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I have been stuck on the proof of a major theorem in my forthcoming paper for about two weeks. Since offline help is limited, I'm trying to see if there is more help online. But I have two concerns.

  1. I am the author of the paper and am supposed to work out the heavy-lift proof. I want to see if someone has worked on similar proofs who may point me in the right direction. What if someone comes up with the complete proof, rather than just suggestions/comments? Should I offer co-authorship or just appreciate their help in the acknowledgement?
  2. For others to offer concrete help, I may have to disclose the proposed algorithm as they suggested. So there is the risk, even remote, that someone just uses it for their publication purpose before my paper is submitted.

How should I proceed, keep grinding hard on the proof offline, or solicit help online with the still-to-be-published algorithm, or something in between?

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sinoTrinity
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    This comment does not answer the question, but I've struggled with a proof for months (and even that is not a long time). Don't give up! Try different approaches. Try to find a counterexample (is the statement even true?). Maybe someone with more experience can suggest a proof strategy or insight (your advisor)? – mrm Sep 25 '14 at 14:39
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    I have been stuck on the proof...for about two weeks — I have published proofs after I was stuck for years. Two weeks is nothing. (But have another problem to work on in parallel while you're stuck on that one.) – JeffE Sep 26 '14 at 02:11
  • Depending on the expected difficulty of the proof and what kind of mathematical/CS expertise is required, you might consider checking whether there is some expert on this matter at your own university. – Wrzlprmft Sep 26 '14 at 08:21

2 Answers2

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This answer is meant to apply to graduate students (like the OP).

Talk to your advisor. She is the designated person to help you with your research. You should do so because:

1) Your advisor will probably have some help to offer. Give them more than one chance to do so, and be clear in communicating how much help you want/need at any given time.

In your case I believe you said earlier that you talked to your advisor and got some but not enough help, because she only thought about your problem when you were there. To me that is the opposite of strange: being a thesis advisor myself, I recognize it as the classic tension inherent in the job. How much help do you give the student? How much time you do spend thinking about the student's problem so as to be able to give help? There are usually no easy answers to these questions. Even in dealing with the same student, over time I often find myself: giving too much help; giving too little help; being put in a situation where the question they ask is too hard for me to give an answer on the spot and then having to try to find time for outside thoughts about their question.

When you talk to your advisor, make sure she understands that you feel so stuck in your research that you are considering seeking outside help.

2) If you get outside help without telling your advisor about it, it could be embarrassing to her.

It can be tough to ask the same person for help on the same thing more than once. But it is part of the advisor/advisee relationship. If my students showed up on SE sites asking questions that I feel that I could have answered, I would not feel great about it. (Most of my negative feelings would be directed to myself rather than at them, but still: not great.)

3) If you are truly stuck, your advisor needs to know. It very often happens that the best thing to do is to switch to working on a different aspect of the problem, or perhaps a different problem entirely. Your advisor is the one to help you with that.

Let me also say that in the realm of mathematically-related research, two weeks is a fairly short amount of time to be working on something. If after two weeks you are completely out of ideas and don't even know what else to try, then you should address that. If you simply haven't proved the "major theorem" yet: join the club. To prove a major theorem usually takes me at least two months; two years is not at all unheard of, and is not a maximum. As long as you're making some progress thinking about the problem, I don't necessarily see anything wrong here.

Added: The lack of directness of my answer was intentional, but let me add one comment. In my opinion the greatest risk in asking in the internet community for help in solving your mathematical research problem is....that someone will solve your mathematical research problem. As mentioned above, your advisor is optimally briefed in the matter of how much help to give you / how any one question fits into the larger scheme of your research program, and still advising a student is a matter of successive errors and corrections (i.e., helping too much and too little). Being a PhD student has a highly egoistic aspect to it: you are not just trying to find solutions to problems; you are trying to find them yourself. There is a real risk that the right expert will simply leave you without a problem to be working on. This is why talking your advisor is so critical: she may in fact decide at some point that asking for help is best, but in that case she will know exactly what and whom to ask in such a way that the rug is least likely to get pulled out from under you. This is very important!

