Polymath Project

[William Dupere]

ThePolymath Project is a collaboration among mathematicians to solve important and difficult mathematical problems by coordinating many mathematicians to communicate with each other on finding the best route to the solution. The project began in January 2009 on Timothy Gowers's blog when he posted a problem and asked his readers to post partial ideas and partial progress toward a solution.[1] This experiment resulted in a new answer to a difficult problem, and since then the Polymath Project has grown to describe a particular crowdsourcing process of using an online collaboration to solve any math problem.

In January 2009, Gowers chose to start a social experiment on his blog by choosing an important unsolved mathematical problem and issuing an invitation for other people to help solve it collaboratively in the comments section of his blog.[1] Along with the math problem itself, Gowers asked a question which was included in the title of his blog post, "is massively collaborative mathematics possible?"[2][3] This post led to his creation of the Polymath Project.

Since its inception, it has now sponsored a "Crowdmath" project in collaboration with MIT PRIMES program and the Art of Problem Solving. This project is built upon the same idea of the Polymath project that massive collaboration in mathematics is possible and possibly quite fruitful. However, this is specifically aimed at only high school and college students with a goal of creating "a specific opportunity for the upcoming generation of math and science researchers." The problems are original research and unsolved problems in mathematics. All high school and college students from around the world with advanced background of mathematics are encouraged to participate. Older participants are welcomed to participate as mentors and encouraged not to post solutions to the problems. The first Crowdmath project began on March 1, 2016.[4][5]

This project was set up in order to try to solve the Erdős discrepancy problem. It was active for much of 2010 and had a brief revival in 2012, but did not end up solving the problem. However, in September 2015, Terence Tao, one of the participants of Polymath5, solved the problem in a pair of papers. One paper proved an averaged form of the Chowla and Elliott conjectures, making use of recent advances in analytic number theory concerning correlations of values of multiplicative functions. The other paper showed how this new result, combined with some arguments discovered by Polymath5, were enough to give a complete solution to the problem. Thus, Polymath5 ended up making a significant contribution to the solution.

I am very happy to say that I have recently received a generous grant from the Astera Institute to set up a small group to work on automatic theorem proving, in the first instance for about three years after which we will take stock and see whether it is worth continuing. This will enable me to take on up to about three PhD students and two postdocs over the next couple of years. I am imagining that two of the PhD students will start next October and that at least one of the postdocs will start as soon as is convenient for them. Before any of these positions are advertised, I welcome any informal expressions of interest: in the first instance you should email me, and maybe I will set up Zoom meetings. (I have no idea what the level of demand is likely to be, so exactly how I respond to emails of this kind will depend on how many of them there are.)

I have privately let a few people know about this, and as a result I know of a handful of people who are already in Cambridge and are keen to participate. So I am expecting the core team working on the project to consist of 6-10 people. But I also plan to work in as open a way as possible, in the hope that people who want to can participate in the project remotely even if they are not part of the group that is based physically in Cambridge. Thus, part of the plan is to report regularly and publicly on what we are thinking about, what problems, both technical and more fundamental, are holding us up, and what progress we make. Also, my plan at this stage is that any software associated with the project will be open source, and that if people want to use ideas generated by the project to incorporate into their own theorem-proving programs, I will very much welcome that.

The Polymath experiment is still very much in its infancy, with the result that we still have only a rather hazy idea of what the advantages and disadvantages are of open online multiple collaboration. It is easy to think of potential advantages and disadvantages, but the more actual experience we can draw on, the more we will get a feel for which of these plausible speculations are correct.

Ten years ago on January 27, 2009, Polymath1 was proposed by Tim Gowers and was launched on February 1, 2009. The first project was successful and it followed by 15 other formal polymath projects and a few other projects of similar nature.

Briefly showing that is the Rieman Hypothesis, and it is known that . Brad Rodgers and Terry Tao proved an old conjecture that . The purpose of the project is to push down this upper bound. (The RH is not considered a realistic outcome.)

