I don’t have much to say mathematically, or rather I do but now that there is a wiki associated with polymath1, that seems to be the obvious place to summarize the mathematical understanding that arises in the comments on the various blog posts here and over on Terence Tao’s blog (see blogroll). The reason I am writing a new post now is simply that the 500s thread is about to run out.
So let me quickly make a few comments on how things seem to be going. (At some point in the future I will do so at much greater length.) Not surprisingly, it seems that we have reached a stage that is noticeably different from how things were right at the beginning. Certain ideas that emerged then have become digested by all the participants and have turned into something like background knowledge. Meanwhile, the discussion itself has become somewhat fragmented, in the sense that various people, or smaller groups of people, are pursuing different approaches and commenting only briefly if at all on other people’s approaches. In other words, at the moment the advantage of collaboration is that it is allowing us to do things in parallel, and efficiently because people are likely to be better at thinking about the aspects of the problem that particularly appeal to them.
Whether there will be a problem with lack of communication I don’t know. But perhaps there are enough of us that it won’t matter. At the moment I feel rather optimistic that we will end up with a new proof of DHJ(3) (but that is partly because I have a sketch that I have not subjected to appropriately stringent testing, which always makes me feel stupidly optimistic). In fact, what I’d really like to see is several related new proofs emerging, each drawing on different but overlapping subsets of the ideas that have emerged during the discussion. That would reflect in a nice way the difference between polymath and more usual papers written by monomath or oligomath.
Finally, a quick word on threading. The largest number of votes (at the time of writing) have gone to allowing full threading, but it is not an absolute majority: those who want unrestricted threading are outnumbered by those who have voted either for limited threading or for no threading at all. I think that points to limited threading. I’ve allowed the minimum non-zero amount. I can’t force you to abide by any rules here, but I can at least give my recommendation, which is this. For polymath comments, I’d like to stick as closely as possible to what we’ve got already. So if you have a genuinely new point to make, then it should come with a new number. However, if you want to give a quick reaction to another comment, then a reply to it is appropriate. If you have a longish reply, then it should appear as a new comment, but here there is another use of threading that could be very helpful, which is to add replies such as, “I have a counterexample to this conjecture — see comment 845 below.” In other words, it will allow forward referencing as well as backward referencing. Comments on this post will start at 800, and if yours is the nth reply to comment 8**, then you should number it 8**.n. Going back to the question of when to reply to a comment, a good rule of thumb is that you should do so only if your reply very much has the feel of a reaction and not of a full comment in itself.
Also, it isn’t possible to have threading on some posts and not on others, but I’d be quite grateful if we didn’t have threaded comments on any posts earlier than this one. And a quick question: does anyone know what happens to the threaded comments if you turn the threading feature off again, which is something I might find myself wanting to do?