Pete L. Clark
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  • The "opposite of strange" and "no easy answers" parts really strike a chord. One the one hand, I want to prove it by myself independently. One the other hand, I feel I need someone more experienced (e.g., my advisor) to nudge me a little in the right direction after days of stagnation. I'll go ahead to solicit more help from my advisor. Thanks a lot for the prompt reply. – sinoTrinity Sep 25 '14 at 15:10
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    @sinoTrinity "days of stagnation" ... you are expecting progress too fast. – xLeitix Sep 25 '14 at 15:25
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    LOL, I directly assumed the supervisor would only provide disservice (just like mine). If your supervisor is competent then I don't even understand why is the question here... – Trylks Sep 25 '14 at 15:34
  • @Trylks He is offering constructive feedback. It's just I thought it would be better if I could get additional suggestion online so I can move faster. At least it does not hurt. That's the intention of my question. – sinoTrinity Sep 25 '14 at 15:58
  • If your supervisor is doing his work, then you can keep on that. You may propose collaborations with other people, but that's up to him. Usually the collaborations will be with people in the community, going to the wilderness of the online world is not common. After you have gone to a few conferences/workshops you will probably notice the faces are the same in all of them, AFAIK this happens in most areas, otherwise they would simply fragment into smaller areas, because people like to attend to talks in their very same area, hence only a few people can be there (it's very specific). – Trylks Sep 25 '14 at 16:18
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    @Trylks: For whatever it's worth: this past weekend I was a speaker at a small conference attended by about 40 people. Number of faces I had seen before: 0. Number of people with whom I had interesting, content-related conversations and would like to learn more about their work and keep in touch: at least 10. Moral: there are many more people doing interesting work in my academic subject (mathematics) than I yet know or have met. – Pete L. Clark Sep 25 '14 at 17:07
  • @PeteL.Clark well, I guess your mileage may vary, on your age and the broadness of your subject. "Mathematics" is definitively very broad, but it must be hard to research without more focus. I'm thinking about the famous illustrated guide to a Ph.D. – Trylks Sep 25 '14 at 18:20
  • @Trylks: I guess I don't really understand your last comment. I was responding to your "people like to attend to talks in their very same area" by mentioning my own recent experience. If every conference I went to had the same people and concerned the same topic, I wouldn't see the point of continually going to conferences. (I should point out that in my field, conferences and publications are almost entirely separated.) Also, the number of people who can help me on my work -- and vice versa -- is quite large: in fact, I don't know many of the people who have insight into my work. – Pete L. Clark Sep 25 '14 at 22:34
  • The bit about breadth and focus confused me: certainly "mathematical research" means, at any given time, research in some very specific parts of mathematics rather than mathematics as a whole. But the same person can be interested in and do work on more than one thing and in more than one subfield, especially as the years roll on but most of my PhD students, for instance, have written at least one paper outside of the narrow topic of their thesis. The number of mathematicians with whom I share some common research interest is effectively infinite. – Pete L. Clark Sep 25 '14 at 22:40
  • I'm confused as well. In short, to advance the state of the art in some topic requires to become quite familiar with it, which usually requires a lot of background (and time to acquire it before actual research can be done) even for very narrow topics, i.e. deep knowledge about the topic. Breadth and depth don't use to go well together, that's all. In any case, if it works for you, all I can do is sincerely congratulate you. – Trylks Sep 26 '14 at 09:43
  • "Breadth and depth don't use to go well together" I would rather say that breadth and depth are both extremely desirable, and the fact that it is hard to acquire both is what makes research hard. To make an important breakthrough requires a new insight or idea. Always talking to the same people about the same things is not the best way to generate new ideas (in my opinion and experience). – Pete L. Clark Sep 26 '14 at 13:24
  • Moreover, often an important component of the solution of a big problem is a connection made with a different area of mathematics: the proof of Fermat's Last Theorem connected Diophantine equations and modular forms. The proof of the Poincare Conjecture connected low dimensional topology and geometric analysis. So learning what people who are not standing right next to you are doing and asking how it can help with what you're doing is a really good strategy: http://wiki.genexus.com/commwiki/servlet/hwiki?The+Feynman+Method+to+be+a+Genius. Maybe this would be worth a followup question... – Pete L. Clark Sep 26 '14 at 13:28
  • On point 2: I've never been a thesis advisor, but I have spent some time on that side of a mentor-mentee relationship, and I would not be offended or embarrassed if someone I was mentoring sought qualified outside help to solve a problem I had given them. Rather I would applaud their initiative in doing what was necessary to get the solution. (If it were something akin to a homework exercise and they used outside resources to shortcut the educational value of the problem, that'd be another matter, but still, I wouldn't be offended or embarrassed, just disappointed.) – David Z Sep 27 '14 at 02:51
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I think the answer depends on this:

Do you hope to become an expert at mastering the kind of difficulty you're facing with the proof?

If your answer is:

  • Yes: Then go ahead and work it out on your own. (In particular if you're a grad student and this is your thesis problem... and in this case also consult your adviser as @PeteL.Clark advised.)

  • No, I usually focus on a different subarea, and I realistically expect to keep it that way: Then it may be a good idea to ask online:

    • Use a "minimal working example". That is, don't describe your entire project but reduce your difficulty to a simple to state problem that you feel is far away from your own specialty.

    • Offer co-authorship to whomever helps you out.

Bjørn Kjos-Hanssen
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