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Just a short post to note that this year’s Abel prize has been awarded jointly to Hillel Furstenberg and Grigory Margulis for “for pioneering the use of methods from probability and dynamics in group theory, number theory and combinatorics”. I was not involved in the decision making process of the Abel committee this year, but I certainly feel that the contributions of both mathematicians are worthy of the prize. Certainly both mathematicians have influenced my own work (for instance, Furstenberg’s proof of Szemeredi’s theorem ended up being a key influence in my result with Ben Green that the primes contain arbitrarily long arithmetic progressions); see for instance these blog posts mentioning Furstenberg, and these blog posts mentioning Margulis.

As part of the polymath1 project, I would like to set up a reading seminar on this blog for the following three papers and notes:

- H. Furstenberg, Y. Katznelson, “A density version of the Hales-Jewett theorem for k=3“, Graph Theory and Combinatorics (Cambridge, 1988). Discrete Math. 75 (1989), no. 1-3, 227–241.
- R. McCutcheon, “The conclusion of the proof of the density Hales-Jewett theorem for k=3“, unpublished.
- H. Furstenberg, Y. Katznelson, “A density version of the Hales-Jewett theorem“, J. Anal. Math. 57 (1991), 64–119.

As I understand it, paper #1 begins the proof of DHJ(3) (the k=3 version of density Hales-Jewett), but the proof is not quite complete, and the notes in #2 completes the proof using ideas from both paper #1 and paper #3. Paper #3, of course, does DHJ(k) for all k. For the purposes of the polymath1 project, though, I think it would be best if we focus exclusively on k=3.

While this seminar is of course related in content to the main discussion threads in the polymath1 project, I envision this to be a more sedate affair, in which we go slowly through various sections of various papers, asking questions of each other along the way, and presenting various bits and pieces of the proof. The papers require a certain technical background in ergodic theory in order to understand, but my hope is that if enough other people (in particular, combinatorialists) ask questions here (and “naive” or “silly” questions are *strongly* encouraged) then we should be able to make a fair amount of the arguments here accessible. I also hope that some ergodic theorists who have been intending to read these papers already, but didn’t get around to it, will join with reading the papers with me.

This is the first time I am trying something like this, and so we shall be using the carefully thought out protocol known as “making things up as we go along”. My initial plan is to start understanding the “big picture” (in particular, to outline the general strategy of proof), while also slowly going through the key stages of that proof in something resembling a linear order. But I imagine that the focus may change as the seminar progresses.

I’ll start the ball rolling with some initial impressions of paper #1 in the comments below. As with other threads in this project, I would like all comments to come with a number and title, starting with 600 and then incrementing (the numbers 1-599 being reserved by other threads in this project).

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