What if your advisor does not know anything about your research field?

Well, then go and find another advisor. Do not get me wrong, I don't mean a new one, I just mean someone else to talk to. With a PhD, you are supposed to show that you are able to do independent research, that you are able to do more than just work on minor problems that dropped out of some paper/publication by your advisor (as it is often the case in Bachelor or Master thesises).

I think a PhD (at least in Mathematics, I don't know that much about other fields) where you only worked with a single advisor for multiple years is really uncommon. Go find other people, other professors, etc. to talk to. Your advisor should help you with the following points in case you get stuck there (list might not be complete...):

• How and where to find research material, maybe giving you access to institute libraries that are normally closed to regular students or access to the universities subscriptions to online journals.
• Establishing contacts to other researchers on the field, giving you advice on who to talk to, maybe even recommending you for a research stay of a few months at another university for a project related to your research.
• Administrative processes. That also includes questions on "How and where to best publish this result I came up with?" or "Which conferences should I visit, should I maybe even present something there?"

During a PhD, it is not the advisors job to do your research. You yourself have to decide what you look into, what you do, how much time you spend on one problem until you deem it too hard and move on, etc. If your professor could answer all your questions, then that would mean that your research is not really original and/or rather trivial and thus might not yield a PhD (or at least not an excellent one, and I'm sure you are aiming for that :) ).

So don't worry if the professor can't help you solve your problems. Maybe they will turn up to stay unsolved for centuries, who knows? Try to rephrase your questions, don't go and ask "How can I solve this?" but rather "Do you know of experts on this topic that might be willing to talk to me (maybe even at the same university)? Can you put us into contact?" or maybe asked more generally like "Do you have a suggestion on which direction to look from here on? Should I continue down this path or maybe try a different approach?".

So long story short: Even if the professor knew your research field, he would not be able to always assist you with your problems.

PS: As I am also writing a PhD in math just now I'm a little curious: What is your research field, for which there are no research groups and your professor has no idea about? :)

In mathematics at least, the number one job of the advisor is to actually assist you in formulating a problem (or series of problems, even better), which if resolved would result in a reasonable/acceptable PhD thesis.

If the advisor has no idea about your field, I don't see how he can help you do that. It may happen that you are lucky/capable enough to choose and solve good problems on your own, but a very small percentage of grad students are in that category. So you may end up biting much bigger than what you can chew, or aiming too low and end up with small problems which make no impact on the field.

For instance, how will your advisor judge if/when you are ready to graduate ? Is he simply planning to count publications/preprints that make up your dissertation and put a stamp on it ?

At the very least, I suggest getting a co-advisor who knows your area. Doesn't have to be at your university, professors from other universities can be approached.

I had to work on projects my supervisors were clueless about. They had the money to support my work, but not the time to go into any details, and help me, even with a reference.

What I ended up doing, was ask friends if they knew anyone in the field who was using the methods I needed to learn and I got introduced to a few people, some of whom I still collaborate with. I visited my new collaborators and worked on my projects while I was there. Their postdocs or graduate students helped me with details and so on. In Europe, there are some grants which support us going to collaborators to work there for longer periods of time.

In your case, it is possible that your field is too specialized, which might be a reason you say there are no groups working in that direction right now. But, isn't your field a subfield of a larger field? Maybe you can find groups working in that field and using methods you need to use in your work.

You can ask your adviser, or other faculty you know to help you connect with any of those groups who might help you and see if they are interested in your problems, and/or they can help you. If not, you can try to connect yourself, via email (some answer, some don't), or at conferences.

You don't need to leave your adviser if he's not a specialist in your field of choice. He allows you a lot of freedom which is both desirable and dangerous for your progress, as you are finding out. On the other hand, if he doesn't support your going to conferences, doesn't pay your stipend and doesn't help your research in any way, you should consider this. A more experienced adviser might help you more, but might also force you to solve other problems they find interesting.