After my tentative Polymath proposal, there definitely seems to be enough momentum to start a discussion “officially”, so let’s see where it goes. I’ve thought about the question of whether to call it Polymath11 (the first unclaimed number) or Polymath12 (regarding the polynomial-identities project as Polymath11). In the end I’ve gone for Polymath11, since the polynomial-identities project was listed on the Polymath blog as a proposal, and I think the right way of looking at things is that the problem got solved before the proposal became a fully-fledged project. But I still think that that project should be counted as a Polymathematical success story: it shows the potential benefits of opening up a problem for consideration by anybody who might be interested.
Something I like to think about with Polymath projects is the following question: if we end up not solving the problem, then what can we hope to achieve? The Erdős discrepancy problem project is a good example here. An obvious answer is that we can hope that enough people have been stimulated in enough ways that the probability of somebody solving the problem in the not too distant future increases (for example because we have identified more clearly the gap in our understanding). But I was thinking of something a little more concrete than that: I would like at the very least for this project to leave behind it an online resource that will be essential reading for anybody who wants to attack the problem in future. The blog comments themselves may achieve this to some extent, but it is not practical to wade through hundreds of comments in search of ideas that may or may not be useful. With past projects, we have developed Wiki pages where we have tried to organize the ideas we have had into a more browsable form. One thing we didn’t do with EDP, which in retrospect I think we should have, is have an official “closing” of the project marked by the writing of a formal article that included what we judged to be the main ideas we had had, with complete proofs when we had them. An advantage of doing that is that if somebody later solves the problem, it is more convenient to be able to refer to an article (or preprint) than to a combination of blog comments and Wiki pages. (more…)