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First off, I was wondering what level of mathematics and physics is sufficient to do an REU summer program. I've emailed a few programs, and they have constantly stated that there are no specific requirements and that freshman are generally not taken. So, does anyone have any idea about a general set of classes that one should take before applying?

Currently, I am taking Calculus III (MV Cal at other colleges) and Modern Physics. Next semester I'll take Math Methods in Physics ( or Intro. to Differential Equations) and Quantum Mechanics I. Btw, does anyone think I should definitely take Diff before Math Methods? Also, I will hopefully start doing some Physics research next semester.

Second, I want to improve my Math and Physics problem-solving. Does anyone have any ideas on how to improve problem-solving skills? Are there any seminal books that I should buy? (I have Problems in General Physics by that Russian guy, btw)

Third, how important is your undergraduate institution when applying to top-rated Physics programs like MIT, Caltech, Princeton, etc.?