On productivity in math

Inspired by Terry Tao's career advice page and by a recent interview on Math-Life balance, I collected here some thoughts on productivity in math. The recent interview was with Saul Glasman and in it he mentioned, more as a side note, that math research is hard because one often gets stuck while working on a problem. It's just very hard to focus for a long time on a single problem, seemingly without any progress. But the crucial insight here is, in my opinion, the fact that it only seems as if there is no progress.

According to Glasman, when it happens that you work too long on a problem and then take a break, you can't really get yourself to work on it again and start feeling bad because you think you must work on that one problem until there is a breakthrough. This leads to a downward spiral because you think that it is your job to work on or solve this problem and you can't even get yourself to do your job. But this is probably false and Terry Tao knew this for a long time.

This is wrong because:

1. nobody tells you which problem you should solve
2. there rarely is a discrete single step from not solved to solved when working on a problem (i.e. you take many steps and can publish intermediate results)
3. When you run out of ideas on one problem then you can work on another one

Terry Tao urges you to be flexible and to try to not announce too early which problems you're working on:

Another corollary is that it is generally not a good idea to announce that you are working on a well-known problem before you have a feasible plan for solving it, as this can make it harder to gracefully abandon the problem and refocus your attention in more productive directions in the event that the problem is more difficult than anticipated. From https://terrytao.wordpress.com/career-advice/be-flexible/

Another important topic that Tao mentions is to stay in (or only slightly out of) one's own comfort zone.

I believe that the optimal way to develop one’s talents is to invest in the middle ground between these two extremes, thus adding new challenges and difficulties to your research program in carefully controlled amounts. Examples of such research objectives include:

1. Looking at the easiest problems of interest that you can’t quite completely handle with your existing tools, for instance by taking an unsolved problem and making various assumptions to “turn off” all but one of the difficulties;
2. Taking a known result and reproving it by “tying one hand behind your back”, by forbidding yourself to use a method which is effective for that result, but does not extend well to more difficult problems; or
3. Taking a known result and generalising it to a situation in which most of the steps in the standard proof of the existing result look like they will extend, but which have just one or two parts which look tricky and will require some modest new idea, trick or insight. [...] This is somewhat analogous to exploiting the power of compound interest in long-term investing; imagine, for instance, what your mathematical abilities would be like in a couple decades if you were able to improve your range by, say, 10% a year. From https://terrytao.wordpress.com/career-advice/continually-aim-just-beyond-your-current-range/

Tao also has advice for what to do when one does not want to work on that major problem that one has not managed to solve in spite of so many (fruitless) attempts. These are Learn and relearn your field and ask yourself dumb questions (and answer them). These approaches provide you with an endless supply of new problems and exercises to work that also make you feel more productive. But it is also important to pick each task according to the energy level of the day (or hour), as he explains in his thoughts on time management. And with all things you're working, be it big, important or small (and ultimately equally important) problems: always value partial progress to keep motivation up