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== The Problem ==
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{{RightTOC}}
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This is the wiki for ''polymath'' projects - massively collaborative online mathematical projects.  The idea of such projects originated in Tim Gowers' blog post [http://gowers.wordpress.com/2009/01/27/is-massively-collaborative-mathematics-possible/ Is massively collaborative mathematics possible?]
  
Let <math>[3]^n</math> be the set of all length <math>n</math> strings over the alphabet <math>1, 2, 3</math>.  A ''combinatorial line'' is a set of three points in <math>[3]^n</math>, formed by taking a string with one or more wildcards <math>x</math> in it, e.g., <math>112x1xx3\ldots</math>, and replacing those wildcards by <math>1, 2</math> and <math>3</math>, respectively.  In the example given, the resulting combinatorial line is: <math>\{ 11211113\ldots, 11221223\ldots, 11231333\ldots \}</math>.  A subset of <math>[3]^n</math> is said to be ''line-free'' if it contains no lines. Let <math>c_n</math> be the size of the largest line-free subset of <math>[3]^n</math>.  
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Many polymath projects will be proposed, planned, and run at [http://polymathprojects.org/ This Blog].
  
'''<math>k=3</math> Density Hales-Jewett (DHJ(3)) theorem:''' <math>\lim_{n \rightarrow \infty} c_n/3^n = 0</math>
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A Polymath [[logo]] is being trialled.  If you have more suggestions, please add them to the [[logo]] page, or add to the discussion at [[Talk:logo]].
  
The original proof of DHJ(3) used arguments from ergodic theory.  The basic problem to be consider by the Polymath project is to explore a particular [http://gowers.wordpress.com/2009/02/01/a-combinatorial-approach-to-density-hales-jewett/ combinatorial approach to DHJ, suggested by Tim Gowers.] Some background to this project can be [http://gowers.wordpress.com/2009/01/30/background-to-a-polymath-project/ found here], and general discussion on massively collaborative "polymath" projects can be [http://gowers.wordpress.com/2009/01/27/is-massively-collaborative-mathematics-possible/ found here].  A cheatsheet for editing the wiki may be [http://meta.wikimedia.org/wiki/File:MediaWikiRefCard.png found here].
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The wiki is currently locked down due to a major influx of spam (July 29, 2013)Please email mn@michaelnielsen.org if you'd like an account set up, and I'll do my best to reply quickly.  
  
== Threads ==
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== Existing polymath projects ==
  
