Archive for the ‘polymath1’ Category

Polymath paper published

April 23, 2012

I’m glad to be able to report that “A new proof of the density Hales-Jewett theorem” has recently appeared in Annals of Mathematics. Unfortunately it’s behind a paywall, but you can find an almost final version on the arXiv.

I might add that my enthusiasm for this way of working is undimmed. The reason there has been no Polymathematical activity on this blog for quite a while is that I’ve been busy with more conventional projects, but in the not too distant future I’d like to do some more open research. Also, Gil Kalai and I have a plan to try soon to revive the EDP project. I won’t say any more about that now, but it seems a good moment to mention it.

DHJ latest

February 17, 2010

A quick post to say that earlier today I put a new version of the write-up of Polymath1’s proof of the density Hales-Jewett theorem on the arXiv. Very soon it will be submitted for publication. I will not say more at this stage (since I don’t want the journal to evaluate the paper in the public eye) but will report back when I know whether it has been accepted.

Update: it is now submitted.

Miscellaneous matters

October 20, 2009

Michael Nielsen and I have written an Opinion Piece for Nature about the Polymath project and related matters. Thanks almost entirely to Ryan O’Donnell, a preprint at last exists that contains Polymath’s proof of the density Hales-Jewett theorem with all the details. It will be posted on the arXiv very soon and I will update this post when it is.

Update: it can be found here. Owing to a misunderstanding, it was posted before I had any input into it, but in any case, the full proof is here, even if the version that is submitted for publication will have some changes.

The Notices of the AMS have published five back-to-back reviews of the Princeton Companion to Mathematics. They are by Bryan Birch, Simon Donaldson, Gil Kalai, Richard Kenyon and Angus Macintyre.

From Quomodocumque I learned of a new website, Math Overflow, where you can ask and answer mathematical questions. It seems to be very active, with a lot of users, rating systems for comments and commenters, and the like. So in principle it could be another mechanism for pooling the resources of mathematicians with the help of the internet. For example, if you need a certain statement to be true and do not know whether it is known, then my guess is that you could find out pretty quickly if you post a question there. For more discussion, see a post over at the Secret Blogging Seminar.

DHJ write-up and other matters

June 25, 2009

This short post is in response to Jozsef Solymosi’s request for a new DHJ thread, since the previous one has become rather long and unwieldy. We’ve stopped numbering comments now, and the main purpose of the post is so that people can continue the discussion of the write-up of the proof of DHJ(k). Thanks mostly to the efforts of Ryan O’Donnell, we now have a complete draft. See also this write-up of DHJ(3) by Jozsef.

While I’m writing, I thought I’d take the opportunity to say that I am not intending to post much over the next two or three months, either here on on the Tricki. That’s because I have three more or less completed research projects that need to be properly finished (one of which is DHJ) and I owe it to my coauthors to get them done. So the plan is to clear my backlog over the summer and then come back, refreshed and ready to go, in the autumn. At that point I plan several Tricki articles (more advanced than most of the ones I’ve written so far). I also plan to start a new polymath project. Or rather, I have a file in which I have written plans for ten polymath projects, so what I’ll probably do is explain briefly what they are and try to get some idea of what appeals to people most. I am excited about several of these possible projects, so whatever we do I will be disappointed about the ones we don’t do. I may well have an online vote about it, but first I have to decide what the results of the vote will be.

Can Polymath be scaled up?

March 24, 2009

As I have already commented, the outcome of the Polymath experiment differed in one important respect from what I had envisaged: though it was larger than most mathematical collaborations, it could not really be described as massive. However, I haven’t given up all hope of a much larger collaboration, and in this post I want to think about ways that that might be achieved.

