When to look for a new PhD position? TL; DR: Should I re-apply for another PhD program? What could I do instead?

Since April, I am a PhD student in mathematics after having finished my master's studies last year. Although I spent most of my attention on doing abstract algebra doing my masters, I felt like shifting towards applied topics and found myself a PhD position in topological data analysis.

However, after having started to work, my enthusiasm has dimmed significantly in the last few months. It seems to me that I am dissatisfied with the following particular aspects:


The working group is extremely small. We do not have a research seminar or any lectures that are relevant or of interest for me. There is little exchange between the few people in our group, since everybody does something unrelated to the others' work.
I am somehow disappointed by subject: I have the impression that all the papers out there gather some more or less obvious statements, or sell things as new that are known to other fields since long time. I am uncertain whether I should expect any significant theory to be developed in the future.
My supervisor does not seem to have a particular idea where he wants to lead the research of his group. When I talk to him on what to work on, he usually comes up with some half-baked ideas that barely make some occupation for a day or so. When discussing a possible problem to work on, he quickly makes the conversation get lost in unimportant details, such that eventually I have no plan what to occupy myself with.
At the moment, I am working on some specific problem neither I nor my supervisor have the expertise for. To me, our approach seems rather non-mathematical. It is not what we have agreed upon when I applied what I would be working on; however, it seems that he has no ideas where to employ my knowledge.


These points make me wonder whether I should quit and apply for a new PhD position, or try to somehow cope with what it is like now. In the last months, I have worked on an article on some ideas from my master's thesis. With the temporal distance, these topics do not seem so frightening anymore, and by now, I can manage to identify with what I worked on at the time; hence, I would try to return to abstract algebra if I quit.

Questions: However, I now that I am an uncertain and insecure person, and in particular the third point makes me wonder if I am  qualified for pursuing a PhD: 


Can I expect my supervisor to provide me with problems to start working on, especially if I am new to the field? 
Should I consider myself unqualified for a PhD if I am not able to identify relevant questions and work on these completely on my own?
Should I expect things to improve when I relocate?
Is newly applying for other PhD positions the right strategy to deal with the above problems?
While this problem is very personal I can offer a bit of guidance rather than an answer to your problem. You have a number of things coming together to bring you to this state. However, it isn't a unique state and others have resolved it in various ways. 

I suspect that your advisor is a bit inexperienced at this and is genuinely unable to give you the guidance you need. That isn't uncommon. Some doctoral students in math come up with their own problems and some advisors provide problems to students. I don't know which is more common, actually. I was once refused by a potential advisor who told me he didn't think he had the experience to guide me properly. But he got me connected to a (scary scary) senior professor and it worked out fine. That professor was full of problems he was more than happy to share. 

Switching sub-fields isn't that uncommon either. Many students come to doctoral studies relatively "unformed", even with a master's degree behind them. But, working in a field that you like is a big help. For me it was an issue of insight. I had deep insight into classical real analysis and some aspects of topology. But actual insight into algebra was always hard for me. In analysis I knew what I could expect to be true before starting a study. I never had that sense in algebra. 

No, you aren't a failure. Whether you stay or go, do it for a reason related to what you really want to do. 

You may find a new situation to be similar or not, but you have some control over that. A more experienced advisor in a field you love, could be explored before you make any decisions. You can search for an alternative without making a prior commitment to actually change.