How to deal with an advisor that offers you nearly no advising at all? 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.
 A: I've gotten here late, but I'd like to emphasize one point that I don't think has been made: I don't know what it is like in Germany, but from a US perspective, you have already done enough for a Ph.D. Many of us graduate with no papers yet published, their dissertations containing their only published results. You are already doing substantially better than many recent American Ph.D. graduates. The only requirement for a Ph.D. is having done independent research (beyond the level of a master's degree, whatever that means), and you have already done that. I know of people whose dissertations were a few published papers tied together with some interstitial wording—this is obviously not ideal, but I state it to point out it is an option: it has been done.
The fact your advisor has been too uninvolved to propose a thesis problem actually grants you a great deal of freedom, because you don't need to pursue some impossible white whale of a problem to please him. He can't fault you for choosing and answering your own problem. All you need to do to graduate is decide one of these independent projects, ideally a harder one, is the one you want to be your dissertaion, write some additional background chapters you wouldn't put in a paper, and submit that, and it sounds like you have plenty of time to do that. So by no means quit.
Unfortunately, unlike the Ph.D. itself, there are no standards required to be an advisor, and many of us get rather little advice. There's not much you can do to fix this problem, which is on his end, but it sounds like you're already doing the right thing, pursuing collaborations with people whose research is of interest to you and your own projects.
As far as career-related advice goes, many of us have had advisors who aren't terribly involved or aren't tremendously worldly, and what one usually does is to ask advice of postdocs you've made friends with, other established people in the department you trust, your peers, to compare, some of your older coauthors who you feel you have some rapport with, etc. It's unfortunate your advisor is unwilling to advise, but you have coauthors who want to see you stay in mathematics, as well as surely friends who have your best interest at heart, and it is always OK to ask. (And if all else fails, there's the Academia StackExchange.)
A: Whatever you do, don't quit because of your supervisor.  If you want to quit, quit, but not because of your supervisor, maths is too good for that!
For what it's worth, I also struggled with a supervisor offering no help (actually no help would have been better, the little that my supervisor did say was always counterproductive and often simply wrong). I don't know if it's a German thing or if it's institute dependent (I was in Göttingen), maybe a bit of both. I'll try give a longer reply later with more concrete things that helped me and so hopefully can help you, but for now just take my first two sentences! :)
A: The following suggestion may be naive, and I would say it definitely depends a lot on the personalities involved, so you should consider how things might play out before taking the following advice. But especially at an American university, I would advise a student in your position to discuss your situation with someone with a bit more authority at your university that you think may be sympathetic. I would probably begin by thinking about any teachers you've had at that university that you felt you got along with and who you might trust to handle a situation diplomatically. Alternatives might be your department chair or graduate program director (at least those would be the right people in the US - I apologize for not knowing if the German universities have direct equivalents). The goal is to find someone who knows your advisor well enough to offer some good advice and assistance but in a tactful way. Again, depending on the individuals, I could imagine one outcome where this professor is willing to talk with your advisor and say "look, I'm concerned your student might be struggling a bit and might need some more attention," though hopefully without pinning this too much on you. Alternatively, you might find someone who honestly says "yes, this person is not a great advisor - I suggest you switch to someone else." Such a person might even have a good recommendation for someone else to switch to who might be good for continuing on your current work at your current location. If your advisor is simply inadvertently neglectful, these approaches might work out. On the other hand, if you're concerned that your advisor might be vindictive about you talking about this behind their back, then of course you should tread more carefully. On the other hand, if you're considering giving up your Ph.D. altogether as an alternative, then trying to speak to someone at your university is probably at least worth a shot.
A: 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

And I also feel that this whole situation damages me mentally to a point where I frequently get anxiety attacks.

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.)
A: In the German tradition a PhD advisor is called a "Doktorvater/Doktormutter" --- a thesis parent. It might help to think about your relationship with your advisor along those lines. Many parents like to leave their kids on their own to explore the world, and see their role as "cheer leader" --- your biggest fan. From what you describe, "He gave this thesis the best grade possible", I do get the impression that your advisor admires you and is supportive, while letting you explore the world of mathematics on your own. And you definitely seem to be making good progress in that exploration, initiating collaborations, co-authoring papers. These are all signs of a great scientific maturity, you can be proud of how far you have come. Whether you continue in academia or not, these accomplishments will stand out on your CV.
If your advisor would "force" you to work on problems that do not interest you, I would definitely recommend you to seek another advisor. But you seem to have an advisor who lets you pursue your own research interests and the scientific environment in which you find yourself does seem to be intellectually stimulating. Perhaps it's true that you are climbing a mountain via the steepest route, but that may well be the most rewarding route.
