The Domineering Collaborator: making your ideas heard.
Krishnan Shankar was asked to write about collaboration because he has been known for completing excellent papers with top notch mathematicians. In his writing here he addresses some of the issues which arise when working with powerful mathematicians and also the benefits.
I have been in the game for a few years now (1999 Ph.D) and I have written about 14 or so mathematical papers. With two exceptions, every one of my papers has been in collaboration with one or more people. Evidently this is because I really enjoy working with other people on mathematics. To me the subject comes alive when I am able to talk through my ideas with someone else.
Collaboration is a tricky thing and one needs several ingredients to make it work. For me I discovered the critical ingredients are humor and patience. I have also been very lucky; with one exception I have gotten along very well with my collaborators and have gained valuable friendships in the process. But humor and patience are essential. In grad school I was a good student but there were easily some extraordinary students a few years ahead of me. One in particular was absolutely brilliant and also equally caustic whenever I asked him a question. I don't think I have ever taken so much verbal abuse at the hands of one person, but through it all I never got a sense of condescension from him, just frustration. He would get upset that I did not know something simple (like the Pontrjagin signature formula) but instead of lacerating me and then dismissing me, he would sit and make sure I understood before letting me go. So I put up with it and learned a lot. He and I are good friends now and I learned a valuable lesson: as long as there isn't malice or condescension, if one can disregard extraneous details like personality quirks, then it leads to a great collaboration and perhaps even a friendship.
To be honest I believe that every one of my co-authors has been and is a top notch mathematician. That is another secret: try and collaborate with people you consider better than you. This may not always work out since they may not want to waste their time. But as I said I've been lucky in the people I knew. In most instances, they listened carefully to my input or ideas even if 9 times out of 10 it was really dumb. Occasionally someone would totally disregard or ridicule my input or my ideas. My usual response to this (whether in mathematics or in real life) is to turn it into a joke at their expense. This is a tricky thing because you don't want to make everything a joke and trivialize it. But you also want to let the other person know that you are not there to simply nod your head and take abuse. Hitting the right note in these circumstances takes a lot of practice and being able to laugh things off. Here is an example: I asked one of my collaborators a question about Riemannian manifolds satisfying a certain geometric property. He is of course very good and his immediate response was and I quote, "Who in the world studies crap definitions like that?" I smiled and told him that the definition originally came from problems in polygonal billiards and the specific question I asked was posed in a paper by another excellent mathematician. Then I said, "Are you sure the definition is crap or did my asking it color your view of it?" He was a little taken aback and agreed that he just said it to put me down. I just smiled and said, "Here is another crap definition..." and carried on.
Ultimately, you should weigh whether working with a difficult collaborator is worth the stress. In most instances for me the process was smooth and enjoyable. In some instances there would be bumps on the road: in one instance, after several months on a project one of my collaborators spoke rudely and rejected every one my suggestions out of hand. I usually don't take things personally but there came a point when I decided that this wasn't healthy. So I said very politely that perhaps I was not equal to the task and that I was sorry he felt he had to put me down so much. Finally I said, sincerely with no malice, that we should part ways and he should write the paper himself and I was happy to accept some thanks or acknowledgment in the final paper. He was so stunned that he backtracked and the rest of the collaboration was smooth and trouble-free. That is my last piece of advice: don't take things personally and unless the paper is on the verge of proving the Riemann Hypothesis, be ready to walk away from it calmly to preserve your good natured steady state.
One last thing I was asked specifically for this webforum: how do you get your ideas in? If you are in the situation where your ideas are not being heard, and you have tried politely several times, then one thing that has worked for me several times is the following: when I can see that an idea of mine will definitely pay some dividends, I will take it away and work out some useful or interesting intermediate result and then show it to my collaborator. In every single instance, this yields the desired effect: they now listen with more attention the next time I suggest something.