Can members of the Princeton's Institute for Advanced Study supervise PhD students? There is a faculty member of Princeton's Institute for Advanced Study (a mathematician), whose research is particularly enticing to me. Do faculty members of the IAS take PhD students? If so, should I just contact him directly?
Are you a Princeton graduate student already? If yes, ask someone local about both the rules and whether that person takes students regularly.

If you're at another nearby school (Penn, Rutgers, Columbia, etc.), then usually you also need a local official advisor in addition to an outside advisor.  Again, seek advice from someone local who works in a similar area.

If you're not already enrolled in grad school then don't worry about this yet.  It is very unusual for people to pick an advisor before going to grad school in mathematics.  I certainly wouldn't bother someone at IAS if you haven't gotten into Princeton yet since most people don't get in anyway (eg me).