<p>It depends on the field. In analysis and logic, there are very active seminars and an enthusiastic group of people who collaborate a lot (also joint seminars with UCLA). The geometers also seem to do a lot of collaborative work. Algebraists and algebraic geometers are more solitary. There are excellent graduate students, but the graduate program at Harvard and Princeton is stronger in math than at Caltech. </p>
<p>In any case, I don’t think the level of collaboration and seminar attendance by the faculty should influence your decision of where to go as an undrgraduate… realistically, even though Princeton has a bigger faculty and more different kinds of math going on, the probability of being able to do some math (through an undergraduate research project) is much higher at Caltech… on an odd and slightly related note, Harvard’s permanent faculty in math is only 3 or so professors biggger than Caltech’s.</p>