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Two quick updates with regards to polymath projects. Firstly, given the poll on starting the mini-polymath4 project, I will start the project at Thu July 12 2012 UTC 22:00. As usual, the main research thread on this project will be held at the polymath blog, with the discussion thread hosted separately on this blog.

Second, the Polymath7 project, which seeks to establish the “hot spots conjecture” for acute-angled triangles, has made a fair amount of progress so far; for instance, the first part of the conjecture (asserting that the second Neumann eigenfunction of an acute non-equilateral triangle is simple) is now solved, and the second part (asserting that the “hot spots” (i.e. extrema) of that second eigenfunction lie on the boundary of the triangle) has been solved in a number of special cases (such as the isosceles case). It’s been quite an active discussion…

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Chris Evans has proposed a new polymath project, namely to attack the “Hot Spots conjecture” for acute-angled triangles. The details and motivation of this project can be found at the above link, but this blog post can serve as a place to discuss the problem (and, if the discussion takes off, to start organising a more formal polymath project around it).

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/

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