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Tim Gowers’ post about Polymath paper published

On January 27, 2009 Tim Gowers’ blog he asked “Is massively collaborative mathematics possible?”. In the blog post he wrote,

“In short, if a large group of mathematicians could connect their brains efficiently, they could perhaps solve problems very efficiently as well.

The next obvious question is this. Why would anyone agree to share their ideas? Surely we work on problems in order to be able to publish solutions and get credit for them. And what if the big collaboration resulted in a very good idea? Isn’t there a danger that somebody would manage to use the idea to solve the problem and rush to (individual) publication?

Here is where the beauty of blogs, wikis, forums etc. comes in: they are completely public, as is their entire history. To see what effect this might have, imagine that a problem was being solved via comments on a blog post. Suppose that the blog was pretty active and that the post was getting several interesting comments. And suppose that you had an idea that you thought might be a good one. Instead of the usual reaction of being afraid to share it in case someone else beat you to the solution, you would be afraid not to share it in case someone beat you to that particular idea. And if the problem eventually got solved, and published under some pseudonym like Polymath, say, with a footnote linking to the blog and explaining how the problem had been solved, then anybody could go to the blog and look at all the comments. And there they would find your idea and would know precisely what you had contributed. There might be arguments about which ideas had proved to be most important to the solution, but at least all the evidence would be there for everybody to look at.”

So, he did just that! He started a polymath project on his blog to tackle a problem that had already been proven but not with an elementary proof, the density Hales-Jewett theorem. Specifically, “a combinatorial approach to density Hales-Jewett, is about one specific idea for coming up with a new proof for the density Hales-Jewett theorem in the case of an alphabet of size 3” which is often referred to in the blog as DHJ(3). In short, combinatorializing the ergodic-theoretic proof of DHJ(3). He wrote,

“Let me briefly try to defend my choice of problem. I wanted to choose a genuine research problem in my own area of mathematics, rather than something with a completely elementary statement or, say, a recreational problem, just to show that I mean this as a serious attempt to do real mathematics and not just an amusing way of looking at things I don’t really care about. This means that in order to have a reasonable chance of making a substantial contribution, you probably have to be a fairly experienced combinatorialist. In particular, familiarity with Szemerédi’s regularity lemma is essential. So I’m not expecting a collaboration between thousands of people, but I can think of far more than three people who are suitably qualified in the above way.”

Things kicked off February 1, 2009 and by March 10, 2009 a solution was being announced! The proof was submitted to the arXiv on October 20, 2009 and now will appear in the Annals of Mathematics.

You can watch it all unfold here: 
http://gowers.wordpress.com/category/polymath1/page/2/

Read the proof here: 
http://arxiv.org/abs/0910.3926

This is the wiki for polymath projects: 
http://michaelnielsen.org/polymath1/index.php?title=Main_Page

This is the polymath blog started by Terence Tao: 
http://polymathprojects.org/

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Rohit Gupta, who has an educational background in Chemical Engineering and describes himself as “a cosmologist who is interested in connections between physics and number theory”, is conducting an online workshop called KNK103 which is going to be a long mathematical expedition in which anyone can participate but for a fee of $100 to attempt solve the Riemann Hypothesis, proposed over 150 years ago and is one of the seven Millennium Prize Problems. Read the article here. Will this attempt at massively collaborative mathematics work? Gowers does mention this case in particular in the blog posted below:

“Now I don’t believe that this approach to problem solving is likely to be good for everything. For example, it seems highly unlikely that one could persuade lots of people to share good ideas about the Riemann hypothesis. At the other end of the scale, it seems unlikely that anybody would bother to contribute to the solution of a very minor and specialized problem. Nevertheless, I think there is a middle ground that might well be worth exploring…”

Gowers’ audience is of course professional mathematicians and that is who is referring to when he says “lots of people”. Gupta, however, is allowing non-mathematicians to participate. We will see if that $100 US will pay off. I for one won’t waste my money.

In reference to the post below, here is the 2009 blog entry about massively collaborative mathematics.

Interesting question:

“In 2009, mathematician Timothy Gowers posed this question to the blogosphere: “Is massively collaborative mathematics possible?” He described an unsolved math problem and asked for help figuring it out. Over the next few hours and days, commenters began to pick at the problem together. They brought up incomplete ideas, which were expanded and incorporated into other peoples’ ideas, until Gowers posted 37 days later that the problem had (probably) been solved.”

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