<p>I'm wondering what my priorities should be, so could you rank them for me in order of importance? By the way, I'm trying to get into Ph.D in pure math</p>
<ul>
<li>GPA</li>
<li>GRE math/verbal</li>
<li>GRE Subjects</li>
<li>Prof Recommendations</li>
<li>Research experience/publications</li>
<li>Personal statement</li>
<li>Undergrad Institution (i.e. would they heavily favour someone from MIT over someone from a second-tier university?)</li>
<li>Putnam Competition Results</li>
<li>Others (please specify)</li>
</ul>
<p>Most important (in no particular order):
- recommendations
- research
- possibly Putnam. (If you have a sub-30 score, say, it won't hurt you by any means; if you have a 50+ score, it can be a big help. I know someone who's going to Cambridge for her DPhil, but who never even took the Putnam.)</p>
<p>Reasonably important:
- subject GRE
- personal statement</p>
<p>Less important:
- general GRE
- institution</p>
<p>I haven't included GPA in these rankings. This is because it definitely varies, depending on where you're applying. Cambridge, for instance, probably along with Oxford and the other good English universities, values GPA very, very highly, since it's not easy to judge American (and I'm assuming that you are American) students on anything other than GPA, since GRE scores aren't considered by English universities. However, GPA isn't quite as important stateside --- I have the impression that here, it can break you, but it can't make you. A professor here at the University of Washington told me that the UW math program, which is good, but not in the top five or ten for graduate programs, is getting so competitive that a 3.7 in upper-level undergraduate courses (algebra, analysis, topology, possibly number theory) will disquality an applicant.</p>
<p>Hope this helped! Feel free to PM me; I'm looking for a math Ph.D as well.</p>