I am a young PhD student (24) at a Germany university and I am not sure whether this is the right place to ask this kind of question. If not feel free to move it elsewhere or delete it completely.
Currently, I a have a half time position in Analysis and my doctoral advisor more and more turns out to be not very involved in my PhD. I started 1 1/2 years ago at the age of 22 and my PhD advisor was at least somehow involved as I wrote my Master thesis but gave me much freedom. He gave this thesis the best grade possible and I felt I also deserved it to an extent. The topic was one that I had chosen myself and I learned much new things writing it; combining different areas that I did not know before. This time was quite stressful for me personally as I tend to pressure myself too hard if I have to perform like this. After that I was quite exhausted but wanted to pursue a PhD at my university in the field I wrote my Master thesis in because I felt like the right thing to do; I really like the people and also the topic I wrote my thesis on.
However, things changed rapidly as I became officially a PhD student. First thing my advisor told me was that he had no time to spend on me as his oldest PhD student had to finish after over four years. I did not wonder and at that time I had the luck to write a paper with a guy, which still is like some mathematical godfather to me (and much more capable than me), who I met on a conference. The paper that we wrote had quite nice new ideas in it, however I felt my part in creating it was minor. But I also did some good work I think.
At the start of the year I investigated some other question on my own and managed to produce a positive result with the tools I learned from writing the first paper. I also extended the question quite a bit to write a paper on my own about the topic; without any advising at all. The only thing I currently do, is to speak with some colleagues of mine over very specific questions. I sent this "paper" to my advisor and the only thing he told me is that he would be too busy to read it in near future.
Currently, I have another collaboration going with the guy I mentioned beforehand and several others who advise me more than my own advisor, although they work on completely different universities. So currently I am quite lucky to have some advision and a perspective in research.
Finally, my PhD advisor didn't give me a question to work one. He just mentioned very vaguely that one maybe could extend some of the concepts used in my Master thesis but he couldn't tell me any possible applications for these abstractions. So I did not feel like this would be promising to work on. He also does not meet up with me on a weakly basis to discuss. Furthermore, my advisor also holds a record on suggesting topics to this PhD students that are completely inept to work on at this stage of their mathematical career. My older "PhD brothers", for example, spend to years working on a big conjecture in one specific field without making any progress, whatsoever. My PhD advisor had also no new idea how to approach that problem; he basically just told them to try it without giving much help.
So I frequently ask myself the following question: "Do I feel it is worth to pursue a PhD under this circumstances?"
I saw how other advisors work with their PhD students and I feel their advisors have a clear initial idea on the "what" and on the "how". Moreover, they meet up and discuss the current problems that arise while pursuing the question. All this I do not have at the moment. I really enjoy teaching courses but I do not have the impression that I move forward in research to much and that really pulls me down. And I also feel that this whole situation damages me mentally to a point where I frequently get anxiety attacks. On the other hand, I know that I could earn good money in the economy with my qualities and my intellectual capacity.
So my question is: Would you advise me to quit my PhD and to try it at another university in my field? Or should I stay and fight? Or should I just skip the PhD and do something that earns my money and gives me more structure?
I know that the "right" answer to this question is not determined in any way and that it might fit in the category "vague question" that we usually try to avoid on this platform. But I do not know where to ask it elsewhere and I would like to get answers from people with more academic experience than I have. I really cannot really pigeonhole my whole situation and do not know what to do at the moment.
Best Answer
I second Nate's suggestion to look at https://academia.stackexchange.com, there are already many similar questions (with answers, some of them specifically from mathematicians) on that site that may help you.
But since "go somewhere else" is not exactly the kind of answer someone in your situation needs, here are a few thoughts from a random person on the internet:
First, it sounds to me as if your PhD is actually going rather well: You've already obtained independent results, written papers, and initiated fruitful collaborations on your own. If that is not what you should demonstrate for this degree, I don't know what is. (Of course it could -- always! -- be going better, and the experience could be more pleasant for you.) So I wouldn't worry about your chances of graduating.
In fact, one possible (but certainly not the only) explanation is that your advisor is thinking the same thing: "They're doing well on their own, they don't need my help, and it's better for their career if they're working independently anyway." (Of course, this can also be a convenient rationalization of laziness or poor time management on their end...) If this is the case, I'd sit down with them, explain to them that you in fact do need their help, and negotiate exactly what kind and on what schedule. If that doesn't work (and you haven't graduated by then), switching advisors or getting a formal co-advisor (either in the same department or a different university) is certainly not unheard of.
Finally (and this is the reason I am writing this answer now), you write
It's completely normal to have doubts and frustrations during your PhD (and the timing seems about on schedule for it, as well), but this is a strong emotional response that you should take seriously and seek help dealing with. Here I don't necessarily mean professional help (although there are certainly professionals that can help with this), but finding a trusted person you can talk to about these issues on a regular basis to prevent them from building up. (Here especially, https://academia.stackexchange.com can give you much better recommendations since this is something that happens in all disciplines.)