<p>Graduating a year early is a very big decision, and there were many factors that went into it for me: I went to a very expensive school (NYU), so a fourth year would have cost me more money; I was already taking graduate courses, so why not go to grad school immediately where I would get <em>paid</em> to study, instead of paying to continue in undergrad; I had already decided a long time ago to graduate in three years, so the vast majority of my friends were graduating with me, and thus I wouldn't have as many friends at NYU had I stayed an extra year; etc. It was a big decision, and there were a lot of other factors I had to consider. Thus, with all of those reasons to leave, I'm very happy with graduating early.</p>
<p>As far as the research I did, I attended the Lafayette College REU in the summer of 2005 and was in a group studying isoperimetric surfaces in specific four-dimensional metric spaces that have relevance to physicists. We ended up making some progress in two of the spaces, which led to giving a talk at the Joint Math Meetings, and we have submitted our paper to a journal (for which it is currently being refereed).</p>
<p>So whereas my research experience is nontrivial (thus giving me an edge with some of the top 50 schools), it wasn't one of the famous REUs (like Duluth), it wasn't one of the NSA programs, and we didn't have a published paper by the time I had applied to graduate schools. So that's something that certainly didn't help me with Harvard and MIT (and Columbia, which had a particularly difficult admissions season this year, admitting only 11 out of over 350). There are enough students who have those edges applying to the top schools that they can exclusively admit from that pool if they wish, so I really never had a shot at them.</p>
<p>Honestly, I don't know if staying an extra year is going to help you out that much with Harvard, MIT, Stanford, and Princeton. If those schools do, as I suspect, admit students primarily from the applicant pool I mentioned above, then an extra year will likely only make a difference if you come out of it with a phenomenal research experience and a phenomenal letter of recommendation. And even then, there's no telling if that's going to help you break the barrier and get into the aforementioned schools.</p>
<p>So really, I wouldn't put any weight about getting into these schools into your decision to graduate early or stay an extra year. Figure out if you're ready to graduate or not, and base your decision on that. Your undergraduate career should be more than just a stepping stone to graduate school.</p>