* (1-199) [http://gowers.wordpress.com/2009/02/01/a-combinatorial-approach-to-density-hales-jewett/ A combinatorial approach to density Hales-Jewett] (inactive)
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* [[Polymath1]]: New proofs and bounds for the density Hales-Jewett theorem. Initiated Feb 1, 2009; research results have now been published.
* (200-299) [http://terrytao.wordpress.com/2009/02/05/upper-and-lower-bounds-for-the-density-hales-jewett-problem/ Upper and lower bounds for the density Hales-Jewett problem] (active)
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* [[Definable Banach Spaces|Polymath2]]: Must an “explicitly defined” Banach space contain <math>c_0</math> or <math>l_p</math>?  Initiated Feb 17, 2009; attempts to relaunch via wiki, June 9 2010.
* (300-399) [http://gowers.wordpress.com/2009/02/06/dhj-the-triangle-removal-approach/ The triangle-removal approach] (inactive)
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* [[imo 2009 q6|Mini-polymath1]]: Solving Problem 6 of the 2009 International Mathematical Olympiad.  Initiated July 20, 2009; five proofs obtained so far.
* (400-499) [http://gowers.wordpress.com/2009/02/08/dhj-quasirandomness-and-obstructions-to-uniformity Quasirandomness and obstructions to uniformity] (inactive)
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* [[The polynomial Hirsch conjecture|Polymath3]].  The polynomial Hirsch conjecture.  Proposed July 17, 2009; launched, September 30, 2010. 
* (500-599) [http://gowers.wordpress.com/2009/02/13/dhj-possible-proof-strategies/#more-441/ Possible proof strategies] (active)
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* [[finding primes|Polymath4]]: A deterministic way to find primes.  Proposed July 27, 2009; launched Aug 9, 2009.  Research results have now been published.
* (600-699) [http://terrytao.wordpress.com/2009/02/11/a-reading-seminar-on-density-hales-jewett/ A reading seminar on density Hales-Jewett] (active)
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* [[The Erd&#337;s discrepancy problem|Polymath5]]: The Erd&#337;s discrepancy problem. Proposed Jan 10, 2010; launched Jan 19, 2010. Activity ceased by the end of 2012, but results from the project were used to solve the problem in 2015.
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* [[imo 2010|Mini-polymath2]]: Solving Problem 5 the 2010 International Mathematical Olympiad.  Proposed Jun 12, 2010; launched and solved, Jul 8 2010.
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* [[Improving the bounds for Roth's theorem|Polymath6]]: Improving the bounds for Roth's theorem. Proposed Feb 5, 2011.
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* [[imo 2011|Mini-polymath3]]: Solving a problem from the 2011 International Mathematical Olympiad. Proposed Jun 9, 2011; launched and solved, Jul 19, 2011.
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* [[imo 2012|Mini-polymath4]]: Solving a problem from the 2012 International Mathematical Olympiad.  Proposed, Jun 3, 2012; launched, July 12 2012.
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* [[The hot spots conjecture|Polymath7]]:  Establishing the Hot Spots conjecture for acute-angled triangles.  Proposed, May 31st, 2012; launched, Jun 8, 2012.
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* [[Bounded gaps between primes|Polymath8]]: Improving the bounds for small gaps between primes. Proposed, June 4, 2013; launched, June 4, 2013. Research results have now been published.
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* [[Discretized Borel Determinacy and P=NP|Polymath9]]: exploring Borel determinacy-based methods for giving complexity bounds.  Proposed, Oct 24, 2013; launched, Nov 3, 2013.
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* [[The Erdos-Rado sunflower lemma|Polymath10]]: improving the bounds for the Erdos-Rado sunflower lemma.  Launched, Nov 2, 2015.
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* [[Frankl's union-closed conjecture|Polymath11]]: proving Frankl's union-closed conjecture. Proposed Jan 21, 2016; launched Jan 29, 2016.  Concluded, Jan 17, 2017.
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* [[Rota's conjecture|Polymath12]]: proving Rota's conjecture. Proposed Feb 28, 2017.
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* [[Intransitive dice|Polymath13]]: Intransitive dice.  Proposed Apr 28, 2017.
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* [[linear norm|Polymath14]]: Classifying homogeneous norms on groups. Initiated Dec 16, 2017. Results submitted, Jan 11, 2018.
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* [[De_Bruijn-Newman constant|Polymath15]]: Upper bounding the de Bruin-Newman constant.  Proposed, Jan 24 2018; launched Jan 27 2018.
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* [[Hadwiger-Nelson problem|Polymath16]]: Simplifying the lower bound proof for the Hadwiger-Nelson problem.  Proposed, Apr 10, 2018; launched, Apr 14, 2018.
  
[http://blogsearch.google.com/blogsearch?hl=en&ie=UTF-8&q=polymath1&btnG=Search+Blogs Here is a further list of blog posts related to the Polymath1 project].  [http://en.wordpress.com/tag/polymath1/ Here is wordpress's list].
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== Polymath-like projects ==
  
A spreadsheet containing the latest upper and lower bounds for <math>c_n</math> can be found [http://spreadsheets.google.com/ccc?key=p5T0SktZY9DsU-uZ1tK7VEg here]. Here are the proofs of our [[upper and lower bounds]] for these constants.
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* Scott Aaronson's "philomath project": "[http://scottaaronson.com/blog/?p=453 Sensitivity vs. Block sensitivity]" (see also [http://mathoverflow.net/questions/31482/the-sensitivity-of-2-colorings-of-the-d-dimensional-integer-lattice this Math Overflow question]).  Launched Jul 13, 2010.
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* A wiki page clearinghouse for the [[Deolalikar P vs NP paper]].  Launched Aug 10, 2010.
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* <del>[http://researchtrends.wikia.com/wiki/Main_Page Math Research Trends Wiki] "research in the middle"</del> This project is recommended to be discontinued in favor of <b>[https://conference.portonvictor.org/wiki/Research_in_the_middle Research in the middle project] at [https://conference.portonvictor.org Virtual scientific conference]</b>.
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* The page for the [[ABC conjecture]] contains links and information about Mochizuki's claimed proof of this conjecture.
  
We are also collecting bounds for [[Fujimura's problem]].
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== Proposed polymath projects ==
  
Here are some [[unsolved problems]] arising from the above threads.
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* [http://gilkalai.wordpress.com/2009/03/25/an-open-discussion-and-polls-around-roths-theorem/ The cap set problem].  Proposed March 25, 2009 (see also these [http://gilkalai.wordpress.com/2009/05/11/around-the-cap-set-problem-b/ two] [http://gilkalai.wordpress.com/2009/05/18/the-cap-set-problem-and-frankl-rodl-theorem-c/ followup] posts).
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* [[Boshernitzan’s problem]].  Proposed July 27, 2009.
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* [http://gowers.wordpress.com/2009/09/16/possible-future-polymath-projects/ Possible future polymath projects].  Discussion opened September 16, 2009.
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* [http://rjlipton.wordpress.com/2011/05/12/a-possible-polymath-project/ A possible polymath project:] Proposal by Richard Lipton to attack a conjecture due to Erdos, about a class of Diophantine equations.
  
