Warning: Extremely long post. Not really even the full story but one must stop somewhere.
My problem is the following:
I am a pure mathematics PhD student in (a remote part of) Europe and am part of a Doctoral Programme, which means that financially I am in a good position (relative to other grad students anyway). This is the same university in which I completed my undergraduate education. I was chosen together with a handful of other applicants from a pool of over hundred candidates (all natural scientists, not just mathematicians). I'm on my first year.
During my stay I have struggled to find a suitable problem (In fact I have overheard graduate students leave work early in my university because the feel that they didn't have much to work on). Basically my supervisor suggested some problems, I didn't like them, I tried desperately to invent my own problems to fill the void and to find something in the intersection of my interests and my supervisor's interests (and expertise). I realize now that with many of those problems neither I or my supervisor could really judge whether the problem was good or not (or whether it was known or not). During the summer the possibility that this arrangement is nonsensical started to dawn on me and I started to wonder if I should just consider moving to another university in another country to do a PhD.
My interests have sharpened and shifted somewhat during my stay as I got a better understanding of the current state of the field. On a couple of occasions I tried to tell him about my interests. Quite recently my supervisor told me that I need to start working on something and offered some problems that I didn't care about. Inspired by some advice of Sir Michael Atiyah ("Advice to a young mathematician") and AMS blog post "Tips for New Grad Students" I brought up the possibility of changing supervisor/university/country. He was certainly not relieved, and was concerned about the reputation of his research group, among other things.
I have discussed my concerns with doctoral programme coordinator and the person in charge of the PhD studies in our faculty. I am considered extremely fortunate to be part of the doctoral programme and losing a student would be unfortunate for the faculty. Changing supervisors or universities is something that doesn't generally happen here.
To an outsider the differences between my interests and my supervisor's would probably appear like splitting hairs. Indeed this is the position of my supervisor. The topics are likely to be outside of his expertise. He has offered to study together some of the subjects that I am interested in and to later study abroad with a more knowledgeable professor (the supervisor does have connections). He claims he will eventually find suitable problems if we study the topic together. Basically I envy students who get to study in a big active university or get to work with a guru who dispenses expertise and problems matching students' interests. I would be glad if I was offered even a low hanging fruit problem if it at least would teach me something in a modern and interesting field. It seems backwards to me that to learn a field together with the supervisor, instead of learning the stuff from the supervisor.
Our relationship has suffered as a result of this debacle. I think he is frustrated about my doubts.
So should I: Go seek the frontier where the actions seems to be (I believe the things that interest me are being actively studied):
Pro: Potentially more efficient and stimulating PhD experience.
Pro: Potentially more prestigious university and/or supervisor.
Con: Changing universities is risky, the fact that I got funding in one university doesn't mean I will get it elsewhere, etc.
Stay where I am and try to make it work:
Pro: Safe in the short term.
Con: Uncertain if I will get suitable problems and sufficient knowledge to tackle such problems and continue a career in research.
Con: I have learned that people doing their undergraduate and graduate studies in the same university do less well.
Con: The track record of the supervisor to get PhDs to good places is not great.