I'm trying to figure out which undergraduate colleges are most conducive to graduate school applications, but it seems to me that (with few notable exceptions) my two main criteria that lead to this sort of success are mutually exclusive. Are there any schools in the U.S. that have both an extensive mathematical curriculum and a focus on their undergraduates?
Schools known for being strong in math (such as MIT, Harvard, and Stanford), seem to be very heavily focused on their graduates, with comparatively less attention payed to their undergrads. I'd like to avoid classes taught be TAs and be able to learn from and get to know my professors, both for the improved learning experience and the improved graduate school recommendations that will inevitably eventually result. Perhaps even more importantly, it seems that research as an undergraduate tends to improve students' chances at graduate schools; thus, colleges that have significant research at the undergraduate level are most appealing.
However, schools with the above desirable criterion tend to be less mathematically rigorous, especially in pure math. I've found a handful of schools that will even keep a math major occupied for a whole four years if they've taken only introductory multi-variable calculus, linear algebra, and differential equations. I don't want to have to sacrifice an education for attention -- I also want to go to a school considered highly for graduate school that has extensive course options.
Is the first criterion a fair voting system and the second a dictator? Is there any intersection of the above specifications? My own rudimentary research suggests a few options, but these hardly make a list even the best students can expect to be comfortable applying to.
I want to make sure I have the best options for graduate school in the future, both in being a qualified applicant with the opportunity for stellar recommendations and in having learned the extensive mathematics required to give me a head start for future research.