# Exciting New Blogs and New Posts

There are new exciting blogs and all sort of nice things on old blogs. In Avzel’s journal you can find a post with the words and a link to the performence of a beautiful Leonard Cohen’s song “everybody knows”;  and you can find there as well as on Computational Complexity links to some great songs of Tom Lehrer. Noam Nisan (who encouraged me to start this blog) has now a blog of his own called “Algorithmic game theory” of with interesting posts on the econ-CS interface. One post is about Fourier theoretic proof of Arrow’s theorem and the recent paper by Noam, Ehud Friedgut and me on Quantitative Gibbard Satterthwhaite theorem.  (BTW, are you aware of blogs devoted to learning?, or to mathematical programming/optimization?) Dick Lipton has a new blog  called “Godel’s lost letter and P=NP,” where he covers in a very very nice style, the major developements that occured in theoretical computer science, (and from time to time outside it,) and the people that were involved in these developements. There is a spirit of light skepticism regarding the common wisdom about some of the central problems in some of the posts. This reminds me that scientific skepticism and how it should be practiced (if at all) is a great topic for discussion. Economist Al Roth has an extremely prolofic blog called “Market design.” Roth have created (with the help of Aaron his son,) very eraly on (in 1995,) a scientifically useful home page. And finally Terry Tao had a beautiful post on sailing into the wind in higher speed than the wind. And post-finally Scott Aaronson, breaks yet again new grounds, and asks four questions for Passover about corn, rice, and wheat.

And Frank Morgan (whom both me and Noga TA’d in Calculus 18.001  18.011 in 1983) has a lovely post on optimal paths in Baseball based on work with  Davide Carozza and  Stewart Johnson, and Jeff Elly has an econ blog with the good name “cheap talk”. And here is an excellent post on the 15 greatest statisticians by Yosi Levy (in Hebrew).

And Terry Tao illuminatingly rescaled the US budget to match incomes and expenses of a single family.

# Polymath1: Success!

polymath” based on internet image search

And here is a link to the current draft of the paper.

Update:  March 26, the name of the post originally entitled “Polymath1: Probable Success!” was now updated to “Polymath1: Success!” It is now becoming clear that there are  three (perhaps four) new emerging proofs of DHJ. (April 2: See this post by Terry Tao. As this update was also based on briefly talking with Terry,  Terry’s new post gives a better description on the state of affairs and relations between the different proofs.)

The proof directly emerged from the project indeed looks conceptually different and simpler than all other proofs, and may indeed lead to the simplest known proof for Szemeredi’s theorem. (But for this we will have to wait for the details.) In addition, there is a new Ergodic proof by Tim Austin, which was partially inspired by and which used (among several other ingredients developed by Austin) some ideas and  results discovered in the polymath project. Both the original  ergodic proof and Austin’s proof were (at least roughly) “combinatorialized”.

In what sense was it a massive open collaboration? It is true that in the crucial phases of polymath, the phases where two concrete strategies for proofs were considered, the number of pivotal participants was not large. But there was an initial phase were probably more than a hundred mathematicians took part as observers and as commentators. The comments in this early phase played some role in the later developments but what is more important is that this stage have led to the (emerging) selection of the team that developed the proof. Among the hundreds, those who felt they have ideas that can be crucial, or methods that could be helpful, and were smart or lucky to be correct,  and had the persistence to follow these ideas and how these ideas can be combined with other ideas, became the pivotal players. The team that played the game was not so large, but the main massive ingredient of the project which accounts for its accessive mathematical power was in the “draft”. The team emerged from a massive number of participants. (So if you believe there were 10 pivotal players out of a hundred, think about the emergence of the team among ${{100} \choose {10}}$  possible (but, of course, not equally plausible) teams, as a point were  polymath had extra power.)

Two related posts: Tim Gowers raised inthis post  interesting questions regarding the possibility of projects were the actual number of provers will be massive. Here on my blog we have a post  with an “open discussion”  on what are the correct bounds for Roth type problems.  The emphasis is on “small-talk discussion” and not on actual “hard-nose researching”.

We took the opportunity to spend three days of “Purim” visiting northern Israel. Coming back I saw two new posts on Tim Gowers’s blog entitled “Problem solved probably” and “polymath1 and open collaborative mathematics.” It appears that “polymath1” has led to a new proof for the density version of Hales-Jewett’s theorem for k=3 which was the original central goal! Also it looks like the open collaboration mode (while not being a massive collaboration) was indeed useful.

Perhaps the most important thing is to make sure that a complete proof for the k=3 case is indeed in place (as these famous problems sometimes “fight back,” as Erdos used to say).  The outline is described here. If everything is OK as Tim and other participants expect, there are already some discussions or even plans about an extension to the general k case. This seems to be the next major step in the project.  There are also other fruits from the various threads of the polymath1 project. Overall, this looks very exciting! The mathematical result is a first-rate achievement, and the mode of cooperation is novel, interesting and appears genuinely useful.

Let me quote what Tim writes about it: “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.” sababa!

# Mathematics, Science, and Blogs

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 $k=3$ “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 $k=3$ 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 $k=2$ 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 $n$-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 $k=2$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

# Combinatorics, Mathematics, Academics, Polemics, …

My name is Gil Kalai and I am a mathematician working mainly in the field of Combinatorics.  Within combinatorics, I work mainly on geometric combinatorics and the study of convex polytopes and related objects, and on the analysis of Boolean functions and related matters. I am a professor at the Institute of Mathematics at the Hebrew University of Jerusalem and also have a  long-term visiting position at the departments of Computer Science and Mathematics at Yale University, New Haven.

Gosset polytope– a hand drawing by Peter McMullen of the plane projection of the 8-dimensional 4-simplicial 4-simple Gosset polytope. Continue reading