<p>Hermes, to give you a small idea from my knowledge of a single field – MIT’s math program supposedly told some students who asked (note: this is something I heard from someone else) that if you’re a student from a small, lesser-known school, if you want any chance at all, you should be the best student they’ve seen in 5 years or so. For a big name school, you should be one of the very topmost in your area of interest to stand a real chance. </p>
<p>If you’re an IMO gold medalist becoming a subsequent Putnam fellow in a US university who’s published groundbreaking work in your field of interest, I anticipate you’re in without any question at all, unless you’re a criminal on the run or something – it’s not super unpredictable in these cases. But obviously these are incredibly rare applicants. Those I know personally who made it in to these schools had really impressive applications</p>
<p>As for graduate study, to comment a little:</p>
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<p>I think it depends on the individual, because some individuals are actually quite flexible in regards to what they would be willing to specialize in, and go in with the opinion that they want to learn about various fields, and make real contributions to at least one. So sometimes, depending on how flexible you are, you could just go to a great school which has something you want, as long as you’re fairly sure you want to do real professional research in that area.</p>
<p>I don’t think you should bank on being too flexible though, because to sit through graduate school in a discipline is really grueling work, and unless you care about the minute details, you won’t make it…important to consider, because a lot of things appear attractive when presented in a refined form during a course. For all you know, if you don’t have an idea what research is like, you may find it’s very boring, as interested in the field as you may be.</p>
<p>All this said, I think I end up coming to the same conclusion as Mollie, which is you shouldn’t play the whole “Oh, I’ll head here for undergrad and there for grad, no big deal” game and expect to be happy with the results.</p>