Two great Peter Frankls
Tim Gowers launched polymath 11 aimed at Peter Frankl’s conjecture asserting that for every union-closed family there is an element that belongs to at lease half the sets in the family. Here are links to Post number 0 and Post number 1. (Meanwhile polymath10 continues to run on “Combinatorics and More.”)
Here is the link to a mathoverflow question asking for polymath proposals. There are some very interesting proposals. I am quite curious to see some proposals in applied mathematics, and various areas of geometry, algebra, analysis and logic.
Dinesh Thakur David Speyer
A beautiful polymath proposal by Dinesh Thakur was posted by Terry Tao on the this blog. The task was to explain some remarkable, numerically observed, identities involving the irreducible polynomials in the polynomial ring over the finite field of characteristic two. David Speyer managed to prove Thakur’s observed identities! Here is the draft of the paper. Congratulations to Dinesh and David!
Polymath10 has started on my blog. The aim is to prove the Erdos-Rado sunflower conjecture (also known as the delta-system conjecture). Here is the wikipage.
The main objectives of the polymath8 project, initiated by Terry Tao back in June, were “to understand the recent breakthrough paper of Yitang Zhang establishing an infinite number of prime gaps bounded by a fixed constant , and then to lower that value of as much as possible.”
Polymath8 was a remarkable success! Within two months the best value of H that was 70,000,000 in Zhang’s proof was reduced to 5,414. Moreover, the polymath setting looked advantageous for this project, compared to traditional ways of doing mathematics. (I have written a post with some more details and thoughts about it, looked from a distance.)
An update on the status of the Polymath4 paper on finding primes. I’ve received a referee report from Mathematics of Computation on the submission, which can be found here. The referee liked the result but wanted a fair number of expository changes before he or she was willing to recommend acceptance, so the editor has asked for a revision. I will be happy to make the relevant changes, but if there are any other changes that other participants would like to make, now would be a good time to suggest them. (The most recent version of the paper can be found at the Subversion repository or at this link; see also the arXiv version.)
One change requested is to add a list of participants to the project. In analogy with what we did for Polymath1, I therefore started a “signup sheet” on the wiki at
for people to self-report their participation, contact information, and grant information for the project. There is the usual problem of trying to decide who is a “main participant” of the project, and who is a “contributor” (though I think I can safely add Ernie, Harald, and myself as participants); as with Polymath1, I will leave it to each of you to self-report what level of participation you feel is appropriate.
After some discussion and a lengthy hiatus, the Polymath3 project (on attacking the polynomial Hirsch conjecture via combinatorial means) has officially started with a new research thread on Gil Kalai’s blog (which, for now, can also double as the discussion thread, given that the activity level is still quite low), and a Polymath wiki page.
I am proposing the sixth question for the 2010 International Mathematical Olympiad (traditionally, the trickiest of the six problems) as a mini-polymath project for next month. Details and discussions are in this post on my other blog.
[Update, June 27: the project is scheduled to start on Thursday, July 8 16:00 UTC.]
Timothy Gowers and Michael Nielsen have written an article “Massively collaborative mathematics“, focusing primarily on the first Polymath project, for the October issue of Nature.