<p>Hello, I’m a current math major who can answer some of those -</p>
<p>Do undergraduates receive a lot of attention from faculty?</p>
<p>Very much so. This might be the single greatest thing about the department in my opinion. You have to initiate it though, but they’re more than willing to speak/work with you.</p>
<p>Are faculty approachable? Or students are mostly left to their own devices?</p>
<p>Again, definitely. They always have weekly office hours for any given class and I’ve had plenty of success just emailing someone asking to talk about something. </p>
<p>Do faculty involve undergrads in their research?</p>
<p>I don’t know about in “their” research. I know that I wouldn’t understand most of it at this point 
You could talk to them about it though, of course. I did have a professor who was constantly citing papers he’d written for “further reading”. </p>
<p>How common it is for undergrads to do research during the academic year and during summer?</p>
<p>Very common, especially during Junior and Senior years. After all, you have to write a senior thesis, and Junior Papers are also an option each semester (that or a seminar). What you write is largely up to you, so if you want to do research then you will. </p>
<p>Is it easy to get funding for summer research?</p>
<p>I know they give ~20 grants for students to do research with a professor during the summer on campus. They also have an application for funding to do research outside of Princeton, but I’m not sure how often that is used. </p>
<p>Do any undergrads travel abroad to study math or do research?</p>
<p>I don’t think this is particularly common, but it happens.</p>
<p>Is there work study opportunities for undergrads at the department (e.g., grading problem sets, tutoring).</p>
<p>Yep. Some students grade for classes they’ve completed, and there’s a few tutoring opportunities.</p>
<p>In general, what is the culture in the department? Is there collaboration among students? </p>
<p>Collaboration is definitely the name of the game. Just so long as everyone writes their own answers, they encourage collaboration, and without it problem sets would really be quite unwieldy. </p>
<p>I can’t help directly with Princeton vs. Stanford since I have no experience with Stanford.</p>