First, let me say what I think is the main rather general reason for the failure of Polymath1 to be genuinely massive. I had hoped that it would be possible for many people to make small contributions, but what I had not properly thought through was the fact that even to make a small contribution one must understand the big picture. Or so it seems: that is a question I would like to consider here. (more…)

DHJ(3) and related results: 1050-1099

March 16, 2009

I’m not sure how many more comment threads we will have, but we are running out of the 1000-1049 thread, so it’s time for a new one. The main news to report since the last post is that progress is being made on writing up the proof of DHJ(3) and DHJ(k). At the moment it is more like preparatory sketches, but they are pretty detailed and can, as usual, be found on the wiki. It looks as though some of the more technical parts will end up very streamlined thanks to work of Ryan O’Donnell: the final proof of DHJ(k) should be surprisingly short (though I hope that we will write it up with plenty of accompanying explanation so that it is not too compressed and hard to understand).

Polymath1 and open collaborative mathematics

March 10, 2009

In this post I want to discuss some general issues that arise naturally in the light of how the polymath experiment has gone so far. First, let me say that for me personally this has been one of the most exciting six weeks of my mathematical life. That is partly because it is always exciting to solve a problem, but a much more important reason is the way this problem was solved, with people chipping in with their thoughts, provoking other people to have other thoughts (sometimes almost accidentally, and sometimes more logically), and ideas gradually emerging as a result. Incidentally, many of these ideas are still to be properly explored: at some point the main collaboration will probably be declared to be over (though I suppose in theory it could just go on and on, since its seems almost impossible to clear up every interesting question that emerges) and then I hope that the comments will be a useful resource for anybody who wants to find some interesting open problems. (more…)

Problem solved (probably)

March 10, 2009

Without anyone being particularly aware of it, a race has been taking place. Which would happen first: the comment count reaching 1000, or the discovery of a new proof of the density Hales-Jewett theorem for lines of length 3? Very satisfyingly, it appears that DHJ(3) has won. If this were a conventional way of producing mathematics, then it would be premature to make such an announcement — one would wait until the proof was completely written up with every single i dotted and every t crossed — but this is blog maths and we’re free to make up conventions as we go along. So I hereby state that I am basically sure that the problem is solved (though not in the way originally envisaged).

Why do I feel so confident that what we have now is right, especially given that another attempt that seemed quite convincing ended up collapsing? Partly because it’s got what you want from a correct proof: not just some calculations that magically manage not to go wrong, but higher-level explanations backed up by fairly easy calculations, a new understanding of other situations where closely analogous arguments definitely work, and so on. And it seems that all the participants share the feeling that the argument is “robust” in the right way. And another pretty persuasive piece of evidence is that Tim Austin has used some of the ideas to produce a new and simpler proof of the recurrence result of Furstenberg and Katznelson from which they deduced DHJ. His preprint is available on the arXiv.

Better still, it looks very much as though the argument here will generalize straightforwardly to give the full density Hales-Jewett theorem. We are actively working on this and I expect it to be done within a week or so. (Work in progress can be found on the polymath1 wiki.) Better even than that, it seems that the resulting proof will be the simplest known proof of Szemerédi’s theorem. (There is one other proof, via hypergraphs, that could be another candidate for that, but it’s slightly less elementary.)

I have lots of thoughts about the project as a whole, but I want to save those for a different and less mathematical post. This one is intended to be the continuation of the discussion of DHJ(3), and now DHJ(k), into the 1000s. More precisely, it is for comments 1000-1049.

DHJ(3): 851-899

March 2, 2009

Once again there is not a huge amount to say in this post. Since the last post there have been a few additions to the polymath1 wiki that may be of some use. In particular, there is now a collection of fairly complete write-ups of related results (see the section entitled “Complete proofs or detailed sketches of potentially useful results”) to which I hope we will add soon. Also on the wiki is an account of the Ajtai-Szemerédi proof of the corners theorem, which seems to have some chance of serving as a better model for a proof of DHJ(3) than the proof via the triangle-removal lemma. Meanwhile, progress has been made in understanding and to some extent combinatorializing the ergodic-theoretic proof of DHJ(3), ideas from which have fed into the discussion. As with the last post, this one is mainly to stop the number of comments getting too large. We’re now down to 50 comments per post (except that it was 51 for the last one and will be 49 for this), since with the new threading we seem to be averaging at least one reply per comment.

Brief review of polymath1

February 23, 2009

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. (more…)