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Background

Polymath projects are a form of open Internet collaboration aimed towards a major mathematical goal, usually to settle a major mathematical problem. This is a concept introduced in 2009 by Tim Gowers and is in line with other forms of Internet mathematical research activity which include MathOverflow.

Former and current projects

The polymath wiki page gives a description and links to former polymath projects and much additional information. So far, there were about 10 polymath projects of which 6-7 led to intensive research, and among those 3-4 were successful. (There were several MathOverflow questions motivated by running polymath projects, especially questions related to polymath5.) Those projects ran over Gowers's blog (polymath1, polymath5 and others), Tao's blog (polymath8), the Polymath Blog (administered by Tao, Gowers, Nielsen, and me) (polymath4 and polymath7), and my blog (polymath3).

Updates (Before Nov 2016) There were a couple of additional polymath-type projects. (Nov '15, 2016) Currently, polymath10 on Erdős-Rado delta system conjecture is running on my blog.(New, Dec 29, '15) Terry Tao posted (on behalf of Dinesh Thakur) an interesting proposal for a polymath project regarding identities for irreducible polynomials Update: problem solved by David Speyer. ( January 31, 2016) Tim Gowers launched on his blog polymath11 on Frankl's union-closed conjecture.

Updates (Before January 2018) : Timothy Chow launched polymath12 on Rota's basis conjecture (February 24, 2017). (It was proposed as an answer to this question here.) (May 14, 2017) Tim Gowers is running a polymath-like project polymath13 on "Intransitive dices". (Dec 24 2017) A spontaneous polymath project, polymath14, over Tao's blog: A problem was posed by Apoorva Khare was presented and discussed and openly and collectively solved. (And the paper arxived.)

Update (January 25,2018) A new polymath project is emerging on Tao's blog: Polymath proposal: upper bounding the de Bruijn-Newman constant. Update: This is polymath15 which seems very active and quite successful. (wikipage)

Updates (April 14, 2018, June, 2019) Dustin Mixon and Aubrey de Grey have launched Polymath16 over at Dustin’s blog. The project is devoted to the chromatic number of the plane (Wikipage) following Aubrey de Grey's example showing that the chromatic number of the plane is at least 5. See also a proposal post and discussion thread over the polymath blog, and a proposal over here. Polymath 16 was now concluded.

Update, June 2019 Terry Tao initiated a sort of polymath attempt to solve the following problem conditioned on some conjectures from arithmetic algebraic geometry: Is there any polynomials $P$ of two variables with rational coefficients, such that the map $ P: \mathbb Q \times \mathbb Q \to \mathbb Q$ is a bijection? This is a famous 9-years old open question on MathOverflow.

Update, March 2020: On Terry Tao's blog, Polymath proposal: clearinghouse for crowdsourcing COVID-19 data and data cleaning requests. The proposal is to: (a) a collection of public data sets relating to the COVID-19 pandemic, (b) requests for such data sets, (c) requests for data cleaning of such sets, and (d) submissions of cleaned data sets. (Proposed by Chris Strohmeier after discussions among several mathematicians.)

Former proposals for future projects

There were also 10-20 additional serious proposals. A few proposals of various nature (from which polymath5 was selected) are gathered in this post on Gowers's blog, and several that appeared on various places are summarized on the polymath Wiki and also on the polymath blog. The polymath projects so far consisted of an attempt to solve a specific open problem but some of the proposals were of different nature.

More background

So far, polymath projects, while getting considerable attention and drawing enthusiasm, (and some controversy,) were limited in scope within mathematics and among mathematicians.

In most cases a small team of participants were the devoted contributed and in some cases those devoted participants were experts in the relevant area. Thus projects may apply primarily to experts in a specific field of mathematics. In all existing examples the project itself had some general appeal.

For a polymath project, in addition to the main task of trying to reach or at least greatly advance the goals of the specific project there are secondary goals of trying to understand the advantages and limitation of the polymath concept itself, and of trying to openly record the thought process of different participants towards the specific goal.

The question

The question is simple:

Make additional proposals for polymath projects.

Summary of proposals (updated: September 24, 2017)

1) The LogRank conjecture. Proposed by Arul.

