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So I was trying to sketch some plans about what to do in college. One option would be Engineering here in Brazil, in my city we have a great Engineering school, which is known for giving its students a ton of work opportunities - it wouldn't be too hard for me to get a fairly stable/wealthy life. But, after some soul-searching I realized I really like Mathematics and Physics and maybe I could succeed on one of that fields. Then I self studied some Calculus and Linear Algebra (no fancy AP courses here) and enjoyed it (MIT Open Course Ware was a huge help). Now, the thing is that, while I could study (almost 100% of getting accepted) one of these subjects here in what is known as the best university in Latin America (São Paulo University), I am seduced by the idea of studying in a top international university. Furthermore, after some searching, I found that my dilemma between Physics and Mathematics could be solved by doing some kind of hybrid course (double major), which seems possible in some universities. Is that reasonable?

However, we all know that getting admitted in MIT/Harvard/etc is pretty hard. While here in Brazil very few students learn Calculus, Linear Algebra and other college-level stuff in high school, this seems to be rather common in the US. And my application is probably not that much competitive - I haven't done any community service (and don't intend to - I don't have the time and I would be doing it merely for interest, which doesn't sound honest at all), I am not an amazingly creative and awesome person like everyone in MIT seems to be etc... And the big problem is that those colleges are very expensive (the São Paulo University here is a free public school)! If I do Physics or Math I won't be earning much (how poor are graduate students?), so I really cannot afford paying that much (even with financial aid the fees are still pretty darn expensive...).

Despite all these problems, I'll apply anyway. So which universities have the top Math or top Physics programs? Even better, which of them allows you to major in Physics, while having a strong mathematical background (allowing you to specialize in Mathematical/Theoretical Physics?).

I have done some research in the internet and MIT seems to fit what I search for - It seems possible to double major in Course 8/18 (Physics/Math). Then, I would take the Physics requirements alongside with mathematics like Abstract Algebra, a nice proof-based Analysis, an introductory Topology course and maybe Complex Analysis, applied math courses etc. I really don't know if that would be reasonable - so what are the mathematics needed for Mathematical/Theoretical Physics? Should I focus more on Pure or Applied math?

The Physics major in Caltech seems rather interesting since there are various undergraduate courses on math/theoretical physics and maybe I could fit in Abstract Algebra and Real Analysis (the double major doesn't seem possible).

I have heard well of Stanford, Harvard, Princeton etc.

Anyway, I am really sorry for having made you read this rant, but I could really use some advice on which universities have nice Maths/Physics programs around the world. If they happen to be less expensive, even better. And I would really like to know if you think that doing a double major / an mathematical physics program is a good idea or not (maybe it would be better if I focused in one of the fields since like it or not, they are related to each other so I would have to study physics in a math major and vice versa).

Thank you!