<p>Gallo_Pallatino – good question. I actually have a slightly unusual perspective on this question, since I had the good fortune of taking math courses at Princeton while I was a high school senior through Princeton’s generosity before becoming a math major at Caltech.</p>
<p>In any case, I’ve found the Caltech curriculum to be somewhat more organized and complete. There is a definite triad of three courses, each a year long — in algebra, analysis, and topometry (topology and geometry) that everyone takes. Since they’re year-long courses, there is a certain thematic integrity to the material. On the plus side, it’s easier to mix and match at Princeton and Harvard (with semesters), but it’s also easier to leave gaps in your knowledge.</p>
<p>Math 55 at Harvard is certainly the hardest and most breakneck-paced course of the introductions offered anywhere. I haven’t taken the course, but I have read portions of the notes. I have mixed feelings about this. On the one hand, being exposed to truly advanced concepts right away is a lot of fun and a good challenge, but can also be confusing and demoralizing. Personally, I prefer the extreme thoroughness of Math 5 and Math 108 to the beautiful quilt of everything that is Math 55. After two years of algebra and two years of analysis, any talented math student will know enough to teach Math 55, so the real question is what pace do you prefer – do you like a whirlwind tour followed by more depth (55 approach) or a more meticulous step by step approach? This is a matter of taste.</p>
<p>Needless to say, the math majors at Harvard, Caltech, MIT, and Princeton are of very high quality (excluding yours truly). Harvard tends to attract a disproportionate number of Olympiad winners (though Caltech also has plenty of IMO gold medalists). I haven’t found the level of students at Caltech to be lower than that of the students at Princeton – in some places the Techers seem to work a little harder, and the Princetonites were a little more fun to talk to :-)</p>
<p>That’s my brief stab at a comparison. Let me know if you have more questions.</p>