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I am a second year mathematics PhD student (Austria) in computational geometry, although I have obtained my Master's in algebra. However, I have the impression that I must have done my stuff severely wrong since I started my PhD.

I haven't published anything so far, or developed to an extend that deservers being published. I have certainly learned less than in any semester of my studies. I waste a lot of time because I either have no idea what to work, or, if I occasionally find something I could try, I have no ideas how to work on that problem. This is particularly surprising to me because I had been used to working very efficiently during my studies, not to wasting tome, to knowing what I do, which material I want to understand, whom to talk to if I struggle etc.

My supervisor (halftime of is tenure track, no gradudated PhD student until now) doesn't care very much. His style of "guidance" has been very hands-off from the very first day; according to him, doing a PhD is about becoming an independent researcher, so I should not expect anyone telling me what to work on, which attempts to try, or anyone explaining me particular mathematical topics. Probably, this is due to his background being in geometry. He has accepted me as his student because I am supposed to work on algebraic problems arising from his fields of interest. I wasn't aware that this means that I am supposed to cover knowledge he does not have.

Questions.

  • How do I succeed in a PhD without relying on anyone else to ask for explanations/problems/advice…?
  • How do I find promising problems, acquire the relevent knowledge and eventually solve them on my own?
  • How can I spot and learn relevant things in self-study as fast as I did when taking lectures during my studies?
  • How do I get rid of my imaginary need for exchange with experienced researchers fellow students?
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    Argh. In mathematics, in the U.S., this situation is a problem. I'll write some opinions either later today, or tomorrow... In brief: my sympathies; you are not really supposed to be so independent... More later. I'm sure others will have things to say. – paul garrett Jun 27 at 22:12
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    This kind of question comes up a lot here. Most of the time, urging students to be independent to this degree is just an excuse for being a lazy or unable advisor. – knzhou Jun 28 at 12:38
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    "according to him, doing a PhD is about becoming an independent researcher" - that's fair enough, but surely it's not about being an independent researcher already, which he apparently expects you to be! – Lewian Jun 28 at 14:08
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    I agree with @paulgarrett; I'm also a computational geometer. Yes, the goal of a PhD program is for you to become independent, but you have to learn how to be independent, and it's your advisor's job to teach you how to be independent. If your advisor isn't interested in... um... advising, find another advisor, or at least other collaborators. Fortunately, there are lots of Austrian computational geometers who like to collaborate. – JeffE Jun 28 at 17:35
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There are two sides to this; I believe that indeed you should have more supervision than you have, and that your supervisor doesn't give you remotely as much help as you deserve and should have a right to receive. On the other hand you may be right about not yet being independent enough and that there's some skill set that you need to learn.

There is a huge variety of supervision styles and supervisor behaviour and your supervisor is certainly far on the "hands off and rely on student's independence" side of things. There are supervisors who support their students much more, however your supervisor is by far not alone with his interpretation of his role. Unfortunately, for many PhD students this will not work (you are absolutely not alone with this problem).

"According to him, doing a PhD is about becoming an independent researcher" - that's fair enough, but surely it's not about being an independent researcher already, which he apparently expects you to be! A good responsible PhD supervisor should lead their students into such a position rather than assuming that the students get there on their own!

Not sure to what extent you already tried to get more help out of your supervisor. Surely "independence" also means that if your supervisor tells you "leave me alone and find your own problem" you shouldn't necessarily just say "Yes, supervisor!" Being "independent" as a researcher doesn't mean not to talk to anyone, not to admit where you are struggling, and not to ask anyone for help!

Maybe you can "milk" your supervisor better by telling him that and in what way you are struggling, and to ask him to talk more with you about the research he is doing and how a problem could be found for you that he is actually interested in and that may help his work somehow. Try to get out of him as much as you can, and if that means not accepting his initial idea about supervision, so be it. Also try to connect things that you already know to what he says. What do you like in mathematics, in algebra or elsewhere? What kind of problem do you like to work on (for a moment ignoring whether that would fulfill requirements for a PhD)? Can you somehow make a connection between this and what your supervisor is interested in? (You can also ask him whether he sees a connection.) There are by the way textbooks that have exercises that in fact are research problems in disguise, so looking at textbook exercises in the area might help.

Once you have a problem, obviously the first step is to look for literature and to find out how other people solved similar problems, and whatever you can find is connected.

In case you don't manage to get more out of your supervisor, I agree with another posting that you should contact somebody else in the department. Isn't there a person such as a PhD tutor to which PhD students can go with issues? Somebody more senior you trust? Student representatives? Ultimately the department should make sure you get enough supervision, and there should also be some point to go to for students who have problems.

The side of your own attitude is somewhat difficult to assess. Let's say I'm somewhat surprised that it only dawns on you 1+ year into your PhD programme that you have no idea what to work on. Would it be like this even in algebra or the area of math you are most familiar with/like most? Have you done a Masters project or something like that? How was it? Have you ever worked on a problem, even a set exercise or something, that made you wonder about whether something more can be done there than you were asked for, or where you even had an idea what else to ask or what else to do connected to the problem? You surely need some proper curiosity and drive for doing a math PhD. The problem of matching your interests with the supervisor better can in all likelihood be solved, as can the problem of generating suitable problems "between" your interest and the supervisor's. If your supervisor is not the best person for helping you in that respect, talk to others. However, if you don't have a genuine interest for research and some curiosity and drive, your position isn't that good.

I think in your position, without supervisor help, I'd have grabbed a sufficiently sophisticated textbook about something somewhat related to computational geometry and something that seems interesting to me, and would've started to work through the book, trying to understand more or less everything, do some exercises, with particular awareness for any hint what could be a direction for my own research.

