can compare you to other strong students from your institution.