<p>I do not have a lot of time, but I definitely wanted to reply to this:</p>
<p>a) I highly doubt you get any more rigorous than Princeton in these subjects. I am a physics major that has taken many math classes, and it is amazing.</p>
<p>b) The faculty are always willing to talk. I have stopped by professor’s offices many times in the past and they are always willing to talk. The math department even has a tea every day where professors come and talk with the students.</p>
<p>c) I have never known of a person rejected to help a professor with his or her research. One summer, over 50 students were being paid for their research at Princeton, just in the Physics department.</p>