Last remark about "waste of time" - it's an experience, and many PhD stories involve being stuck and feeling that it's not going forward for quite some time. If you manage to dig yourself out of such a hole, you will feel the success and with hindsight will probably think that this phase was a valuable part of the experience. Don't forget that as long as you're studying just the stuff others tell you about and solve the exercises they have solved already, you may learn, but you don't push science forward. If after three years you have spent half the time in a hole trying to find your way out, and half the time doing something that is really your own project, this is quite a bit more productive than learning stuff that was spoonfed to you and passing exams on it, be it with flying colours. Many PhD projects, particularly in math, don't lead to published papers in the first two years. When I got my PhD after 3 1/2 years or so, I only had a paper on some earlier work from my Masters project; it took almost a year more to get something from my PhD out - this was the old days when there was less pressure to publish though, but really still it can pay out to take time to try to get into things in some depth rather than going for anything that could be publishable on the quick.

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During my time in grad school I found it quite essential to work on something closely related to what my adviser has written paper(s) on. This is because if I had worked on something quite far off from my adviser's expertise, I would not have had the resources to work on the problem. Also other people might be thinking about a problem that you stumble upon on your own.

Independent could mean that you don't have a boss saying exactly what to think about and when. But an adviser can still suggest a toy problem to work on, which the adviser can solve if he/she thought long enough. Such a result would probably not be a significant result, but it would be something, and you would be able to put it on the arXiv. Then with your new knowledge you can continue to other problems now that you have more experience. So, small steps. Even with such a problem there will be a significant gap between what you know, and what you need to know to solve the problem. Then you can ask your adviser what books or articles he/she recommends to learn some topic.

Being an independent researcher does not mean that you are not allowed to get help. Actually, the more help you can get, the more efficient you will be. Ultimately reading papers is one way to get help. If you can't ask people for help, then why would you be able to consult expository books? You are not asking people to do the problem for you.

The thing about finding your own problem...it can be great if you do find one that is

-interesting,

-non-trivial,

-that nobody else is thinking about, and

  • is doable in the time you have.

But it is unlikely that a beginner in the field will be able to find a good problem by oneself. In fact, many papers are written in which the acknowledgments say, "We thank (blah) for suggesting this problem." And these are papers written by well-established mathematicians.

Let me be clear: you are supposed to seek help. Talk to your grad student friends. Talk to postdocs and faculty. It helps to have a small team of experts. Mathoverflow exists for a good reason. This is important especially because there are "well-known" facts that the experts know but are hard-to-find or not even stated in the literature. Example: that the poles of Eisenstein series are given by their constant terms.

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I had a similar experience to yours, I did my PhD with a junior faculty, I was her first PhD student and she hired me to work on something she wasn't particularly interested on, while she was looking for funds to work on her passion project. She was not hand off, by any means, and she was always meeting with me, discussing the project, giving me ideas, etc, but she gave me a lot of freedom to do what I wanted.

I must admit that's difficult and you see friends that get plugged into a lab that has projects ready to go, years of expertise and accumulated information, and they basically start getting into publications very early. It does require a lot of self discipline, because the research must be interesting to you, and you alone should have the drive to move forward. This is true for any PhD, but in your case, you usually don't have the support people in more established labs have. In my case, as my advisor built her program and recruited for her project, there was a postdoc, couple of students, meanwhile I continued to be the only student working on the other project.

I finished my PhD and I must admit I owed a lot to my advisor pushing the right buttons. I do appreciate she trusted me to come up with a project myself, I learned a lot about thinking globally about projects, because you usually think about experiments in a piecemeal fashion, and that's not always the best way to do things. And I understand where you're coming from, as people here has commented, what that entails is that you have to create the group yourself. Talk to other people working on similar problems, engage a lot. The support will come from somewhere else, and use your advisor wisely.

Good luck with this, I know it's difficult, especially if you're not the self-starter kind of person, but you'd learn way more about academia than someone that just gets plugged into a massive lab, gets a project from upstairs and that's it.

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While supervision styles differ, a good supervisor will adapt their style to the needs of the student. Some (very very few) students in mathematics come out of their masters fully independent, full of ideas what to work on, able to find their own literature and not relying on their supervisor knowing more than they do. Most students have to reach that point in the course of their Ph.D. studies. An important difference between Ph.D. student and postdoc is that you should not be expected to fill your supervisor's expertise gaps.

My main piece of advise: I would recommend you that you escalate your concerns higher in the departmental hierarchy. You are saying that your supervisor has not yet supervised any students to graduation. That means that whatever their views and expectations are, and whatever legitimate variance in those views there may be among academics, your supervisor does not really have much of a right to have high confidence in their views, and must be open to mentoring. It is the department's responsibility to ensure that adequate mentoring to your supervisor and safety nets are in place. In many departments I know, there would be an obligatory co-supervisor assigned to you. In any case, just like it is your supervisor's responsibility to help you gain the skills you need to become an independent researcher and get a Ph.D. degree, so it is the department's responsibility to help your supervisor become an effective supervisor. Neither is a set of skills that anybody is born with.

There is a lot of advice out there on how to become an independent researcher (specifically in mathematics), how to find good problems to work on, how to learn to absorb new literature quickly, etc. Some searching will get you lots of reading. But nevertheless, I maintain that throwing people in at the deep end when they cannot swim, while it sounds romantic and may sometimes work, will regularly lead to avoidable deaths. Insist that your advisor is only permitted to do that if they know what they are doing, and if they secretly have a safety buoy up their sleeve.

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