Handing a Mathematical Divorce:
When Mathematicians Decide to Stop Working Together.
The following stories were told to me from the perspective of Mathematician A.
Mathematician A and B are having a falling out because Mathematician B believes he or she has all the good ideas. Mathematician A feels that he/she is being treated as the one who merely verifies and types up the results. Mathematician A is particularly upset because his or her ideas would complete the project sooner and there are other projects demanding attention. Both A and B are young and are concerned about promotion and jobs. There is a senior mathematician that both respect. Mathematician A explains his/her concerns to this senior mathematician, and can do so without risking any accusations which would have a negative impact on B. The mediator has sufficient expertise in the field to evaluate the merit of A's suggestions and explicitly advises B to apply A's suggestion. Mathematician B does this, the paper is completed, and they never coauthor again.
Note that in this story, the fact that Mathematician B doesn't trust other people's ideas has not been addressed. On the one hand, it was not necessary to deal with this larger issue in order to mediate the problem and to allow both A and B to complete their paper. On the other hand, while this served A very well, mediation may have failed to address a problem which will affect B's future mathematical relationships.
Mathematician A and B have a multiyear project with no completed papers but many theorems. Both mathematicians are equally stressed about promotion and have been presenting their joint work without producing preprints. The delay in the production of preprints is due to the fact that neither mathematician really trusts the other mathematician's proofs. They are in different fields and cannot verify each other's work. Each has begun accusing the other of not being rigorous. As an outsider judging these subfields, I would say that they are characterized by an abundance of tersely written papers for an audience of experts. Rather than mutually agreeing that they each need to put in the extra effort to make their work intelligible to both fields: these two mathematicians have reached the point of directly attacking one another personally. Finally Mathematician B declares that the research need never appear in print.
Both mathematicians complain to their doctoral advisors but they do not find a common mediator. The advisors tell each author that this work is important, must appear in print, and that both names should appear on all work produced from this project. Mathematician A's advisor suggests trusting Mathematician B's ability to write a proof. Mathematician B's advisor suggests Mathematician B write up everything he or she can prove independently and make that a joint paper but not to trust Mathematician A. Both mathematicians follow their respective advisor's advise, but the conflicting advise does not, of course, help to resolve the impasse.
Mathematician B writes a paper containing some of their joint work, which A agrees to co-author. The paper is submitted to, and quickly accepted by, a journal in Mathematician B's field. After A writes up the rest of the results, there follows a long period of heated arguments while B claims to be checking the paper, and A's write up is finally submitted and accepted for publication. In the end, a sufficient number of the results are published, and they papers have a tremendous positive impact on each of A's and B's careers. But the two are no longer on speaking terms and some of their results may not ever be published. It is evident from the publication record that both A and B should now trust one another to write up a coherent account of the mathematics, but they are, unfortunately, unable to reach an understanding.
To their credit, these mathematicians have not publicly complained about their coauthoring troubles. It is immediately obvious to the mathematical community who wrote up which paper, but this is not seen as a sign of a problem. Coauthors will often divide results according to subfields and publish in specialist journals. In fact, these two might have avoided the divorce in the first place, if they had agreed on this approach right from the start.
Mathematician A and B have a multi-year project which has not yet produced any papers. Mathematician A is tenured, with no looming deadlines for grants or promotion and finds him(her)self unable to continue the collaboration in a timely fashion. He(she) decides to bring the projects to a close. Mathematician B, however, is preparing for promotion to tenure and wants to continue the joint work. Mathematician A proposes the following solution: A will essentially abandon authorship claim to the part of the project which is nearest to completion and pursue the longer term aspects of the project later on his(her) own. Mathematician B agrees to the plan and completes the first paper with a new coauthor. Then B even offers A co-authorship as a gesture of good will, which Mathematician A declines. So they just cite each others' work and acknowledge each other for important ideas leading to the results.
This story ends with the two mathematicians on good terms and able to communicate with one another and support each other's work. Any divorce of a senior mathematician from a junior mathematician can be stressful to the junior mathematician. Here, the senior mathematician has chosen to be generous and sensitive to the timing issues of the junior mathematician, even while being clear that the joint work had come to an end. The choice to abandon the project could have happened for any number of non-mathematical reasons, including health issues, relocation, and choices about prioritizing work.
Finally, I would like to mention a question that was asked on the live panel at the Joint AMS meetings from a young mathematician had been through three mathematical divorces in a row and wondered why. Most likely this person was exceptionally unlucky. However, it is worth discussing the scenarios in more detail with someone to look for a common theme. It is always helpful to step back, after a mathematical divorce, and ask if there was anything that could have been done differently.