<p>My 2 cents, coming from a beginning graduate student in math who’s just been through the application process herself:</p>
<p>I agree with broken_symlink that your list is too top-heavy. Even the Ivies make their top students apply to several graduate programs outside of the top 20 and as someone not from a top university, you will need that safety net even more. </p>
<p>My current understanding of the admission process is that the two most important factors are your math background and your letters of recommendation. Successful applicants to the top 10 programs usually have a solid 2 year’s worth of graduate coursework under their belt. (That is, a foundation in graduate-level algebra/analysis/topology and additional coursework in a narrower specialty like representation theory or low-dimensional topology.) Of course it’s not surprising that the content of your letters of recommendations is important, but it also seems to matter who they are from. The top math graduate programs seem to swap students among each other and you might be at a very significant disadvantage if your professors have no relationships at the schools you are applying to.</p>
<p>As a concrete example, 18 of the 21 American admits to the math PhD program at MIT this past year had their undergraduate degree from a top 20 math program. The remaining 3 were from Duke, the University of Illinois at Urbana Champaign, and Bryn Mawr College (though practically an undergraduate at the University of Pennsylvania) - all very strong universities too!!!</p>
<p>Out of the graduate programs I visited, MIT, Princeton and Stanford seemed very exclusive and prestige-oriented. Berkeley, Michigan, Austin, Columbia and Cornell were more open to bright students from “normal” universities, though most of their accepted students still had some graduate-level training as undergraduates. Penn accepts a lot of students from liberal arts colleges, where graduate courses are not commonly available.</p>
<p>Hope this helps.</p>
<p>P.S. Applying to PhD programs is not about getting into the most prestigious program but rather finding the best adviser for your interests. The top programs usually have a high concentration of good advisers, but not every program is strong in every discipline. Conversely, there are some fantastic and famous advisers at less prestigious programs. Work with your undergraduate advisers to find graduate programs which will meet your academic needs; don’t just take the x most prestigious ones.</p>