For reasons that I have already gone into, I do not think that the origin-of-life project is suitable as the next Polymath project on this blog, though there seems to be enough enthusiasm for it that I am quite serious about giving it a try at some point in the not too distant future. The complexity-related project also no longer seems such a great idea for now. That leaves three candidates from amongst the problems that I have posted about recently: the project related to the polynomial-DHJ problem, the project related to the Littlewood problem, and the project to solve Erdős’s discrepancy problem. My impression is that for each of these three projects there is already a small group of highly interested people, and certainly my level of enthusiasm for each of these three problems is enough for me to be ready to devote plenty of time to it. (A theory I want to test is that if I post regularly and am not completely stuck, then that will be enough to keep the project feeling active and attracting contributions from other people as well, even if a proof does not appear to be round the corner.)
To help me decide which of the above three problems to go for, here are four polls. The first is a straightforward choice between the three problems. However, there are drawbacks with such a poll, because the success of a project depends not just on the level of general interest, but also on finding a nucleus of people who are prepared to work seriously on the problem. So I have three more polls, designed to test, for each problem, whether there would be such a nucleus. I have tried to identify three categories of potential participation. An active and enthusiastic participant with relevant knowledge is somebody who expects to follow closely the progress of the project, to think about it hard, and to attempt, with a reasonable probability of success, to make posts that will constitute genuine advances. An occasional contributor is somebody who will not necessarily try all that hard to keep up with the discussion, but will certainly check on its progress from time to time and will make comments if they can think of useful ones without too much effort. (For example, such a contributor might happen to know the answer to a question that somebody has asked. If so, they would provide it.) An interested non-participant is somebody who would be enthusiastic about a project but would not expect to be able to make genuine mathematical contributions to it. If you feel no enthusiasm at all for a given project, then you can express that by not voting for any of the three categories. And you are of course welcome to express more detailed opinions by commenting on this post. Those who followed polymath1 will recall that there was quite a bit of discussion about the danger of steamrollering all over somebody’s existing work on a problem. I hope these problems will avoid that, but if for any reason you are anxious that a given problem should not be chosen and want to express your reservations in confidence to me, then drop me an email. If somebody has put a lot of work into a problem and appears to be close to a solution, then I will avoid treading on their toes. This may be a difficult judgment to make, but with luck I won’t in fact be called upon to make it.
One other thing is that I have not cast any votes myself: obviously I would be happy with any of the three projects and would be an active participant.