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I'm currently a first year PhD student at a respectable but not top-ranked math program. I'm also confident that I am among the top three first year students in my program. As an undergrad I went to a somewhat prestigious liberal arts school and was valedictorian of my graduating class. I did a little bit of research and took a fair number of upper level classes (although no grad courses) along the way.

In hindsight I can see where the problems lay with my application. As the title of this post suggests, I got a mediocre score on the math GRE. In addition to this, I was kind of a late bloomer as a mathematics student. I didn't start taking many upper level courses until my junior year. The people who wrote my recommendations were not necessarily all that famous either.

So I got into all the programs that I thought I had a good chance of getting into but none of the programs that I thought would be more of a stretch. The one I'm currently attending gave me a good fellowship and I've been doing very well here so far (passing qualifying exams, taking lots of classes and doing well in them). But I honestly feel like I'm not being challenged quite enough.

Long story short, I screwed up on the GRE when I first took it and I want to try to transfer into a top-ranked program now. I've heard plenty of times that top programs will throw out applications with low math GRE scores. But does the same standard hold for grad students applying to transfer? Will programs throw out applications from current grad students if their GRE scores are too low? To be honest I do feel a bit silly thinking about retaking it. Of course, any advice on transferring between programs in general will be greatly appreciated.

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    I think it would be safe to aim for a high GRE mathematics subject score. I don't know if graduate admissions committees actively "throw out" applications with low mathematics subject GRE scores (although there are rumours of this), but it definitely seems rare to be admitted to the top programs with scores less than 70%, for example. I do know of some people being admitted to the best programs with scores between 70% and 80%, but the safe option would be to aim for a score above 80%. – Amitesh Datta May 1 '14 at 8:29
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    However, I will add that it seems recommendation letters are one of the most important components of the application (assuming the other components of the application, such as the GPA and GRE scores, don't raise a red flag). I don't think the recommenders need to be famous but, practically speaking, one would imagine it would help if they are known to the admissions committee. – Amitesh Datta May 1 '14 at 8:40
  • Although this PDF is written from the point of view of admissions to Master's programs in computer science (at the top programs such as Stanford, MIT, UC Berkeley etc.), I think a lot of it is quite relevant to the admissions process for PhD mathematics. I think it is worth having a look as it is written by someone who has been on the admissions committee at Stanford. – Amitesh Datta May 1 '14 at 8:46
  • I think I will be able to get a recommendation or two from some pretty well known people this time. At my undergrad liberal arts school there weren't nearly as many well-known researchers as at my current school. – user14221 May 1 '14 at 8:47
  • Yes, I think that would help, as long as the letters are good. Also, it might be worth taking some (if possible, advanced) graduate level classes and doing well in them. In conjunction with a high score on the mathematics subject GRE (as well as good scores on the general GRE), I think that would be a competitive application. – Amitesh Datta May 1 '14 at 8:53
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Among elite and maybe even top-20-or-so math grad schools, many use the math subject test GRE as an easy, automated filter... even while recognizing to some degree that it doesn't really test what we'd like... because even after applying that slightly unreasonable filter, there are still enough good candidates.

So, yes, if you have a really-mediocre score, there would be advantages in re-taking.

But/and, as in comments, letters of recommendation are far more important once you've gotten past some initial filtering. The letters should be from people who can be believed to understand what a serious grad program in math is about, e.g., by having gone through one themselves, and they should say that you'll excel.

Meanwhile, in contrast to undergrad institutions, there really is no notion of "transferring" to another grad program in math. Nothing gets easier. You will be expected to have excelled in the year you've spent already, which it seems you have. You'd not get to really compete directly with people with fresh B.S. degrees, since, in effect, "performance/showing-potential" is prorated on what chances you've had, how much time you've had, to demonstrate it. This is reasonable, in many regards.

In summary: yes, re-take the GRE, to overcome initial filters. After that, it's about letters of recommendation, and (for me) your personal statement. That is, explaining that you got a late start (which can easily happen to sensible people) can account for some things. True, some admissions committee people might be looking for people who've "always known" that they wanted to do mathematics, but that can't be helped.

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