I'm a professor in pure mathematics (at a university with a Ph.D. program), in the United States. I'm a little bit biased, since I went through the system, but

I'm a professor in pure mathematics (at a university with a Ph.D. program), in the United States. I'm a little bit biased, since I went through the system, but that said -- > Would that basically be the equivalent of what a PhD is supposed to signal? Signal to whom? Probably not, in my estimation, but maybe -- it depends on what you're attempting to accomplish. Enrolling in a formal, traditional Ph.D. program has lots of benefits; getting to work under a mentor is only one of them. Perhaps the most significant is that you become part of a network of peers: you surround yourself with people whom you can bounce ideas off of. Also, there will generally be lots of visitors passing through, with lectures and opportunities to go out to dinner with the speakers afterwards -- also invaluable learning perspectives. Unfortunately interdiscplinarity is a difficult thing: most faculty only know their own discipline. (University administrators love interdisciplinary research, but impressing them is irrelevant before you have a tenure-track position.) > and publish a few noteworthy papers Noteworthy to whom? It sounds like you want to pioneer a new research area that nobody works on. Would you be satisfied to do work that, in some abstract sense, is worthy of taking note of? Or is it important to you that other people actually notice? If the latter, then you almost certainly want the backing of an advisor and of an institution. In general, professors tend to appreciate students who take the initiative and seek out their own research projects. Depending on your long-term goals, I would recommend seeking out professors who work in one of the topics you mentioned, and depending on their response, consider enrolling in their Ph.D. programs. Best of luck!