Michael Nielsen wrote a lovely essay entitled “Doing science online” about mathematics, science, and blogs. Michael’s primary example is a post over Terry Tao’s blog about the Navier-Stokes equation and he suggests blogs as a way of scaling up scientific conversation. Michael is writing a book called “The Future of Science.” He is a strong advocate of doing science in the open, and regard these changes as truly revolutionary. (The term “Science 2.0″ is mentioned in the remarks.)
Michael’s post triggered Tim Gowers to present his thoughts about massive collaboration in mathematics, and this post is also very interesting with interesting follow-up remarks. Tim Gowers mentioned the n-category cafe as a place where a whole research programme is advanced on a blog. Terry Tao mentioned comments on posts on his open-problems series as having some value. He mentioned, in particular, the post on Mahler’s conjecture. (Also I think some discussions over Scott Aaronson’s blog had the nature of discussing specific technical math (coming from CS) problems.)
Tim actually proposes an experiment: trying to solve collectively a specific math problem. This would be interesting!!! I suppose we need to give such an effort over a blog a longer than ususal life-span – a few months perhaps. (And maybe not to start with a terribly difficult problem.) (What can be an appropriate control experiment though?)
Ben Webster in “Secret blogging seminar” mentioned, in this context, earlier interesting related posts about “Working in secret“.
Christian Elsholtz mentioned on Gowers’s blog an intermediate problem (called “Moser’s cube problem”) where you look not for combinatorial lines (where the undetermined coordinates should be 1 in x 2 in y and 3 in z), and not for an affine line (where it should be 1,2,3 in x y and z in any order), but for a line: it can be 1 in x 2 in y and 3 in z or 3 in x 2 in y and 1 in z.
Update: Things are moving fast regarding Gowers’s massive collaboration experiment. He peoposes to study together a new approach to the “density Hales -Jewett theorem”. A background post apears here. Hillel Furstenberg and Izzy Katznelson’s proof of the density Hales-Jewett theorem was a crowning achievement of the ergodic theory method towards Szemeredi’s theorem. Like the case for Furstenberg’s proof of Szemeredi’s theorem itself the case was considerably simpler and had appeared in an earlier paper by Hillel and Izzy. The recent extensions of Szemeredi regularity lemma that led to simpler combinatorial proof of Szemeredi’s theorem did not led so far to simpler proofs for the density Hales-Jewett case. If you look at Tim’s background post let me ask you this: What is the case of the density Hales-Jewett’s theorem? It is something familiar that we talked about!
Here is a particularly silly problem that I suggested at some point along the discussions: How large can a family of subsets of an -element set be without having two sets A and B such that the number of elements in A but not in B is twice the number of elements in B but not in A?
Update: This problem was completely reolved by Imre Leader and Eoin Long, their paper Tilted Sperner families contains also related results and conjectures.
Massive collaboration in art (click the picture for details)
Q: What is the case of the density Hales-Jewett’s theorem? A: It is Sperner’s theorem! (that we discussed in this post.)
I will keep updating about news from Tim’s project. [Last update is from October 21]. More updates: Tim’s project is getting quickly off the ground. A useful wiki was established. More update: It is probably successful!
There are several new posts in Gowers’s blog describing the project, its rules and motivation, and interesting discussions also over What’s new and in In theory. The project itself is fairly focused; Let me mention another connection which is a little beside its defined scope. Combinatorial lines for are simply three vectors which in some coordinates they agree, and in some others they are ’1′ in ’2′ in and ’3′$ in . If instead you ask that form an affine line or, equivalently an arithmetic progression (in ) getting a density theorem is easier. This just means that in every coordinate where x y and z are different one of them is ’1′ one is ’2′ and one is ’3′ but you don’t insist which is which. (There is a famous problem regarding the density needed.) This problem is reviwed by Terry Tao here. I am not sure if the regularity lemma approach works for this problem; Hillel and Izzy proved using the ergodic theory appproach density results for affine lines before they moved to the more complicated (less structured) case of combinatorial lines.
And more:there are already almost 100 comments in Gowers’s main post. I will try to keep updating about some highlights in this post from time to time; in my opinion, a good time scale to examine it will be, say, once every 1/2 year. I find it interesting in various ways, and exciting, and nevertheless I am also somewhat skeptical about some of the perhaps too strong and too romantic interpretations.
And more (7/2): The discussion is now divided into three threads. The original project of finding a proof of density Hales Jewett using some form of a regularity lemma is continued as a seperated thread. (The plan is much more detailed than that.) The same thread also discusses issues related to Sperner’s theorem which is sort of a baby case for HJT. Terry Tao hosts a thread about upper and lower bounds on DHJ and related problems. A third thread about obstructions to uniformity and density-increment strategies is forthcoming alive and kicking on Gowers’s blog.
And more (2/3): The discussion moves on in several directions. A wiki was built to describe some background, variants, approaches, proofs of partial results, approaches, links to the original threads and more. (Also a sort of time-table.) Tim Gowers launched a slower-going polymath2 aimed to reach a useful notion of “explicitely defined” Banach space. A well-known example of Tsirelson (described in the post) is the archi example of a “non explicitely defined” space and Tim wants to collectively reach a conjecture that “explicitely defined Banach spaces” contain some simple classical Banach spaces. (Now if Tim will launch polymath3 dealing with expanders, property T, growths in groups, and the congruence subgroup problem these three projects will come close of covering major interests of many of my colleagues here.)
Sort of update (July 2009): Tim Gowers plans to propose a list of ten problems for one to be chosen in a polymath project in October; A polymath3 dedicated to the Hirsch conjecture is proposed over this blog. A mini polymath is taking place on Terry Tao’s blog. Terry set (with Tim Gowers Michael Nielsen and me) a new polymath blog. Probably I will stop updating at this post. When the project starts I was both skeptical and enthusiastic; The success of polymath1 suprised me, it went well beyond my expectations. I think we should remain skeptical and enthusiastic. (August 2009) polymath4 dedicated to finding primes deterministically was launched over the polymathblog. (Polymath4 was very active for several months. It led to some fruitful discussions but not to a definite result.) (September 2009) Tim Gowers wrote a preliminary post about possible problems for the October 2009 polymath.
(October 09 ) The paper with the DHJ proof is now uploaded. Ryan O’donnell who preform the lion share of writing left a little mystery: what Varnavides’ first name was? (A theorem of Varnavides’s played a role in the new proof and the only reference to his first name was the initial P. ) Terry Tao asked it on his blog and in a short time the mystery was resolved when Thomas Sauvaget found the answer. The next day another participant Andreas Varnavides wrote: “Yes his first name was Panayiotis, born in Paphos Cyprus. He was my uncle.”
(January 2010) Polymath5 devoted to Erdos discrepency problem is launched on Gowers’s blog.
(January 2010) A draft of a second paper devoted to the study of DHJ and Moser numbers can be found here.