2) The circulant Hadamard matrix conjecture. Proposed by Richard Stanley.

3) Finding combinatorial models for the Kronecker coefficients. Proposed by Per Alexandersson.

4) Eight lonely runners. Proposed by Mark Lewko.

5) A problem by Ruzsa: Finding the slowest possible exponential growth rate of a mapping $f:N→Z$ that is not a polynomial and yet shares with (integer) polynomials the congruence-preserving property $n−m∣f(n)−f(m)$. Proposed by Vesselin Dimitrov.

6) Finding the Matrix Multiplication Exponent ω. (Running time of best algorithm for matrix multiplication.) Proposed by Ryan O'Donnell.

7) The Moser Worm problem and Bellman's Lost in a forest problem. Proposed by Philip Gibbs.

8) Rational Simplex Conjecture ( by Cheeger and Simons). Proposed by Sasha Kolpakov.

9) Proving that for every integer $m$ with $|m| \le c(\sqrt{n}/2)^n$ there is an $n \times n$ 0-1 matrix matrix whose determinant equals $m$. Proposed by Gerhard Paseman.

10) Proving or disproving that the Euler's constant is irrational. Proposed by Sylvain JULIEN.

11) The Greedy Superstring Conjecture. Proposed by Laszlo Kozma.

12) Understanding the behavior and structure of covering arrays. Proposed by Ryan.

13) The group isomorphism problem, proposed by Arul based on an early proposal by Lipton.

14) Frankl's union closed set conjecture (Proposed by Dominic van der Zypen; Also one of the proposals by Gowers in this post). (Launched)

15) Komlos's conjecture in Discrepancy Theory. Proposed by Arul.

16) Rota's Basis Conjecture. Proposed by Timothy Chow. Launched on the polymath blog.

17) To show that $2^n+5$ composite for almost all positive integers $n$. (Might be too hard.) Proposed by me.

18) To prove a remarkable combinatorial identity on certain Permanents. Proposed by me. Update, Aug 6, 2016: settled!

19) Real world applications of large cardinals Proposed by Joseph van Name. There were a few more proposals in comments.

20) A project around a cluster of tiling problems. In particular: Is the Heech number bounded for polygonal monotiles? Is it decidable to determine if a single given polygonal tile can tile the plane monohedrally? Even for a single polyomino? Proposed by Joseph O'Rourke

21) To prove that $\sum \frac{\sin (2^n)}{n}$ is a convergent series. Proposed by JAck D'aurizio

22) The Nakayama conjecture and the finitistic dimension conjecture (major problems from the intersection of representation theory of finite dimensional algebras) and homological algebra. Proposed by Mare.

23) Major questions in the field of stereotype spaces and their applications, proposed by Sergei Akbarov.

24) The Erdos-Straus conjecture, proposed by Amit Maurya

25) The Collatz conjecture, proposed by Amit Maurya.

26) (New) Indecomposability of image transformations, proposed by Włodzimierz Holsztyński

27) (New) Is there a degree seven polynomial with integer coefficients such that (1) all of its roots are distinct integers, and (2) all of its derivative's roots are integers?, Proposed by Benjamin Dickman.

28) (New) The Cartan determinant conjecture for quiver algebras, proposed by Mare.

29) (New) The number of limit cycles of a polynomial vector field, Proposed by Ali Taghavi.

30) Small unit-distance graphs with chromatic number 5, proposed by Noam Elkies. Became Polymath16, see above.

Proposed rules (shortened):

All areas of mathematics including applied mathematics are welcome. Please do explain what the project is explicitly and in some details (not just link to a paper/wilipedea). Even if the project appeals to experts try to give a few sentences for a wide audience. Please offer projects that you genuinely think to be potentially suitable for a polymath project.

(Added) Criteria that were proposed for a polymath project.

Joel David Hamkins asked for some criteria that have been proposed for what kind of problem would make a good polymath project?

I don't think we have a clear picture on criteria for good polymath projects and there could be good projects of various kind. But the criteria for the first project are described by Gowers (I modified the wording to make them not specific in one sentence), and they seem like good criteria for a first project in a new field be it algebraic geometry, algebraic topology, group theory, logic, or set theory (to mention a few popular MO tags).