Constructive Summer

The first week after finishing a class I am usually kind of a wreck. I stress through the end, turn in the grades, and collapse into a heap. Then there’s this weird time when I want to relax and recreate, but I don’t quite know what to do without all the responsibilities of teaching. I dream all semester about having nothing that needs to be done for tomorrow, but when it happens, I feel vaguely anxious and at loose ends. So it was a surprise this summer when that didn’t happen at all (not yet, anyway). I feel great! What was different? I think it was that, instead of determining to do absolutely nothing for a while like I usually do, I scheduled a bunch of research related meetings as soon as teaching ended. This turned out to be a great choice.

Often, serious research just sounds too daunting as I finish teaching. Probably because so much of my math research life involves being stuck. Things are going along okay, until I hit a snag. I think I can fix it one way, it doesn’t work, I try something else, it doesn’t work, and at some point I feel truly stuck. Then there is nothing to do but put it aside for a while. Maybe when I pick it back up I get a new idea, but it’s a little scary to pick it up again because I’m so stuck on it.

Skype meeting fun! Me, Katie Haymaker, and Gretchen Matthews in a Skype meeting this week.

I was willing to jump right in this summer because I actually had a few projects on which I wasn’t totally stuck (yet). The first is a collaborative project in coding theory with Katie Haymaker and Gretchen Matthews, which we’d talked about briefly in person but hadn’t had time to flesh out. Gretchen came to visit for few days right after finals week, and starting with this meeting set a great tone for me. All that energy I had been using to grade papers could suddenly go in to math. This was a brand new project, but we were starting with an existing construction and could get going right away by computing some examples. We all brought different backgrounds to the project, so what was basic for someone would be really interesting and new to someone else. By the time we’d done a bunch of examples, we had some ideas of how a paper could work and realized the kinds of things we still needed to work out. Spending several days together got us through the confusing, unsure parts. We got to ask each other questions that would have been hard to figure out alone but were often very easy for the others to answer. We got to a place where we could really start writing up our work and each add some new ideas. Score one for collaboration with great people who know very different things.

Another project I’ve worked on this summer is an expository paper, arising from an email I got a few years ago from a friend’s mom (old blog post here, follow up here). She was asking a question about passing quilts in her quilting group. This simple question led me to learn a lot about a certain kinds of Latin squares. I’d been thinking about this for a few years and finally had started a paper in the fall (again, with my colleague Katie Haymaker), but we just hadn’t had time to do anything with it for months. By the time teaching ended I was just really looking forward to it. This is my first expository paper, and it is fun to work on because I have found that I can’t really get stuck! There is investigation, organization, and writing, but there is no way to go into a mathematical black hole. Unsolved problems are just another section of the paper. Score one for expository papers.

This is the row-complete latin square modeling the above quilts.

Finally, I spent several days working on a long-term project with friends in Colorado. I had been pursuing a particular line of thinking for the last year or so, proving small lemmas and inching closer to proving a conjecture that we had made. One of my collaborators did yet another literature search, and we started reading some new papers. We made some great strides in understanding! We also discovered that our conjecture had already been proven by others. This could have been discouraging, but it fact it was great that we found this out. First of all, our conjecture was totally true! Score one for us! Also, I would have spent many more hours following my same line of inquiry, never realizing that we could just move on to the next stage of the project. Of course, that stage is hard, but hey, at least we still have all summer.

Here’s what I’m taking away from all this:

• Doing research at the beginning of summer can be really rewarding, even though it might seem daunting after a long and difficult semester.
• Again, I am reminded that collaborations really work for me. They are the most fun way for me to do math.
• I should always have at least one project that I can actively work on. By “actively work” I mean compute examples, write background, something concrete that doesn’t require me to have some enormous idea to make any progress at all. That way I can always pick up this project when everything else sounds too hard.
• Expository papers can be a good way for me to organize learning new math. Nobody has agreed to publish our expository paper yet, so it may have no real payoff-towards-tenure, but it was still really fun to write and gave me a great sense of accomplishment.
• If I can’t figure out what to actually do on a problem, it can pay off to keep searching the literature relentlessly. Maybe somebody has already solved this awful part of the problem.

This has been an excellent math summer so far. However, just so I don’t give the wrong impression that I gave up on my dream of doing nothing productive: I have also binge-read five wonderful novels, watched two full seasons of 30 Rock, listened to a lot of music (including some old Hold Steady albums, hence the blog title) and gotten some massive bruises roller skating with my sister. I have still been collapsing into a non-mathematical heap; just for smaller periods. Hooray for summer!

My totally non-mathematical summer reading, so far.

Any thoughts on the best parts of an academic summer, or the best ways to use it?