<p>Your situation is the opposite of the one most Caltech applicants are in. The typical situation is near-perfect SATs, APs, and grades, and nothing really eye-catching in the extracurricular area (e.g. some community service plus one sport). You, on the other hand, have distinctive extracurriculars, are super-committed to science, and are clearly striving for the best opportunities available to you. On the other hand, you have low SAT and AP math scores by Caltech standards, as you yourself noted, as well as one worrisome gradne in math. To answer your most recent question, your independent study and clear willingness to go above and beyond what most people at your school do (with Physics C) is a huge help, but probably will not get you over the line without better numbers.</p>
<p>The bottom line is that with a student like yourself who clearly has a huge amount of passion and independence, Caltech is sometimes willing to overlook one big departure from the normal standard, which is: A- or better in math and science through high school; mid-to-high 700 on the SAT I Math; a 5 or at worst 4 on AP math and science tests (unless the school is clearly incapable of teaching those subjects well); and a score in the high 700s on the math SAT II Math.</p>
<p>With you, I think it’s even possible that two such problems would be overlooked, but it’s absolutely crucial that you get your SAT I Math score up. Apply regular action; take the test after October. Surely someone with your drive and commitment can spend enough time with an SAT I book to nail this. It is also nearly required (especially with a 3 on the AP Calc) that you really nail the Math SAT II. Assuming that you do both of those things, you have a quite serious chance of getting in with your unconventional profile… just because Caltech really loves people who are clearly committed to science. You just need to help yourself over the threshold of the front door with test scores.</p>
<p>Now, of course, whether that’s worth your time is up to you. (Everyone would certainly understand if not.) With your current scores, your chance will be low, so you have to decide whether it’s worth the time. With higher scores, I think it’s definitely worth giving Caltech the chance to admit you, since you would clearly make a cool addition to the community.</p>
<p>P. S. cghen is right that it’s the focus on core math that drives these types of decisions. Often, there is an awesome student whom the committee would really love to admit, but the worry is that the core courses will be so difficult for a student that it would be mean to let him or her in unprepared… that is one of the genuine drawbacks of having such demanding standards coming in.</p>
<p>Anyway, best of luck to you!</p>