<p>Princeton has one of the most incredible research groups working in math today- many of the best young researchers in the world are grad students, assistant professors, post-docs and alums. </p>
<p>As an undergraduate, you will learn from top people in virtually every field- analysis (Fefferman, Stein, Klainerman, Sarnak), geometry and topology (Gabai, Kollar, Katz, Pandharipande (?)), number theory (Bhargava, Wiles)… I’m sure I’m leaving a number of important profs out… my point is, you can learn standard undergrad math from the classic textbooks and work out proofs and problems on your own. But having some of the most talented peers around (at places like Princeton and MIT) should spark an interest and motivation to tackle some very difficult subject material as an undergraduate. </p>
<p>Also, you need to know a lot of math to start writing and one of the advantages with the junior paper and senior thesis is that it forces you to think like a research mathematician quite early in your career.</p>
<p>I would say senior year for a Princeton math major is a lot closer to the first and second year of a PhD program than it is for other majors and for math majors at most other schools.</p>