Collaboration station

I think a lot about why I'm a mathematician.  (Welcome to my identity crisis!)  I enjoy doing research and teaching.  It's not clear to me that pure math research is necessarily the best use of such a huge source of brainpower, but I think teaching can be quite meaningful.  I also greatly enjoy the flexible schedule.  But the thing that brings me the most happiness is the relationships I've built in mathematics, especially many of the ones that I've built over the past 15 years since I started grad school.  Through the course of your career you get to see how people grow as mathematicians and as people, which is really lovely.  These relationships exist in all aspects of my math life (collaborators, colleagues at NCSU, mentors, former mentees, etc).  There's a lot more blog posts I want to write about personal relationships in math, but I wanted to write something about collaborations in this entry.  

Here's a few quick opinions I have on various aspects and stages of collaborations.  Hopefully something is useful.  (I might return to discuss more aspects of this in a later post, since this is only a small part of my views on collaborations.)

- Be willing to talk to people!  This is often how collaborations start.  It applies to people in your department, visiting seminar speaker, some awkward grad student at a conference, Professor FancyPants giving the distinguished lecture, etc.  It can be scary to talk to someone because then they might find out what you don't know, but you can always just introduce yourself or ask them a simple question about their talk or what they have been working on.  If you plan to be in the field for a long time, then it pays to constantly be investing in these relationships: the next time you see this person, they might talk to you more, you may feel more comfortable and be ready to talk to them about more stuff.  This all might lead to more in-depth discussions, getting invited to give a seminar, etc which all could turn into collaborations.  Many successful collaborations I've had came after a number of previous conversations with that same person that led nowhere.

- Someone might have what you want!  In general, it's bad to treat people like objects.  However, *sometimes* it's good to think about what you want to get out of a person.  An obvious goal is to try to get a result or paper, which might mean trying to get Person X to use their skills to complement yours to prove the theorem.  However, another valuable thing is learning *how* to use Person X's signature math trick.  I can trace a number of things I use regularly now to what I've learned back to regular meetings with people: e.g. knot Floer complexes from Jen Hom and Allison Miller; Seifert fibered spaces and their Floer homology from Cagri Karakurt; instantons from Ali Daemi and Juanita Pinzon-Caicedo; and the list goes on.  On the other hand, sometimes a collaboration can be valuable because it's fulfilling to you in whatever way is important to you - they're super fun, you feel comfortable expressing yourself to that person, you have compatible working styles, they live in a city you visit a lot, etc.  It's important to mention there's also lot of value in just shooting the shit mathwise with someone, and that can lead in many interesting directions.  But it's helpful to know if that's what you're doing / what you want to be doing.  I've definitely benefited from those discussions as well, and not every discussion needs to have the goal of proving a theorem. 

- Know your strengths and everyone else's.  This is a little similar to the previous item, but I think it can be helpful to think about what you can bring to a collaboration and what you need from a collaborator.  Sometimes a collaboration takes many people with the same strengths using those all at full force; sometimes there's an elegant dance between complementary skills (e.g. one person can do the topology and the other cranks on the algebraic obstruction).  For example, my strengths tend to be more bigger picture (knowing what tools might be useful for solving which problems); but I have a lot of weaknesses, for example pinning down anything technical carefully (e.g. signs/orientation), and I have terrible intuition for contact/symplectic topology.  I sometimes have an idea for a problem but don't have the technical skills and go looking for an expert to do exactly that.  Some people have powerful technical abilities they don't know where to implement them and can go asking around for what to do with their newest ideas.

- Think about and ideally discuss expectations.  Do you only work on this project and spend all of your time on it, while your collaborator has three other projects and is teaching 4 classes?  It makes sense they can't spend as much time on the project, so don't expect them to be putting in the same effort you are.  Similarly, if someone is going full blast but you are drowning in a summer TA, then let them know you can't work as hard on things at the moment.  (This is preferred to just disappearing for 4 months without a trace.)     