Pavel Patk presented a lemma from one of his papers that might be useful. Let M be a matroid of rank r and let S be a sequence of kr elements from M, split into r subsequences, each of length at most k. Then any largest independent rainbow subsequence of S is a basis of M if and only if there does not exist an integer s Finally, let me make a few remarks about the directions of research that were suggested in my previous Polymath 12 blog post. I was initially optimistic about matroids with no small circuits and I still think that they are worth thinking about, but I am now more pessimistic that we can get much mileage out of straightforwardly generalizing the methods of Geelen and Humphries, for reasons that can be found by reading the comments. Similarly I am more pessimistic now that the algebro-geometric approach will yield anything since being a basis is an open condition rather than a closed condition.

Let me also mention The PolyTCS Project aimed for proposing projects in theoretical computer science. There are so far three very interesting proposals there, and the first proposal is the Friedgut-Kalai Entropy/Influence conjecture. For various proposals, see also the polymath blog administered by Tim Gowers, Michael Nielsen, Terry Tao, and me, and this MO question.

The conjecture states that if and are any two real numbers, and , then there exists a positive integer such that . Famously, Einsiedler, Katok and Lindenstrauss proved that the Hausdorff dimension of the set of counterexamples to the conjecture is zero. Gowers had ideas for an elementary approach, and his ideas are described in this later post. This project was not launched and I am also not aware of progress related to the problem (but I am not an expert).

Conjecture: For every and every positive integer there exists such that if is any subset of of density at least , then contains a combinatorial line such that the wildcard set is of the form for some subset .

Let be a positive integer and let be a graph. Erds and Hajnal conjectured that there is a constant such that if is any graph with at least vertices that does not contain any induced copy of , then either or contain a clique of size .

Tim asserted that the simplest open case is where is a pentagon. This special case was recently settled by Maria Chudnovsky, Alex Scott, Paul Seymour and Sophie Spirkl. They rely on recent results by Janos Pach and Istvan Tomon. See this videotaped lecture by Paul Seymour at IBS Discrete Mathematics Group, South Korea.

Tim wrote: This is a notorious question, and possibly the least likely to yield to a Polymath approach (it feels as though there might be a burst of ideas, none of which would work, followed by disillusionment, or else, if we were very lucky, a single bright idea from one person that essentially solved the problem, but I could be wrong).

This question is due to Erdős and Rado. A delta system is a collection of sets such that all the sets (with ) are equal. Equivalently, there exists some set such that for every , and the sets are disjoint.

Erds offered 1000 dollars for a solution to the following problem: does there exist a constant such that for every and every system of at least sets of size , there must exist three of them that form a delta system?

I devoted polymath10 to this problem. I tried to promote certain homological approach. This has not led to progress but did lead to some interesting observations on the problem and also some refinement and better understanding of my approach.

While still open since 2009, there were major breakthroughs regarding the problem that we described here and here, most notably the problem was nearly solved by Ryan Alweiss, Shachar Lovett, Kewen Wu, and Jiapeng Zhang.

One comment mentioned the then very recent computer programs for playing Go (based on machine learning). Let me mention that deep learning led to a revolution in this area around 2015. (And also that we had a guest post by Amir Ban on chess playing computer programs.)

This rather tentative suggestion, was to try to come up with a model that would show convincingly how life could emerge from non-life by purely naturalistic processes. It was further discussed in this post. There was a lot of excitement around it but it did not lead to a polymath project, and I am not sure about related progress after 2009.

3.a A theory of convex hulls of real algebraic varieties. One project in this spirit that I already proposed is to try to develop polymathly a theory of convex hulls of real algebraic varieties.

3.b Extending algebraic shifting to a wide variety of combinatorial structures. This is a project with Hlne Barcelo from the mid 90s that at the time did not get off the ground and it could be very nice to explore it collectively.

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