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Although I am an active user of SE, I have chosen to remain anonymous because of the nature of this question. I am currently a graduate student at a university that lands in the mid 70s among U.S. News' top graduate schools in mathematics, which by my understanding is very poor. My grades were quite good and although I didn't do terrifically on the subject GRE, I had a few offers from schools in the mid 40 range which I turned down. This was partially because I think I undervalued the importance of choosing a better school, and for a variety of personal reasons. My university has some unique opportunities in applied math, but is otherwise unremarkable.

As I enter my second year, I am increasingly convinced that a career as an active research mathematician is what I want for myself. However, I've heard so much contradictory information that I'm unsure if this is a realistic goal given the status of my university. While a variety of people have told me that there are plenty of desirable jobs available in "industry," the general consensus is that graduate students from low-ranking schools tend to end up as community college professors. The fact that most universities (including my own) are full of doctors from strictly high ranking schools seems to support this.

I have the chance to receive a terminal masters at the end of this year, rather than continue on for another four and complete my PhD, so I'm considering trying to retake the GRE and reapplying (I am unclear whether this is frowned upon in the math community in general, although at my school I have been reassured it is not). On the other hand, my university has a few professors who seem to be well known within their niche, mainly because their particular specialties are very unique.

My general questions are these: How important is the quality of the university when it comes to having a full career in pure mathematics research? Is a well known adviser a factor, possibly outweighing the school? Is this significantly different among applied mathematicians?

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First, let me answer an easy question. It is not frowned on to get a masters at one school, then transfer elsewhere for a Phd. In fact, this is often encouraged. Next, let me confirm that yes, the ranking of your PhD institution really does matter for a future career as a research mathematician.

In some sense, Gerry is right when he says that the ranking of importance is (1) quality of dissertation, (2) reputation of adviser, and (3) ranking of department. One big problem though, is that many of the people evaluating you won't know how to evaluate criteria (1) and (2). Hiring committees often include many (all?) people who are not familiar with your research area. So they will often rely on what they (at least think that they) know, which is quality of department.

I have written elsewhere about why you should go to the strongest program that you can get into, particularly if you want a career as a research mathematician. I don't know all the details of your personal situation, but I strongly encourage you to at least take the GRE again and reapply. It may be possible for you to wait to see whether you get accepted to a school that you prefer before you decide whether you'll leave with a terminal masters or not.

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My opinion, which is worth what you are paying for it:

The most important thing is the quality of your dissertation.

The second most important thing is the reputation of your thesis advisor.

The ranking of the department only comes in third.

Of course, these things are not independent of each other. It may be that at a higher-ranked department you'll get a better education and write a better thesis.

Probably best to speak to someone at your current institution (or your undergrad institution) who knows you well and can give you better advice.

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