OP (also) talked about math major and ranking.
UMich’s math ranking is probably graduate student ranking. The question is: would an undergraduate be able to take advantage of those math wizard professor?
Unlikely, for an undergrad in any large school (but not impossible). Chances are the math you will go through will (until Junior year) be taught by grad students. Not sure about Rice (the ability to talk to the math wizard professors).
What are the chances a college student wanting to do math (X) at a university will end up doing math (X)? Not in the 90% range. Most UGs change their major, or at least reconsider their major.