- Learn to speak many different math languages.  Not everyone thinks about / uses objects in the same way.  It's helpful to be able to approach problems from a variety of perspectives, and being able to see things the way your collaborators do can give a different perspective.  (For example, if you have a 3-manifold, you might try to think of it in terms of a triangulation, Heegaard splitting, surgery presentation, bounding four-manifold, subgroup of PSL_2(C), etc.  You might think about things from a categorical perspective, hands on examples, intuition from gauge theory, etc.)  If you want to ask someone a question, it can help to think what that person needs to hear in order to understand and/or be interested.  (E.g. if you have a knot theory problem that stems from Fukaya categories, you may want to think about how to pitch this to a knot theorist in a way that piques their interest rather than selling them on the symplectic geometry story.)  The more you do math with people in their language, the more you can work in this framework in other settings.  When you're working on other projects in the future, you will be able to channel your inner *insert desired mathematician here* and try to approach the problem the way your collaborator might.   

- Avoid "talking at" situations.  Everyone has been stuck in a situation where someone is talking at you about who knows what for 45 minutes, but you feel you can't escape the situation.  The math doesn't make any sense and so you politely nod ad nausea.  Try to avoid those situations if you can or manipulate them in some way to be more beneficial.  (It's not necessarily helpful for the other person to just say words that don't mean anything to you either.)  Ask them to back up and explain something or try to move on to a different topic or just say you have to go.  This is true if you're collaborating - it's important to understand what your collaborators are talking about, and if they're good collaborators, they will explain it to you or point you where to look.  Don't let things get away from you for too long or else you might never feel like you're caught up with what's going on.  This all applies to when you're doing the explaining, which we do surprisingly more than we think.  (Just because you're new to an area, it doesn't mean your knowledge is a strict subset of all other mathematicians.)  Mike Miller Eismeier likes to stop in his explanations and say, "does that parse?".   

- Not all collaborations require having a project in mind.  Sometimes it's just valuable to have someone that you talk with about math.  This can be someone you go over papers or seminar talks with, picking a general topic to learn together, etc.  Recently, I've been trying to get in the habit of reading a paper with a collaborator when I visit.  This keeps us from getting stuck on one thing and expands our common knowledge base.  (Don't take my word for it - read the intro to this paper of Dror Bar-Natan, which helped to inspire this practice for me.)  This can then set the stage for finding a project to work on together. 

 - Collaborations should be positive overall.  If a collaboration is consistently miserable or the person is not treating you well there's a couple options, which might depend on your relationship with the person (e.g. peer, senior leader in the field, etc).  One option is stepping away from the project temporarily to reset.  You can explain that you need to focus on other pressing stuff.  (If you're going on the market soon, that's always a good excuse!)  Another thing is to just check in with your collaborators and see if there's things that can be done differently (e.g. does someone need to be doing more paper writing, working through examples, less frequent meetings, etc).  If you're stuck because of a power dynamic in the collaboration, try talking to a mentor to get some advice based on the specific situation/person.

- Talk to other people about collaborations.  There are a million different ways of collaborating (whether it's interpersonal or workflow or whether to use Overleaf vs Github).  It's good to talk to friends, mentors, and anyone else to figure out what approaches there are.  It's important to find what works best for you and your collaborators (which might vary from project to project), and sometimes you might not realize what methods there could be at your disposal.  For some people, progress really happens when everyone is at a board together.  For this, you can apply for a SQuaRE at the American Institute of Mathematics or Summer Research in Mathematics at SL Math which can fund you and your collaborators to travel to one place together to work on math.  (Some of these programs provide support for childcare, etc.)  But it might work best for everyone to work separately and just put all the pieces together.  My favorite approach to collaborations is actually going on long walks with collaborators to shake things up and get new ideas.  You never know what might work and it's really up to you to explore! 

 - Your collaborators are people too.  I have built many nice relationships with a number of my collaborators on a professional and personal level (recently I have been singing karaoke with some of them!).  You don't need to be BFF's with your collaborators (or anyone in math for that matter)...it's ok to keep relationships strictly professional.  However, it's important to keep in mind that your collaborators have their own job goals, life stressors, and mathematical values that don't necessary line up with yours and how you can balance those discrepancies.