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This post is a sequel to: Collaboration or acknowledgment?

The following has come to my attention. A senior mathematician (let us call him or her Alice) suggested a problem to a young mathematician (Bobby) who proceeded to solve it on her own and wrote up the result. Bobby agreed (*) to let Alice be listed as a coauthor, but Alice also insisted to include her PhD student (Charlotte) as a coauthor because they were thinking about the same problem, despite the fact that Alice and Charlotte did not even have partial results.

(*) Bobby had no problem with Alice joining her as a coauthor for the reasons mentioned by Igor Rivin below (I include you as a coauthor, you write me a good recommendation). Thus, the credit was unfairly diluted by including Charlotte who had not contributed.

Question: Is there a way for Bobby to manage such a situation without creating conflict?

Full disclosure: I am Bobby's PhD advisor. I can not interfere directly because Alice is a powerful person in the field known for aggressive backing of her PhD students and I do not want to inadvertently hurt Bobby's career.

Update: Following the advice of Joel David Hamkins, Bobby will be the sole author. There is nothing in the paper that Alice and Charlotte could point to as an idea they already had in mind. Looking back, what bothered me the most was not that Bobby's credit would be diluted but that someone who did not deserve it would be rewarded.

I will award bounty points to JDH for his uplifting Thanksgiving Day Answer, but, of course, any new comments are welcome.

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    $\begingroup$ It seems academia is a more appropriate venue for this question. $\endgroup$ – Andrés E. Caicedo Nov 23 '17 at 14:15
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    $\begingroup$ The junior/senior issue here is a confounding element, but in my opinion, asking a good question is often a good enough reason to be co-author. I have several papers, where someone asked me a question (sometimes on MathOverflow), which I answered, and I asked them to join as co-author. $\endgroup$ – Joel David Hamkins Nov 23 '17 at 14:28
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    $\begingroup$ From your description it is completely possible that the senior mathematician reached the final formulation of the question with the help of the PhD students. In that case, the student certainly deserves credit. Generally, my advice is that it is always better to err in the direction of generosity. Otherwise, people would not like to work with you. $\endgroup$ – Yiftach Barnea Nov 23 '17 at 20:50
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    $\begingroup$ I would like to disagree with the comment made in the edited version of the question. Just because I have voted to close this question, you should not infer anything about my views on the ethics of the situation that you have described but for which I do not have any sight of the actual evidence. en.wikipedia.org/wiki/Rashomon_effect $\endgroup$ – Yemon Choi Nov 23 '17 at 21:23
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    $\begingroup$ I am shocked to see how many people try to protect the status quo, and say things like "we don't have enough information, so we can't judge". Yes, there's enough information! And yes, it's wrong for a senior mathematician to push for a PhD student of his to be coauthor of a paper of someone else! It's wrong, and it's unacceptable. $\endgroup$ – André Henriques Nov 23 '17 at 21:32
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Well, of course the young mathematician should simply discuss the matter with the senior mathematician and perhaps the student until they can come to an agreeable arrangement. My advice is that they should all talk about it. Co-authorship is a matter upon which all authors must agree. What other answer could there be?

If it seemed that the professor or the student had little or no contribution, then the young mathematician should say so and inquire why the professor should be co-author, or why the student should be co-author. If there was not sufficient contribution, then the young mathematician should simply say so and there should be a discussion about it. Perhaps the senior mathematician will point out that the contribution was greater than realized, or that there were other aspects of the collaboration of which the young mathematician is not aware. Or perhaps not, and the senior mathematician will agree that the young mathematician should proceed solo.

But apart from the particular situation described in the post, let me now mention several further reactions that I have more generally to the issues about co-authorship that this question raises.

The first and most important thing to say is that collaborative research is one of the great joys of mathematical life, and I strongly recommend it. To discuss a mathematical idea with another mathematician, who can understand what you are saying and who has thought deeply about the very same topic, gives enormous satisfaction and meaning to one's life as a mathematician. Collaborative research is our mathematical social life. For my own part, I am thankful on today, Thanksgiving Day, for the opportunity that I have had to interact with all my collaborators; I have learned so much from them. (See the list of my collaborators.)

Therefore, my advice is that one should seek out collaborations wherever they are to be found. Often, after one has proved a theorem, then in joint work it becomes much better, strengthened or simplified, or a collaborator finds new applications or uses. If someone asks a question and you answer it, then perhaps the mathematics is not yet finished, but only begun. Aren't there further natural questions arising from the result or its proof? This could be the beginning of a collaboration rather than the end of one.

Another part of my view is that one should be relaxed about collaboration and co-authorship. Except in extraordinary cases, the stakes are low. A mathematician seems to get basically as much respect and credit for a result, whether or not there are co-authors on the paper, and so I question whether there really is any meaningful "dilution" as mentioned in the post. It is simply no big deal to have co-authors or not.

