After some discussion and a lengthy hiatus, the Polymath3 project (on attacking the polynomial Hirsch conjecture via combinatorial means) has officially started with a new research thread on Gil Kalai’s blog (which, for now, can also double as the discussion thread, given that the activity level is still quite low), and a Polymath wiki page.
September 30, 2010
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 29, 2010
The paper is focused on what I think is our best partial result, namely that the prime counting polynomial has a circuit complexity of for some absolute constant whenever and . As a corollary, we can compute the parity of the number of primes in the interval in time .
I’d be interested in hearing the other participants opinions about where to go next. Ernie has suggested that experimenting with variants of the algorithm could make a good REU project, in which case we might try to wrap up the project with the partial result and pass the torch on.
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.
October 27, 2009
The current goal is to find a deterministic way to locate a prime in an interval in time that breaks the “square root barrier” of (or more precisely, ). Currently, we have two ways to reach that barrier:
- Assuming the Riemann hypothesis, the largest prime gap in is of size . So one can simply test consecutive numbers for primality until one gets a hit (using, say, the AKS algorithm, any number of size z can be tested for primality in time .
- The second method is due to Odlyzko, and does not require the Riemann hypothesis. There is a contour integration formula that allows one to write the prime counting function up to error in terms of an integral involving the Riemann zeta function over an interval of length , for any . The latter integral can be computed to the required accuracy in time about . With this and a binary search it is not difficult to locate an interval of width that is guaranteed to contain a prime in time . Optimising by choosing and using a sieve (or by testing the elements for primality one by one), one can then locate that prime in time .
Currently we have one promising approach to break the square root barrier, based on the polynomial method, but while individual components of this approach fall underneath the square root barrier, we have not yet been able to get the whole thing below (or even matching) the square root. I will sketch the approach (as far as I understand it) below; right now we are needing some shortcuts (e.g. FFT, fast matrix multiplication, that sort of thing) that can cut the run time further.