In a few months (the tentative target date is October), we plan to launch another polymath project (though there may also be additional projects before this date); however, at this stage, we have not yet settled on what that project would be, or even how exactly we are to select it. The purpose of this post, then, is to begin a sort of pre-pre-selection process, in which we discuss how to organise the search for a new project, what criteria we would use to identify particularly promising projects, and how to run the ensuing discussion or voting to decide exactly which project to begin. (We think it best to only launch one project at a time, for reasons to be discussed below.)
There are already a small number of polymath projects being proposed, and I expect this number to grow in the near future. Anyone with a problem which is potentially receptive to the polymath approach, and who is willing to invest significant amounts of time and effort to administrate and advance the effort, is welcome to make their own proposal, either in their own forum, or by contacting one of us. (If you do make a proposal on your own wordpress blog, put it in the category “polymath proposals” so that it will be picked up by the above link.) There is already some preliminary debate and discussion at the pages of each of these proposals, though one should avoid any major sustained efforts at solving the problem yet, until the participants for the project are fully assembled and prepared, and the formal polymath threads are ready to launch.
[One lesson we got from the minipolymath feedback was that one would like a long period of lead time before a polymath project is formally launched, to get people prepared by reading up and allocating time in advance. So it makes sense to have the outlines of a project revealed well in advance, though perhaps the precise details of the project (e.g. a compilation of the proposer's own thoughts on the problem) can wait until the launch date.]
On the other hand, we do not want to launch multiple projects at once. So far, the response to each new launched project has been overwhelming, but this may not always be the case in the future, and in particular simultaneous projects may have to compete with each other for attention, and perhaps most crucially, for the time and efforts of the core participants of the project. Such a conflict would be particularly acute for projects that are in the same field, or in related fields. (In particular, we would like to diversify the polymath enterprise beyond combinatorics, which is where most of the existing projects lie.)
So we need some way to identify the most promising projects to work on. What criteria would we look for in a polymath project that would indicate a high likelihood of full or partial success, or at least a valuable learning experience to aid the organisation of future projects of this type? Some key factors come to mind:
- The amount of expected participation. The more people who are interested in participating, both at a casual level and at a more active full time level, the better the chances that the project will be a success. We may end up polling readers of this blog for their expected participation level (no participation, observation only, casual participation, active participation, organiser/moderator) for each proposed project, to get some idea as to the interest level.
- The feasibility of the project. I would imagine that a polymath to solve the Riemann Hypothesis will be a spectacular and frustrating fiasco; we should focus on problems that look like some progress can be made. Ideally, there should be several potentially promising avenues of inquiry identified in advance; simply dumping the problem onto the participants with no suggestions whatsoever (as was done with the minipolymath project) seems to be a suboptimal way to proceed.
- The flexibility of the project. This is related to point #2; it may be that the problem as stated is beyond the ability of the polymath effort, but perhaps some interesting variant of the problem is more feasible. A problem which allows for a number of variations would be more suitable for a polymath effort, especially since polymath projects seem particularly capable of pursuing multiple directions of attack at once.
- The available time and energy of the administrator. Another thing we learned from the minipolymath project was that these projects need one or more active leaders who are willing to take the initiative and push the project in the directions it needs to go (e.g. by encouraging more efforts at exposition when the flood of ideas become too chaotic). The proposer of a project would be one obvious choice for such a leader, but there seems to be no reason why a project could have multiple such leaders (and any given participant could choose to seize the initiative and make a major push to advance the project unilaterally).
- The barriers to entry. Some projects may require a substantial amount of technical preparation before participation; this is perhaps one reason why existing projects have been focused on “elementary” fields of mathematics, such as combinatorics. Nevertheless, it should be possible (perhaps with some tweaking of the format) to adapt these projects to more “sophisticated” mathematical fields. For instance, one could imagine a polymath project which is not aimed at solving a particular problem per se, but is instead trying to understand a difficult mathematical topic (e.g. quantum field theory, to pick a subject a random) as thoroughly as possible. Given the right leadership, and sufficient interest, this very different type of polymath project could well be a great success.
- Lack of conflict with existing research. It has been pointed out that one should be careful not to let a polymath project steamroll over the existing research plans of some mathematician (e.g. a grad student’s thesis). This is one reason why we are planning an extended process to select projects, so that such clashes can be identified as early as possible, presumably removing that particular project from contention. (There is also the danger that even a proposal for a polymath project may deter other mathematicians from pursuing that problem by more traditional means; this is another point worth discussing here.)
Over to the other readers of this blog: what else should we be looking for in a polymath project? How quickly should we proceed with the selection process? Should we decide by popular vote, or by some fixed criteria?