The mini-polymath project to find solutions to Problem 6 of the 2009 IMO is still ongoing, but I thought that, while the memories of the experience are still fresh, it would be a good time to open a parallel thread to collect the impressions that participants and observers had of how the project was conducted, how successful it was, and how it (or future projects) could be made to run more smoothly.

Just to get the ball rolling, here are some impressions I got as a (rather passive) moderator:

  1. There is no shortage of potential interest in polymath projects. I was impressed by how the project could round up a dozen interested and qualified participants in a matter of hours; this is one particular strength of the polymath paradigm.  Of course, it helped that this particular project was elementary, and was guaranteed to have an elementary (and relatively short) solution.  Nevertheless, the availability of volunteers does bode well for future projects of this type.
  2. A wiki needs to be set up as soon as possible. The wiki for polymath1 was an enormously valuable resource, once it was set up.  I had naively thought that the mini-polymath1 project would be short enough that a wiki was not necessary, but now I see that it would have come in handy for organising and storing the arguments, strategies, insights, and ideas that arose through the linear blog thread format, but which was difficult to summarise in that format.  (I have belatedly set a wiki for this project up here.)  For the next polymath project (I have none planned yet, but can imagine that one would eventually arise), I will try to ensure a wiki is available early on.
  3. There is an increasing temptation to work offline as the project develops. In the rules of the polymath projects to date, the idea is for participants to avoid working “offline” for too long, instead reporting all partial progress and thoughts on the blog and/or the wiki as it occurs.  This ideal seems to be adhered to well in the first phases of the project, when the “easy” but essential observations are being made, and the various “low-hanging fruits” are harvested, but at some point it seems that one needs to do more non-trivial amounts of computation and thought, which is still much easier to do offline than online.  It is possible that future technological advances (e.g. the concurrent editing capabilities of platforms such as Google Wave) may change this, though; also a culture and etiquette of collaborative thinking might also evolve over time, much as how mathematical research has already adapted to happily absorb new modes of communication, such as email.  In the meantime, though, I think one has accommodate both online and offline modes of thinking to make a polymath project as successful as possible, avoiding degeneration into a mass of low-quality observations on one hand, and a fracturing into isolated research efforts on the other.
  4. Without leadership or organisation, the big picture can be obscured by chaos. As I was distracted by other tasks (for instance, flying from Bremen back to Los Angeles), and had already known of a solution to the problem, I adopted a laissez faire attitude to task of moderating the project.  This worked to some extent, and there was certainly no shortage of ideas being tossed back and forth, arguments being checked and repaired, etc., but I think that with more active moderation, one could have had a bit more focus on longer-term strategy and vision than there was.  Perhaps in future projects one could be more explicit in the rules about encouraging this sort of perspective (for instance, in encouraging periodic summaries of the situation either on the blog or on the wiki).
  5. Polymath projects tend to generate multiple solutions to a problem, rather than a single solution. A single researcher will tend to focus on only one idea at a time, and is thus generally led to just a single solution (if that idea ends up being successful); but a polymath project is more capable of pursuing several independent lines of attack simultaneously, and so often when the breakthrough comes, one gets multiple solutions as a result.  This makes it harder to do direct comparison of success between polymath projects and individual efforts; from the (limited) data points available, I tentatively hypothesise that polymath projects tend to be slower, but obtain broader and deeper results, than what a dedicated individual effort would accomplish.
  6. Polymath progress is both very fast and very slow. I’ve noticed something paradoxical about these projects.  On the one hand, progress can be very fast in the sense that ideas get tossed out there at a rapid rate; also, with all the proofreaders, errors in arguments get picked up much quicker than when only one mathematician is involved.  On the other hand, it can take a while for an idea or insight obtained by one participant to be fully absorbed by the others, and sometimes the key observation can be drowned out by a large number of less important observations.  The process seems somewhat analogous to that of evolution and natural selection in biology; consider for instance how the meme of “try using induction”, which was the ultimately successful approach, had to first fight among competing memes such as “try using contradiction”, “try counting arguments”, “try topological arguments on the cube”, etc., before gaining widespread acceptance.  In contrast, an individual might through luck (or experience) hit upon the right approach (in this case, induction) very early on and end up with a solution far quicker than a polymath effort; conversely, he or she may select the wrong approach and end up wasting far more time than a polymath would.
  7. The wordpress blog format is adequate, but far from ideal. Technical problems (most notably, the spam filter, the inability to preview or edit comments [except by myself], and the (temporary) lack of nesting and automatic comment numbering) made things more frustrating and clunky than they should be.  Adding the wiki helps some of the problems, but definitely not all, especially since there is no integration between the blog and the wiki.  But the LaTeX support included in the WordPress blog is valuable, even if it does act up sometimes. Hopefully future technologies will provide better platforms for this sort of thing.  (As a temporary fix, one might set up some dedicated blog (or other forum) for polymath projects with customised code, rather than relying on hosts.)