I checked the recent ranking of UT A&M and Stony Brook by QS World University Rankings and saw that the schools were ranked 195 and 382 respectively. So my questions are:
(1) is it advisable to pursue a phd in math in schools in this category (i.e. schools ranked below 100) if one's dream is to walk up the ladder of prestigious schools after one's phd (i.e. to be a faculty in schools like MIT, Stanford etc)?
(2) Can we say these schools belong to third tier universities or are they in a class of their own?
(3) How will I compare with guys from Stanford (in terms of access to quality research adviser, overall exposure to quality phd program etc ) by the time I am through with my phd program?
(4) If I want to work with an influential faculty (i.e one of the leading experts in math) in my field of interest, how do I get to know one since some of them are yet to win the Fields Medal (or any prestigious award known to me) and Wikipedia does not have much on them.(Even if it does, do I need to go through the profiles of all the faculties in about 10 schools that interest me? That will mean reading the profiles of about 400 people!)