<p>First, background:
Before I was in Kindergarten, I was getting math books for presents for Christmas/Easter/B-Day. I'd go to my grandparents' house and ask for math to do. To this day, it's the same. I got math books for Christmas, and my idea of a party at my [residential] high school was to have a group of friends come to my room to work problems on my white board (no exercises allowed ;) ). In addition to having taught myself quite a bit, I've found many subjects in math highly intuitive; I at least think I'm quite good at "asking the right questions" in addition to solving them. I'm currently a freshman at the University of Wisconsin at Madison. By the end of this year, I'll have had (in some form or another) Calc (single/multivar), Linear, Differential Equations, 2 semesters of Modern Abstract Algebra, Introductory Number Theory, Introductory Combinatorics, Analysis I and II, Intro to Probability, Stochastic Processes, Introductory Topology, Introductory Cryptography, Calculus-based Physics, a year of Mathematical Methods in Physics, and Introductory Quantum Mechanics, as well as pieces of other topics (some game theory, classical mechanics, chaos, etc). I'm also pending review for publication in an international journal for my work on nested integrals of arbitrary quantity and the generation of the homogeneous differential equations of which they are solutions. I also have other research. I'm not an IMO medalists (or have qualified for the test); I'm too slow, and I'm dysgraphic. </p>
<p>Problem:
Filling out my college applications last year, I looked for universities at which I could intensely pursue mathematics and music performance. For various reasons, I've decided to drop my music performance portion of my major. I chose UW based on its satisfaction of my aforementioned criterion and because I'd contacted the admissions department and found a way to graduate in the summer of my "Freshman" year; by the end of this year, I'll have tested out of or been accredited 82 hours outside of the courses I'll be/ have enrolled in here. I feel very, very uncomfortable here; I prefer math to alcohol and illegal drugs. This is not to take anything away from the UW math department; I have several phenomenal teachers, and the program (for the upper lvl maths, at least) is quite rigorous. My problems with the place follow:
1) Too many gen ed requirements (6hrs communication, ethnic studies, 6hrs Biology, 12 hrs foreign language, 24 hrs humanities)
2) Too impersonal; I don't like the feel (I can't quite reason out why yet), but I don't like being one of the over 42,000 here.
3) I'm not "normal"; I'm the only freshman in any of my classes (Intro Prob, Elem Topology, Analysis I, Modern Algebra), and I know of no sophomores, and I'm generally the one people go to for help.
4) (the BIG one, follows from 3) I don't feel like I can work out math problems with my peers or do math for fun with them. I don't feel like there are many (if any) others here that enjoy just doing math. That is, I don't fit into the social atmosphere intellectually. Part of this is that I'm not incredibly (read: "at all") outgoing; I'm very friendly, just friends usually wind up coming to me, not the other way around, and so I get acquainted with people much better in a smaller environment.</p>
<p>Situation:
I decided that it would probably be in my best interest to remain an undergraduate student for another year/two/three and become more well-rounded before moving on to grad school and confronting the PhD deadline. There are 3 main things I'm looking for:
1) Least important: Provide a grounding for future work (ie, breadth of math with rigor)
2) Provide resume materials for near future (name of school, research opps)
3) Most importantly: submerse myself in a suitable social environment where I will constantly come in contact with new brilliant "directions" of thinking
I can type up my reasons for these later, as well as my philosophy on education and the "head fake" which it does and should employ.
I'm planning to transfer for the Fall of 2011.
My current considerations are:
CalTech
Cambridge
UChicago
Harvard
MIT
Berkeley
Stanford</p>
<p>And just-in-case:
UCLA
UMich</p>
<p>Funding shouldn't be a problem; I have an internship offer, as well as various other ones which I can apply for. I'm not asking for whether I'll get in, hence me omitting any scores and such. I'm asking for advise on what to do: where to go, how many years to attempt to graduate in, etc. I'm completely open to any alternatives to the universities I've listed. Those are in approximate order of preference. As such, I'm completely open to any alternatives in order I've listed. I'm also open to any change of plans (go to grad school next year instead or whatever). I'm also wondering how exactly (or if) I'd be able to fit into UCambridge as a transfer student; from the research I've done, it appears I've had/ will have had their IA and parts of the rest of the math course. I'm also very interested in a comparison of Cambridge to US institutions if possible; I don't feel as if I have a great wealth of knowledge with which to make that choice. I also know very little about Williams College and am interested to learn more. </p>
<p>Additionally, (other than Cambridge vs US), I'm not as much interested in what I'll learn (though I'd love to hear a comparison of intensity and rigor, it would lead to much flaming) as I am on opinions on how I personally would fit into various institutions. Please also offer any questions to which an answer would help to give more accuracy on a response.</p>
<p>Thanks a ton.</p>