<p>“I don’t think there is a top school realistically for the undergraduate years, and it depends largely on goals.”
I’d say this depends less on goals (given that you want to be a mathematician) than on the environment in which you would like to learn math (and at ~18 years old the latter may be easier to gauge than the former). Some things to consider could be…</p>
<p>a)Number of advanced courses offered. Among the top schools, this varies greatly is pretty weakly dependent on the size of the department. Some of the top departments offer surprisingly few special topics courses–if grad students want to learn subject X they organize a reading group or student seminar, and maybe invite speakers–something undergrads can’t readily do. One thing I loved about Caltech was that (since there are so few undergraduates to teach), they let post-docs teach graduate/special topics courses, so despite the department being quite small, it seemed (at least to an undergrad who doesn’t know much) that there were classes offered (at least every other year) in pretty much every area of math one would fancy. I’m now a grad student at UChicago, where (despite the much bigger department, lots of famous professors and lots of postdocs who will soon be famous professors), it feels like a lot fewer classes are offered. For example, there are (usually) no classes in algebraic geometry offered, despite the algebraic geometry group being one of the best in the world. </p>
<p>b)Methods of assesment/ style of courses. At Caltech most grades are based on HW assignments, so you got regular feedback to check whether you’re on the ball. On the other hand, a lot of people fell into the trap of spending so much time on homework that they had no time to think about the material (I often thought I would have learned more if I was given a textbook and some papers and allowed to just think for a month). At Cambridge, from what I understand, “grades” are based on yearly exams, and for the rest of the year students learn at their own pace, regularly meeting with supervisors to gauge their progress. In retrospect I probably could have learned more under the latter system.</p>
<p>c)Amount of red tape in the department. Some departments heavily regulate who is allowed to register for advanced courses, and heavily enforce prerequisites; some do not. At Caltech prerequisites are just guidelines and you can register for any course you feel prepared for–if in some area you feel you already know the material in the undergraduate sequence you can almost always argue to take the graduate version instead. At Chicago (for instance) it seems to be almost the opposite. For freshmen to take the highest level Real Analysis course they have to distinguish themselves on placement tests; students with poor grades in certain courses are sometimes prevented from registering in more advanced courses in the same area; non-math majors are not always allowed to take advanced math classes, etc.</p>
<p>The ability to get involved with “mathematical research” should probably not be a deciding factor for choosing schools: math is cheap, you can do research anywhere with internet access, and at any decent school there will be faculty willing to guide you through the process if you are interested; besides there are plenty of math REU’s around the country.</p>