November 13, 2011
May 12, 2011
Richard Lipton has just proposed on his blog to discuss the following conjecture of Erdos as a polymath project: that there are no natural number solutions to the equation
with . Previous progress on this problem (including, in particular, a proof that any solution to this equation must have an extremely large value of , and specifically that ) can be found here.
February 5, 2011
For the time being this is an almost empty post, the main purpose of which is to provide a space for mathematical comments connected with the project of assessing whether it is possible to use the recent ideas of Sanders and of Bateman and Katz to break the barrier in Roth’s theorem. (In a few hours’ time I plan to write a brief explanation of what one of the main difficulties seems to be.)
Added later. Tom Sanders made the following remarks as a comment. It seems to me to make more sense to have them as a post, since they are a good starting point for a discussion. So I have taken the liberty of upgrading the comment. Thus, the remainder of this post is written by Tom.
This will hopefully be an informal post on one aspect of what we might need to do to translate the Bateman-Katz work into the setting.
One of the first steps in the Bateman-Katz argument is to note that if is a cap-set (meaning it is free of three-term progressions) of density then we can assume that there are no large Fourier coefficients, meaning
They use this to develop structural information about the large spectrum, , which consequently has size between and . This structural information is then carefully analysed in the `beef’ of the paper. (more…)
July 8, 2010
This post marks the official opening of the mini-polymath2 project to solve a problem from the 2010 IMO. I have selected the fifth question (which appears to be slightly more challenging than the sixth, for a change) as the problem to focus on:
Problem. In each of six boxes there is initially one coin. There are two types of operation allowed:
- Type 1: Choose a nonempty box with . Remove one coin from and add two coins to .
- Type 2: Choose a nonempty box with . Remove one coin from and exchange the contents of (possibly empty) boxes and .Determine whether there is a finite sequence of such operations that results in boxes being empty and box containing exactly coins. (Note that .)
- All are welcome. Everyone (regardless of mathematical level) is welcome to participate. Even very simple or “obvious” comments, or comments that help clarify a previous observation, can be valuable.
- No spoilers! It is inevitable that solutions to this problem will become available on the internet very shortly. If you are intending to participate in this project, I ask that you refrain from looking up these solutions, and that those of you have already seen a solution to the problem refrain from giving out spoilers, until at least one solution has already been obtained organically from the project.
- Not a race. This is not intended to be a race between individuals; the purpose of the polymath experiment is to solve problems collaboratively rather than individually, by proceeding via a multitude of small observations and steps shared between all participants. If you find yourself tempted to work out the entire problem by yourself in isolation, I would request that you refrain from revealing any solutions you obtain in this manner until after the main project has reached at least one solution on its own.
- Update the wiki. Once the number of comments here becomes too large to easily digest at once, participants are encouraged to work on the wiki page to summarise the progress made so far, to help others get up to speed on the status of the project.
- Metacomments go in the discussion thread. Any non-research discussions regarding the project (e.g. organisational suggestions, or commentary on the current progress) should be made at the discussion thread.
- Be polite and constructive, and make your comments as easy to understand as possible. Bear in mind that the mathematical level and background of participants may vary widely.
June 12, 2010
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.]
January 9, 2010
Is taking place on Gowers’s blog!
December 31, 2009
For a description of Erdos’ discrepancy problem and a large discussion see this blog on Gowers’s blog.
The decision for the next polymath project over Gowers’s blog will be between three projects: The polynomial DHJ problem, Littlewood problem, and the Erdos discrepency problem. To help making the decision four polls are in place!
November 20, 2009
Tim Gowers described two additional proposed polymath projects. One about the first unknown cases of the polynomial Density Hales Jewett problem. Another about the Littelwood’s conjecture.
I will state one problem from each of these posts:
1) (Related to polynomial DHJ) Suppose you have a family of graphs on n labelled vertices, so that we do not two graphs in the family such that is a subgraph of and the edges of which are not in form a clique. (A complete graph on 2 or more vertices.) can we conclude that =? (In other words, can we conclude that contains only a diminishing fraction of all graphs?)
Define the “distance” between two points in the unit cube as the product of the absolute value of the differences in the three coordinates. (See Tim’s remark below.)
2) (Related to Littlewood) Is it possible to find n points in the unit cube so that the “distance” between any two of them is at least ?
A negative answer to Littlewood’s problem will imply a positive answer to problem 2 (with some constant). So the pessimistic saddle thought would be that the answer to Problem 2 is yes without any bearing on Littlewood’s problem.
November 8, 2009
A presentation of one possible near future polymath: the mathematics of the origin of life can be found on Gowers’s blog.
July 27, 2009
Another proposal for a polymath project is the following question of Michael Boshneritzan:
Question. Let be a (simple) path in a lattice which has bounded step sizes, i.e. for some C and all i. Is it necessarily the case that this path contains arbitrarily long arithmetic progressions, i.e. for each k there exists with r non-zero such that .
The d=1 case follows from Szemerédi’s theorem, as the path has positive density. Even for d=2 and k=3 the problem is open. The question looks intriguingly like the multidimensional Szemerédi theorem, but it is not obvious how to deduce it from that theorem. It is also tempting to try to invoke the Furstenberg correspondence principle to create an ergodic theory counterpart to this question, but to my knowledge this has not been done. There are also some faint resemblances to the angel problem that has recently been solved.
In honour of mini-polymath1, one could phrase this problem as a grasshopper trying to jump forever (with bounded jumps) in a square lattice without creating an arithmetic progression (of length three, say).
It is also worth trying to find a counterexample if C, d, k are large enough. Note that the continuous analogue of the problem is false: a convex curve in the plane, such as the parabola , contains no arithmetic progressions, but is rectifiable. However it is not obvious how to discretise this example.
In short, there seem to be a variety of tempting avenues to try to attack this problem; it may well be that many of them fail, but the reason for that failure should be instructive.
Andrew Mullhaupt informs me that this question would have applications to short pulse rejected Boolean delay equations.