Take a look at the “Strong Mathematics Graduate Programs” listed by MIT, many of which overlap with the colleges your list. Undoubtedly each has a strong undergraduate program as well.
My similar stats daughter (also a math major) ended up at Cornell, which she enjoys very much. That being said, it’s definitely not the same as attending an urban school.
Good luck!
At most HS’s, clubs have to have a faculty sponsor or advisor. Is that true for yours? And what reason did the faculty member give for not merging the clubs? That seems unreasonable.
Yes, we had a different math teacher advise ours. As for the math teacher running the other club, I love her, but she can be rather stubborn. I guess I’m going off-topic on this thread so I’ll stop 
If you like Amherst, I would recommend looking into Wesleyan; the department is slightly bigger and has the added advantage of a doctoral program in topology, if that floats your boat. Amherst students may also have access to UMass classes, but that’s a whole other discussion in terms of consortiums as opposed to stand-alone LACs.
Amherst and Wesleyan reside in similar sized college towns. My sense is that Amherst students are less food oriented than Wesleyan students and are less likely to dine out, shop for groceries or take a stroll to grab take-out food than Wesleyans - but I could be wrong. Wesleyan’s about a third bigger.
I’m curious what you mean when you say “outside of school.” Are you taking courses at your local community college or online?
Here’s my repetitive mantra: Caltech is a RESEARCH institution.
That means you’ll be doing a lot of independent work.
That means you’re OK with graduate students dispensing tests and grading them.
The professors lecture, and then they’re off to do their own research and write their books.
I did abs alg and real analysis online. AP stats I literally just learned on my own and took the AP exam because I didn’t really feel like taking it as a year-long course at school. If I were to do ODEs, that would be at my local community college or online as well.
You are already so far ahead that I do not think that there is any need to take these outside of school. If you do end up at Caltech or MIT, these classes will be very good classes but will be taught with quite a high level of rigor and will teach these subjects very well. In math it is important to learn each step very well because you will be using it again.
You are clearly a very strong applicant and competitive at any university. Caltech and MIT are of course reaches, but I think that your chances are at least as high as the overall acceptance rate and might be slightly higher. Unfortunately “slightly higher” still leaves your chances in the high single digit or low double digit percentages.
Amherst College is in a rather small but very attractive college town. I do not know whether or not this counts as non-rural.
One small oddity here. The university of Toronto is very good for mathematics, and for a wide range of other subjects. The first gay marriage to be eventually accepted by a modern western nation was performed by a minister who was a U. of Toronto graduate (for his graduate divinity degree – his undergrad was at a small university in eastern Canada). This came about because Ontario (and also Quebec) has a strange system where you can first announce the marriage publicly, then get married, and then go to get a marriage license. A gay couple did the first two, and then when then went to get their license the clerk basically said “I do not know what to do, I think that this is going to have to go to court”. It went to court and eventually I think that the Supreme Court of Canada declared the marriage legal retroactive to the somewhat earlier date of the marriage ceremony. Prior to the marriage ceremony the Governor General of Canada had sent the couple a very kind letter of support, which was also published in a Toronto newspaper. Regardless gay marriage has been legal in Canada for a long time, and very few people seem to think this is anything other than a personal choice.
I think that either Toronto or Waterloo would be very likely for admissions.
One long shot would be the Lester B. Pearson scholarship at the University of Toronto. This is a full 100% merit scholarship for international students. It is exceptionally competitive. You need to get your high school to sponsor you to even be allowed to apply. If you get it, then you would likely end up with a lot of college money left in the bank for graduate school.
Wisconsin is very good for mathematics.
As a math major with a master’s degree, I took a LOT of probability and statistics going on to stochastic processes (multiple years of it). There is a lot here that goes way beyond anything that you are likely to be able to teach yourself. Also the first course in probability theory is a very important basis for all of this, and for a lot of other things. This is also something that I have used on the job multiple times. I am concerned that learning this on your own might not provide as solid of a basis as would be useful to help you do well in these subjects going forward.
You have a long list of schools. I do not know how to trim this down. One thing you might want to think about is how you feel about travel and about cold winters. Both can get old after a while.
For a student in a math-heavy major, AP statistics is more of a way to get a very basic feel of statistics (its content is similar to college introductory statistics commonly taken for general education by humanities majors). It certainly is not a replacement for the first probability theory course for statistics majors.
Thanks for the detailed feedback! I didn’t know about the merit scholarship for UToronto. I’ll look into it. I’ll also take a look at UWisconsin.
Yeah, I could see my senioritis hitting for both of these… ODEs I’ll probably won’t end up taking; it was just a maybe in case I felt up to it. I put down Physics C: Mechanics just because 1) my school doesn’t offer it, and 2) Physics C is essentially Physics 1 with calculus, so it shouldn’t be too much to study for. But again, senioritis.
Yeah…I’m an olympiad kid, but I’d imagine there are a lot of those applying to schools like MIT. Knowing this, would it be worthwhile to EA to them, or would it be better to EA to a school slightly easier to get into? Any suggestions there?
