I.C.
Collaborating with Scientists and Mathematicians Outside of Your Field
My experience is that this can be very rewarding, but there is definitely an art to learning how to do it! The more knowledge that you and your collaborator have of each other's areas, the easier this is. Lately, I have been working in Computational Biology, and many biologists are about as far as you can get from most concerns of pure mathematicians. Yet there is a great need for collaboration and there are many problems of interest to many sorts of mathematicians.
First, it is important to identify the right collaborator. Many biologists have no idea of what mathematicians do, and how they think about problems. They have no idea how to state their problems in such a way that it will make sense to a mathematician. Therefore, if you have never collaborated with a biologist before, particularly if you are junior, I recommend that your first collaborator be an applied mathematician (Ph.D. in math or statistics, not biology) who has him or herself already collaborated successfully with biologists, rather than starting out talking directly with a biologist with no advanced mathematical training. The right collaborator should be able to tell you a problem where they believe the field is stuck because they need a better mathematical definition, or a better mathematical model, or a better mathematical method of finding an optimal solution. They should be able to point you at several papers you can read for background on how the biologists themselves think about the problem and what has been done so far. The best way to find such collaborations is either to seek out such people at your own university (you can probably tell their background and interests from their webpage) or to go to special year workshops at the mathematical institutes, such as the Institute for Mathematics and Its Applications at the University of Minnesota, or MSRI, or wherever, that specialize in bringing people together across themes of broad mathematical interest within interdisciplinary areas.
Second, decide from the outset who your audience might be, though you might change your mind along the way-- or you might have several audiences. It might turn out that the mathematics that is needed to solve the real biological problem is not very deep, or might already be known, just not to the biologists! You might help make an amazing biological discovery, but not advance mathematics very much. On the flip side, a biologist might ask a question about, for example, the number of reversals that are needed to put a string back in sorted order, and might only need to know the answer in some very small cases, but you can easily generalize the problem to a deep combinatorics problem, for example, that is intrinsically interesting to folks in your field. You can go on to solve a generalization of the problem in r dimensions, the biologist only needed r=1, and none of the other cases has any biological meaning, but you were just inspired to produce an interesting and beautiful piece of mathematics. If you make an amazing biological discovery of limited math interest, you will find yourself a co-author on a biology paper; if you do the reverse, your biology friend may be a co-author on your math paper. Sometimes you do something of interest in both arenas but still end up writing for 2 different journals, since the pieces that interest your home communities may really be quite different, which brings me to a related issue which is co-authorship, which I cover next. But I can't stress how important it is to have an audience in mind: is your goal to do work that will be considered important in their field, in your field (both would be wonderful, but won't always happen)-- and do you or your collaborator have the experience and expertise to know how to present your interdisciplinary work so that whichever discipline you are aiming to publish in, will understand and be excited by the contribution?
Third, different scientific fields may have different conventions about co-authorship-- what contributions merit co-authorship, for example or in what order the authors get listed on the paper. Biologists and even applied mathematicians who work in biology don't typically list names in alphabetical order as most mathematicians do on co-authored papers: the last author is the senior researcher with the big grant; the first author is the junior person who did most of the work. Make sure to have a long conversation about authorship credit issues early, to understand everyone's expectations and to avoid hard feelings. If you are going to be writing up a version of overlapping results for a more mathematical audience, state your expectations that you will do this and what journals might be appropriate for you to submit to for such a thing up front. If you are junior person in particular, you need to think about what sort of publications and what kinds of credit you will need to get out of any collaboration. For example, it might be fine if you co-author the "math" version with 3 alphabetized authors and are only stuck in somewhere in the middle as a co-author with 20 biologists on some paper in a biological journal too.
Fourth, agree on shared vocabulary, and understand that sometimes words can have different meanings across fields. I ran into this in another context, once, when I realized that the theoretical computer scientists and the probabilists meant something completely different when they used the phrase "almost surely" back in the mid 1990s-- I don't know if that's still the case, or if one version won out. If you think you are talking past each other, you probably are. I like to joke that when a biologist says "always" they really mean, "I personally am only aware of 1 or 2 exceptions to this rule" which if true, is clearly NOT what a mathematician would guess was meant by the word "always"!!
Collaboration across fields is particularly difficult because of deep cultural differences that can be hard to identify. A contribution that might look to us to be worthy of an acknowledgment (for example, providing a data set) merits co-authorship in some fields. Author order is not alphabetic but very carefully negotiated using a precise set of guidelines. Papers are often written "backwards" with the "results" first and the "methods" last. Even the word processor can be different. Time scales are also often different; papers are written and published very quickly, so a delay of a week in responding to an email might cause ill-will.
These collaborations are incredibly interesting, but be aware of the pitfalls and make sure that everyone understands the need to communicate before conflicts grow.