I’m an incoming sophomore currently trying to decide where to transfer: my options are Cornell, UPenn, Brown, Carnegie Mellon, and Johns Hopkins. Apart from other things I’ve heard (or read) about these schools–and I’ve visited the first three–I’m wondering how the math departments at these schools compare. Does anyone happen to have any relevant knowledge or advice about the rigor of the pure math curriculum at any of these schools? (I’ve been taking upper-division courses, so I’m definitely interested in the depth of a program as well as the quality of graduate courses.)