I'm a third semester math and computer science major. I'm at a decent public school (ranked around 100 in the country). I'm currently taking undergraduate courses in algebra, real analysis, and number theory. I'm doing really well and have essentially perfect grades in these classes (analysis is my worst of the three, and I'm 14 points above the A+ cutoff in that class right now). A twist: I've just been admitted as a transfer student (for January 2010 entry) at Rice University, which is a more prestigious top 20 private college in Houston, Texas. I'm trying to decide to stay here or to go there. My goal is (broadly speaking) to get into the best graduate school for mathematics that I can gain admission to. Of course, there are other factors to consider, but for the purposes of this thread this generic statement should suffice. The following is a hypothetical plan which represents the very best I expect I could pull off at my current school. This plan represents a hypothetical ideal. That said, I do think it's within my grasp to accomplish, but as always if it hasn't happened yet, then it hasn't happened yet. Based on my performance in my classes this semester, I think I could handle graduate level courses next semester. Looking at the graduate curriculum here in a lot of detail, I think I could accomplish the following graduate courses as an undergraduate (not in any particular order): algebra I and II, complex analysis, general topology, algebraic topology, real analysis/measure theory, differential geometry, and algebraic geometry. If I take two courses a semester, I could technically even do more than this, but I'm going to keep the list at this to account for potential scheduling conflicts. Now I also want to go to two REUs (one in each of the next two summers). It's possible that I won't get accepted into an REU for one (or both) of these summers, but let's assume (for the purposes of this thread) that I will get accepted. Say that one publication results from these while I'm an undergraduate. Another possibility is that one semester I will study abroad in Moscow or Budapest. Let's suppose I spend a semester in Moscow at some point. And let's assume (this is a big assumption) that I make As in all these math courses. If I were to carry out this plan successfully, what would my chances be at the very top graduate schools in the country (Princeton, Harvard, MIT, Stanford, Berkeley, etc.) for mathematics and for Part III at Cambridge? Finally, what sort of performance in a competition such as the Putnam would provide a significant (i.e., noteworthy) boost to my chances at the aforementioned schools? What's the worst I could do on the Putnam (say, top 10, top 50, top 100, top 500, etc.) and still get a noteworthy boost out of it? I want to emphasize that I'm just using these schools as a benchmark. I would be okay going somewhere else; I just want to get a feel for very best I could hope for if I achieve the above plan. This will help me in my decision to stay here or transfer. Thanks to anyone who responds! This is a tough decision for me, and I'm trying to weigh my options as thoroughly as possible. The thing is that Rice is going to be a LOT more expensive than my current school, and because I've done so well so far at my current school I think I stand out pretty well here. It's hard to give up acceptance to a great school like Rice, but it's also hard to give up what I have here and the possibilities above that I could realize here.