Update (Jan 21)
j) Polymath11 (?) Tim Gowers’s proposed a polymath project on Frankl’s conjecture. If it will get off the ground we will have (with polymath10) two projects running in parallel which is very nice. (In the comments Jon Awbrey gave a links for a first in a series posts also on Frankl’s conjecture, with the catchy title, Frankl my dear.)
a) NogaFest started a few days ago. It is a wondeful meeting! My lecture entitled “polymath” refers to the older meaning of the word, so appropriate to describe Noga. (I was not aware that the word has a meaning until recently). I talked, among other things, about polymath10. I prepared the talk a week ahead and presented our Conjectures A and B (from polymath10 last post) hoping that perhaps I could add some positive information toward them. Well, just after my presentation was ready, I realized that Conjecture B is false. Here are the slides.
Two quotes from the lecture. First about the birthday boy: the idea of the polymath was expressed by Leon Battista Alberti (1404–1472), in the statement, most suitable to Noga “a man (who) can do all things if he will”. Second, about polymath projects (by Gowers): “a large collaboration in which no single person has to work all that hard.”
b) 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 (perhaps mainly to see what researchers regard as central major projects,) in applied mathematics, and various areas of geometry, algebra, analysis and logic.
c) A very nice polymath proposal by Dinesh Thakur was posted by Terry Tao on the polymath blog. The task was to explain some 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.
d) This reminds me that some years ago David Speyer solved a question that interested me for decades and was presented here on the blog and later on MathOverflow about systems of skew lines in three dimensional vector spaces over division rings (and especially the Quaternions).
e) Related to polymath3, let me mention that Michael Todd proved a small but very elegant improvement of the upper bounds by Kleitman and me from 1992. (The new bound is . The first improvement, I think, in two decades!
f) I have a very nice thing to tell you about polymath4! Shafi Goldwasser abstract for Nogafest talked about a new notion of randomized algorithms: A randomized algorithm to achieve a certain task (for example to find a perfect matching in a graph,) which is guaranteed to reach the same answer with high probability! Such an algorithm is called pseudo-deterministic. It is both an amazing concept, and it is quite amazing that it was not introduced before. The polymath4 question was to find deterministically a prime with n digits and A new challenge (that Shafi asks about) is to find a pseudo-deterministic efficient algorithm. Namely, a randomized algorithm which will find an n-digit prime, but with high probability the same one! (I would guess that it is still hopeless.)
g) And Terry Tao gave a beautiful lecture on Erdos discrepancy problem (the topic of polymath5). I understood a little better the argument (which is similar to Roth density increase argument for 3 term AP,) that allows Tao to use the logarithmically-averaged Chowla inequalities.
h) The old conjecture that centrally symmetric convex sets have nonnegative correlation w.r.t. the Gaussian distribution was proved! Let me refer you to the paper Royen’s proof of the Gaussian correlation inequality for a simple exposition of a proof by Thomas Royen, and more information on the solutions and solvers.
i) The Nogafest participants are invited to a Jazz night at Gilly’s
The third Polymath10 post is active. I hope to post a new polymath10 post in about 1-2 weeks. I hope also to return to various amazing things I am hearing on Nogafest and other places (and also on my ownfest some months ago).