Since you are currently a sophomore, you have some time to thing about this.
There are quite a few universities in the US that are very good for mathematics, and of course quite a few more outside the US. You should be thinking about what you want in a university. This can take some time to figure out.
One thing to look at is the differences between the various top schools. You should try to find a school that is a good fit for you.
While all of these top ranked universities and colleges will be academically very demanding, MIT and Caltech have reputations of being a bit more demanding compared to other top schools.
However, I still remember very well spending 6 hours on a Saturday afternoon at Stanford solving one single problem on one homework assignment (out of 5 problems on that particular assignment, in one class, while taking 5 classes at once). Spending time on weekends doing homework is quite normal in the top schools. Suppose that you were to similarly spend six hours on a Saturday on homework, and managed to solve a very difficult problem that most students in the class were not going to be able to solve. How would that make you feel? Would you be thrilled that you could solve a tough problem? Would you feel that you just wasted your Saturday afternoon? The desire to work this hard and to solve the tough problems that confound your classmates needs to come from inside yourself. You need to want to do it for the top ranked universities to be a good fit for you.
The last time that I looked the top ranked universities in the world for math were (in random order) MIT, Harvard, Princeton, Stanford, Cambridge, and Oxford. However, many, many other schools are also excellent. There are very few secrets in mathematics. The top students at 100 other universities are going to be indistinguishable from the top students at these top ranked universities.
One issue is how large of a school do you want? Caltech and Williams College are relatively small. UC Berkeley and UCLA are large. Stanford is moderately large (with a large and attractive campus).
Do you want to be in a city, or suburban area, or small town? Harvard is right in the middle of Harvard Square, with restaurants and clubs right there. Apparently Passim’s coffee house still has live open mic shows very close nearby. MIT is on the river just a couple of miles from Harvard, across the bridge from Boston. Stanford is in a more suburban area. Williams College is in a small town in western Massachusetts.
The drinking age varies in different countries. In the US drinking alcohol is illegal if you are under 21. Students drink anyway, sometimes to excess, but in the dorm. Often older students will purchase the alcoholic drinks. In Canada or the UK you can drink at either 18 (UK and Montreal) or 19 (Toronto), which creates a market for bars with entertainment (mostly music) near universities. This brings the students who want to drink out of the dorms into the bars. This makes for somewhat more social drinking (students usually do not want to embarrass themselves in public).
The weather will vary from school to school. Wisconsin will have cold winters, as would McGill or Toronto. MIT and Harvard have winters which are milder, but have occasional storms called “Nor’easters”. This is what happens when a low pressure system has its center over the gulf stream east of Boston – the storm picks up moisture from the gulf stream and dumps it on the students at MIT and Harvard (and on other people who are in the area). Caltech and Stanford will in contrast have mild weather and very mild winters – you might want a light jacket. Every few years it might rain (it did not happen to rain at all while I was a student at Stanford).
Sports will vary. Stanford has a wide assortment of sporting teams, and gives sports scholarships. Harvard and Princeton are in the Ivy League. The Ivy League does not allow sports scholarships, so their teams are not as strong as the teams at Stanford or UCLA or Wisconsin. However, the Ivy League is surprisingly good at both football and hockey considering that they do not have any sports scholarships. In contrast both MIT and Caltech have relatively pathetic sports teams.
If you do not qualify for need based financial aid, most of these schools will cost well over $300,000 for four years. McGill or Toronto would be a bit less, but are still not cheap.
Yes, very much so. The top students from U.Mass Amherst or Rutgers or Wisconsin or at least 50 other schools are going to be indistinguishable from the top half of the students at MIT or Harvard or Stanford.
Perhaps the one difference that I noticed: At the very top schools professors might be more inclined to put really, really hard problems in either the homework and/or exams. At MIT I took at class in introduction to probability. One of the homework assignments was an unsolved research problem in probability theory. There was no one in the entire world who knew how to solve the problem. I tried and gave up after perhaps 20 minutes. Every few years someone would solve one of these problems and end up with a joint publication with the professor. At Stanford I remember at least two exams where there were problems that at least 90% of the graduate students at Stanford had no hope in solving. The few students who could solve one of these had a chance to get a + sign after their A in the class (if they otherwise qualified for the A). I remember another exam at Stanford where the class average was in the mid 40’s, and a 75 was the second highest grade (the student with an 85 ended up with an A+ in this particular class). I liked this, but not everyone does.
There is a lot to think about. You have plenty of time.
You might want to read the “Applying sideways” blog on the MIT admissions web site. As I understand it the point is that you should do what is right for you, and do it very well. If you belong at MIT (or other top ranked school) then what is right for you will very likely be the right thing to help you get admitted at MIT or a similarly highly ranked university.