To a casual observer it may look as though the frequency of comments to Polymath5 is dauntingly fast — much faster than it was for Polymath1, say. However, I think appearances are misleading, because many of the comments are very short, and on the whole they are easy to understand because they are not too theoretical. I’ll give the usual roundup later in this post, but I thought I’d begin with a few remarks about the polymath process itself, because I think that Polymath5 has already demonstrated a few things that were not clear, or at least were less clear, before.
The most obvious, given the way the discussion has been going so far, is that the process is ideal for problems where there is the potential for an interplay between theory and experiment. When looking at DHJ, we looked at both theory and experiment, but the two were fairly disjoint. This was due to the nature of the problem: it is not computationally feasible to gather data about DHJ except in very low dimensions, and it is therefore difficult to draw any general conclusions from the results. But for the Erdős discrepancy problem the situation is quite different, and we have learned a huge amount from looking at long sequences that have been generated experimentally. (The main three things we have learned are that they can be surprisingly long, that if they are surprisingly long then they probably have to have some multiplicative structure, and that they have this structure in a approximate sense rather than an exact one.)
Why is this significant? Well, one of the points of polymath-style collaborations was supposed to be that it would greatly speed up the process of making and evaluating those little conjectures that you hope will shed light on a problem and, if you are lucky, turn into useful lemmas. The thought was that some people could throw out guesses, and a group of people each with slightly different expertise would be able to assess the plausibility of those guesses much more quickly than any single individual. So both the generation and evaluation of the guesses would happen much more quickly. But here we have a further speed-up, which is that the experimental evidence already alters the prior plausibility of guesses. For instance, one can conjecture with some confidence that an infinite sequence with bounded discrepancy must have a lot of multiplicative structure: before, that would have been much more speculative. (The fact that random sequences fail miserably suggests that some structure is needed, so I suppose I should add that the key word above is “multiplicative”.)
A second difference between Polymath5 and Polymath1 is that there is an additional tool available now, namely mathoverflow. This may sound like a minor addition, but I think it could turn out to be highly significant. Several people noted that the difficulty of keeping up with the discussion was a notable barrier to participation in Polymath1. Presumably this meant that even if some of the questions that were being asked were self-contained, the people who might know the answers to those questions would not necessarily have the time or inclination to search for them (especially as they might not expect to find them). But now, every time a self-contained question arises that seems as though it might be doable by somebody with the right knowledge, it can be posted on mathoverflow, where people can answer it without having to follow what’s going on in the wider discussion. For the purposes of giving credit where it is due, it might be good to have a page on the wiki that lists all questions related to the project that have been asked on mathoverflow and briefly describes their status. (One can also see what they are by using the polymath5 tag at mathoverflow itself.) Needless to say, any crucial help that we get from mathoverflow will be appropriately acknowledged in any paper that might result from the project.
I find it interesting that the wiki for Polymath5 has a rather different flavour from that of the wiki for Polymath1: it is more focused on the discussion itself and less on background knowledge. It will be interesting to see whether when the more theoretical discussion begins the character of the wiki will change.
A small difference between Polymath5 and Polymath1 is that we are (so far at least) not numbering the comments, and instead are making full use of the threading, which I am still allowing only up to depth 1. I think this has worked rather well, especially when people provide links to comments that they refer to. It means that some of the chronological feel of the discussion, which I think is important, is preserved, but it is also split to some extent into mini-discussions, so that when one looks back at the comments later they are better organized. If we can keep this going, and are also assiduous about putting useful information (experimental results, observations, lemmas, outlines of proof strategies, etc.) on the wiki, then I hope that the barrier to late entry will be much lower than it was for Polymath1.
Returning to the mathematics, in the last few hours it has emerged that there is a striking pattern in the original sequence of length 1124. I am so interested in it that I don’t really have the energy to write about anything else that has been discussed in the last 100 comments, but perhaps I’ll add a brief summary later. (Right now I want to get this post out fast because when I last looked there were 99 comments and probably there are 100 by now, or at any rate will be very soon.)
I now think it likely that we will come up with a formula that gives something very close to the 1124 sequence. If so, then I am keeping my fingers very firmly crossed that it will give an infinite sequence with sublogarithmic discrepancy. But we shall see: some computation is still needed before we will know what the formula is. There is also a chance that we will obtain a formula but with parameters that have to be chosen for each prime, and that we will not be able to see how to choose them in general. In that case, we may at least obtain a very efficient algorithm for finding extremely long sequences of low discrepancy.