A successful research career involves a mix of individual and collaborative projects. Areas of mathematics that people used to consider separate are becoming increasingly interconnected and those connections often lead to interesting new approaches to mathematical problems. This is true not only within applied mathematics. While the connections between different mathematical areas are interesting to develop, they also demand expertise in a larger set of domains, which can be slow or impossible to achieve individually. It is often more efficient to develop collaborations with researchers that have complementary expertise and can contribute to the project through fresh points of view.

In this post I share some thoughts about starting collaborations and developing them in a way that successfully advances mathematics and is of mutual benefit to the collaborators. This may be directed at junior faculty and postdocs but it is also important for graduate students.

Research collaborations are like relationships: they take time and effort to develop and are based on trust and respect.

The points below assume that the topics being discussed are slightly outside your area of expertise and that a collaboration would be more timely than to learn to do the work on your own.

Starting a collaboration:

- Talk to people: A common way in which collaborations start is after meeting other researchers with similar interests at conferences or seminars. After Prof. X’s talk, make a point to introduce yourself and take a few minutes to discuss
*their*work. Questions like “have you thought of doing ____?” are not necessarily the best to ask. Instead ask about what the next steps are or what new questions derived from Prof. X’s work are most important. The goal is to understand Prof. X’s interests and determine if your own work have something to offer that Prof. X probably wouldn’t consider.

- Have something to offer: A collaboration is obviously a two-way street so you must have something to offer. Based on your initial conversation with Prof. X, if you feel that your work presents a new way at looking at their problem – a point of view based on different mathematics expertise – mention it. You shouldn’t give all the details (which you probably won’t have yet) but you can briefly explain the relevant work you have done and the ideas that might be applied to their problem. Especially important is to explain how this approach might open up other research doors.

- Set up a follow-up: If Prof. X seems genuinely interested in your ideas (and is not just being polite), you can end the conversation by asking if it would be ok for you to follow up with an email after you have thought more about your approach and can see how it might work. If Prof. X does not seem particularly interested in your ideas, just say “thank your for your presentation” and that is the end of that.

Become indispensable:

- If you set up a follow-up email, make sure you actually follow up. Send an email with a more concrete vision of how your ideas fit the research questions discussed earlier. You don’t want to give all the details for someone else to take them and do the work; you just need to make the point that your approach will lead to results.
- If Prof. X shares your interests and vision, s/he probably will indicate that you should collaborate on this project. Otherwise, you should ask if they would want to make it a formal collaboration. If the answer is positive, the specific goals of the project with each person’s responsibilities should be outlined.

Efforts you should make:

- Read your collaborators’ research papers on the topic.
- Learn to speak their language. This is especially important when the collaborating disciplines are “far away.”
- Keep deadlines for producing your share of the work.
- Communicate often regarding your progress, new observations, unanticipated snags, etc. Writing up documents, using dropbox (or something similar), and skype (or something similar) are helpful.

Things to keep in mind:

- Collaborations are temporary; they eventually end.
- Some collaborations fail. There are risk associated with pursuing new approaches and thinking ‘outside the box.’ Be aware of this.
- People measure the value of collaborations based on what is important to them (switching math areas, getting tenure, becoming more competitive for funding, etc.). A successful collaboration will be valuable to all parties according to the way they measure value.
- Publication authorship (first author, second author) should be discussed early in the process. If two publications are expected, you may consider reversing the authors order in one of them.
- Don’t be upset if someone rejects your effort to collaborate. It is not personal. It could be bad timing or a change in their research priorities.
- New collaborations can involve some substantial “start-up” time but they pay off in the long run.

Note that in pure mathematics, author order is strictly alphabetical by last name.

It is still a good idea to discuss authorship early in collaborations even if the discussion is simply an agreement to keep with the tradition of alphabetical names.

For me, collaborations have started over a beer discussing a favorite result, asking a colleague a question at tea, describing a favorite challenging problem over lunch, and so many other ways such as responding to an email from a person encountered at a meeting as described in this blog. As an older person looking back, these have made mathematics all the more exciting and enjoyable and they have lead to friendships that have expanded beyond mathematics and endured many many years. The relationships can be as complex as the problems but the expected value of the investment is huge. Sometimes, the work is powerful and fast but often it is challenging and requires persistence and courage….one result took several years and almost 50 revisions of the manuscript as the final results matured. Nevertheless, I can’t think of a single collaboration that I’ve regretted and am currently involved in

projects with individuals that range from undergraduate students to senior colleagues around the world.

Especially for junior colleagues, I would recommend reserving some fundamental facets of your research program for your individual effort and individual publication. Often, I find that evaluators (hiring, fellowships, research proposals, etc) seek concrete evidence of individual contributions and find collaborations difficult to incorporate into their analysis.

With regard to order of authors of mathematics papers, the historical practice has been “alphabetical order” with any departure causing confusion and question. In other fields, this is certainly not the practice and can be quite complex and a matter of important discussion in interdisciplinary research. But, I wonder, has mathematics changed its practice?

The authorship practice is not so strict any more, especially in applied areas of math. I often list myself last on my students’ publications derived directly from their dissertation.

The AWM has an online panel with answers to many questions about collaboration in mathematics: https://sites.google.com/site/awmpanel2012/