Like what ucbalumnus said, AP Statistics is a far cry from college level probability and statistics classes and is definitely not a replacement for them. I just took the AP exam because I already knew enough stats that self-studying for it didn’t take long, and it gives me credit at some of the colleges I’m planning to apply to.
Good point. Winters are honestly fine - I went to HMMT in February and the cold didn’t bother me much. Same thing with travel - my parents are returning to China after I go off to college anyway I’m also thinking about doing some college tours over the summer, so I can get a feel of the campus. That should shave a few colleges off my list.
EA at MIT doesn’t prevent applying anywhere else… it would be the other school that might restrict it (like Stanford REA).
For Stanford specifically, I do not think applying REA is important, unless you are a legacy in which case it is said to matter whether you apply REA. (I am a Stanford alum so my kids discussed this legacy question with me, although they ended up not being interested enough to apply there.)
Others here can make more informed comments than I can on the other private schools on your list.
UCLA and Berkeley (not safeties, but still UCs lol) are definitely my favorites but also the hardest to get into. I’m happy with the others.
RPI checks off almost everything on my preference list, like you said, but from what I’ve read online, RPI focuses heavily on applied math, and pure math tends to get left behind. I’m more of a math purist (even though I would like to take some applied math courses), so I don’t know if that’s really my style 
Yeah, you and the other posters on here make a good point. I’ll probably drop Cornell
Your profile looks very strong!
I have a couple of thoughts.
you may want to look into the math major at UCSB’s college of creative studies (CCS). The CCS offers a graduate-school like experience with a LOT of perks for a handful of majors (including math) within the larger UCSB community. It requires a separate application in addition to the UC application, but might be of interest among the UCs.
Did you take Geometry in middle school or high school? If not, please look into the UCs requirement for geometry, which cannot be validated by higher level math. You might have to take an online UC Scout or similar geometry course. It’s ridiculous but happened to one of my kids.
Please research carefully if you will retain CA residency once your parents move out of California. The rules are not super clear on this.
Editing to ask @Gumbymom her opinion, as she is a UC expert.
Best of luck!
However, high school graduation in California after three years of high school attendance in California qualifies for the AB540 provision even if California residency for tuition is not otherwise retained.
Thank you for posting that!
I’ll be the contrarian on this one. I have family in Ithaca and have been there often. “Rural” is the last word I’d ever think of to describe it although the surrounding area is rural. When you walk around Ithaca, you feel like you are in a small city, which you are (population: 35,000 + 25,000 Cornell students). The city government of Ithaca is very progressive. One example is that they offer lots of government supports to gain access to housing and home ownership. My nephew, a musician, was one of the beneficiaries of this program in getting a mortgage to buy his house. The live music scene is varied and plentiful. The city is very LGBTQ friendly. If you’re comfortable with a place like Amherst, MA, I think you’d be more than comfortable here. It would be a shame to cross Cornell off the list on this basis because the Math opportunities here are superb.
Looking at small liberal arts colleges in non-rural areas which are strong in Math, I’d add Swartmore College ¶. As part of the Quaker Consortium, you’d have access to more courses at Penn in the same way that Amherst College has access to courses at UMass.
I’d also include Barnard College on this same basis. As part of Columbia University, you’d have access to courses at Columbia College.
If you’re going to apply to Harvey Mudd, I’d also consider Pomona, which has a superb Math Dept in its own right. The two have cross registration, so you can take courses at either one regardless of which one you’re enrolled at. I’d compare the two Math departments and see which approach you like better.
I do think sometimes it can be helpful to distinguish the immediate setting from the nature of the general area. And while Ithaca is not a large city, it is certainly larger than some college towns.
Obviously then what you really want to support your preferred four-year experience becomes a very personal question.
I have really nothing more to add about the UC’s, but requirements are requirements even if you do not agree.
You are an extremely accomplished student and the information that all the posters have given you will help narrow down your list.
My only advice about the UC’s is spend time on your PIQ’s since this is where you can set yourself apart from the many competitive and qualified applicants.
Best of luck.
Have you considered Carnegie Mellon? It fits the pattern of schools that interest you (extremely strong STEM in general and math in particular, urban campus, LGBTQ+ friendly). For some majors, especially CS and performing arts, it’s a single-digit-acceptance-rate school, but for math (and the Mellon College of Science in general), acceptance rates are in the mid teens. That might still be too competitive to classify as a match, but it’s not a high reach like MIT/Caltech/Stanford/Harvard/etc. But reputationally, for a math major, I think it compares favorably with those schools.
Seconding Wesleyan as an LAC with robust grad-level math offerings. Acceptance rate is in the mid-teens here too.
Also seconding UCSB CCS Mathematics | UCSB College of Creative Studies
In general, you’re going to have very good options at CA public universities, so IMHO it’s fine if your OOS list is reach-heavy. Good luck!