Therefore, why not be generous? If someone has made a contribution to your project, even when the contribution is light, then invite them as co-author. Few mathematical collaborations are perfectly balanced contributions, and in most collaborations one person has had a more important insight or made a larger contribution than the other. But so what? Perhaps the co-author invitation will be declined, and that is fine, or perhaps they will join and then proceed to make your result even better. I have had many collaborations where at first we had a result, which seemed fine and complete, but then in writing the joint paper we were able together to improve the result or give further applications, which wouldn't have happened without the joint interaction. I think you will often be surprised.

Another point, as I mentioned in the comments, is that asking a good question in my view is often sufficient for co-authorship. I have several joint papers that arose from someone asking me a question (in some cases on MathOverflow), which I answered, and then asked them to join as co-author. And I've had some the other way around as well. I find it more natural in such a case, however, for the theorem-prover to be suggesting the idea of co-authorship, rather than the question-asker, which is part of why the situation in the OP seems wrong to me.

Another point is that it sometimes happens that person A, perhaps a junior person, asks a question that person B, perhaps a senior person, answers, settling the question; but the situation is that person A simply cares more about it or has a stronger vision of what the result can become than person B, who is not as interested. In this case, the solution is that person A should do the work of writing the paper, with person B as co-author, even though the result may have been due to person B. The point is that person B would not bother to write this paper on their own, but the joint authorship brings the mathematical paper into existence. The result can be a great paper, and I know of many papers following this pattern.

In summary, pursue collaborations; be generous about co-authorship; be relaxed about co-authorship; enjoy joint mathematical interaction; make great mathematics.

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    $\begingroup$ "In summary, pursue collaborations; be generous about co-authorship; be relaxed about co-authorship;" Yeah, but *with people of some noticeable strength or, at least, with people you like personally". As far as I understand the question, it is all about "Why should B dilute her credit to give C a free ride to the PhD defense?". I have no answer to that. $\endgroup$ – fedja Nov 24 '17 at 3:50
  • $\begingroup$ You don't like the answer that the authors should discuss the matter? Co-authorship is a matter on which all authors must agree, and if C really made no contribution, then C should not be co-author. This would of course be the central issue to be discussed. In any case, as I mentioned, the latter part of my answer which you quote was not intended to be about the specific situation of the OP, but about co-authorship more generally. $\endgroup$ – Joel David Hamkins Nov 24 '17 at 4:05
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    $\begingroup$ Apparently they have already discussed the matter (perhaps, without C) and it is exactly the result of that discussion that created the current situation. You gave an excellent overview of the general attitude towards co-authorship and I share it in many respects. I'm just trying to say that this special situation is exactly when particulars may outweigh generalities. The way I see it (to put it bluntly) is that A finds C hopeless and desperately wants to create an opportunity for her to defend (I've seen such cases). Maybe I'm wrong but if not, it puts things in somewhat different light. $\endgroup$ – fedja Nov 24 '17 at 5:33
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    $\begingroup$ This is an excellent answer! I have one more thought to add in favour of solving things by amicable discussion: even though Bobby is less senior, he is not as helpless as it might initially seem, and Alice has a strong incentive to find a good and fair compromise. Indeed, if the situation were to sour, and if it were to become public knowledge that Alice had acted in ways that are not correct, then that would be a big thing for Alice... because let's not forget that reputation is an extremely important thing for all mathematicians. $\endgroup$ – André Henriques Nov 24 '17 at 9:58
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    $\begingroup$ I agree with the relaxed attitude on principle, but there is the problem that if a junior person and a senior person collaborate, then the contribution of the junior person is sometimes questioned by hostile members of hiring committees. Also, it sounds like Alice is promoting her own student and exploiting Bobby in a shameless way. $\endgroup$ – CatO Minor Nov 25 '17 at 15:53
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If several people work collaboratively on a problem, they all get to be authors even if all the ideas that finally led to the solution came from only one of them. That is a perfectly ordinary situation.

The question then reduces to that of whether the situation described comes under the description of "collaborative work". I think that needs more detail to judge and there will always be a grey area.

If what actually happened was that the Alice said to Bobby "Charlotte and I are working on an interesting problem, namely ..." and then Bobby went away and solved it, I would call that a collaboration.

OTOH, if what happened was that Alice said to Bobby "here is a problem for you" without mentioning that Alice & Charlotte were working on it too, then Bobby has a good case for being sole author. Whether Bobby should insist on that right is another question.

In a practical situation I would recommend a somewhat greater degree of generosity towards a student than towards a seasoned researcher.

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    $\begingroup$ What if the problem does not come from Alice? If several people have already worked on it without discovering the answer? If Alice mentioned it to Bobby as an interesting problem that people care about? What happens if you discuss a problem with someone at a conference and then solve it without any of the ideas from that conversation? Honestly, I can think of more cases where it is definitely not a collaboration than where it is. $\endgroup$ – user0100661 Nov 27 '17 at 21:48
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Unwelcome requests should be denied. Period.

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