== Bibliography ==
 
  
# M. Elkin, "[http://arxiv.org/abs/0801.4310 An Improved Construction of Progression-Free Sets ]", preprint.
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A (partial) list of proposed projects can be found [http://polymathprojects.org/category/polymath-proposals/ here].
# H. Furstenberg, Y. Katznelson, “[http://math.stanford.edu/~katznel/hj43.pdf A density version of the Hales-Jewett theorem for k=3]“, Graph Theory and Combinatorics (Cambridge, 1988). Discrete Math. 75 (1989), no. 1-3, 227–241.
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# H. Furstenberg, Y. Katznelson, “[http://math.stanford.edu/~katznel/dhj12.pdf A density version of the Hales-Jewett theorem]“, J. Anal. Math. 57 (1991), 64–119.
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# B. Green, J. Wolf, "[http://arxiv.org/abs/0810.0732 A note on Elkin's improvement of Behrend's construction]", preprint.
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# K. O'Bryant, "[http://arxiv.org/abs/0811.3057 Sets of integers that do not contain long arithmetic progressions]", preprint.
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# R. McCutcheon, “[http://www.msci.memphis.edu/~randall/preprints/HJk3.pdf The conclusion of the proof of the density Hales-Jewett theorem for k=3]“, unpublished.
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== Other resources ==
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If you have a tentative proposal for a polymath project, you can either make a post on it on your own blog, or place it [[other proposed projects|here]].
  
* [http://meta.wikimedia.org/wiki/Help:Contents Wiki user's guide]
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== Discussions about polymath ==
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* [http://gowers.wordpress.com/2009/01/27/is-massively-collaborative-mathematics-possible/ Is massively collaborative mathematics possible?] Tim Gowers, January 27, 2009.
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* [http://lucatrevisan.wordpress.com/2009/02/01/a-peoples-history-of-mathematics/ A people's history of mathematics] Luca Trevisan, February 1, 2009.
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* [http://michaelnielsen.org/blog/?p=553 The polymath project] Michael Nielsen, February 3, 2009.
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* [http://www.neverendingbooks.org/index.php/yet-another-math20-proposal.html Yet another math 2.0 proposal] Lieven le Bruyn, February 11, 2009.
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* [http://gowers.wordpress.com/2009/03/10/polymath1-and-open-collaborative-mathematics/ Polymath1 and open collaborative mathematics] Tim Gowers, March 10, 2009.
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* [http://maxwelldemon.com/2009/03/14/polymath/ Polymath] Edmund Harriss, March 14, 2009.
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* [http://science.slashdot.org/article.pl?sid=09/03/18/194228 Massive open collaboration in mathematics declared a success] Slashdot, March 18, 2009.
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* [http://michaelnielsen.org/blog/?p=581 How changing the technology of collaboration can change the nature of collaboration] Michael Nielsen, March 18, 2009.
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* [http://michaelnielsen.org/blog/?p=584 The polymath project: scope of participation] Michael Nielsen, March 20, 2009.
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* [http://gowers.wordpress.com/2009/03/24/can-polymath-be-scaled-up/ Can polymath be scaled up?] Tim Gowers, March 24, 2009.
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* [http://whatisresearch.wordpress.com/2009/03/24/concluding-notes-on-the-polymath-project-and-a-challenge/ Concluding notes on the polymath project - and a challenge] Vilpulniak, March 24, 2009.
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* [http://michaelnielsen.org/blog/on-scaling-up-the-polymath-project/ On scaling up the polymath project] Michael Nielsen, March 25, 2009.
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* [http://numberwarrior.wordpress.com/2009/03/25/a-gentle-introduction-to-the-polymath-project/ A gentle introduction to the polymath project] Jason Dyer, March 25, 2009.
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* [http://blogs.telegraph.co.uk/technology/iandouglas/9656357/tim_gowers_and_the_polymaths/ Tim Gowers and the polymaths] Ian Douglas (the Telegraph), April 29, 2009
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* [http://terrytao.wordpress.com/2009/07/22/imo-2009-q6-mini-polymath-project-impressions-reflections-analysis/ IMO 2009 Q6 as mini-polymath project: impressions, reflections, analysis] Terence Tao, July 22, 2009.
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* [http://polymathprojects.org/2009/07/27/selecting-the-next-polymath-project/ Selecting the next polymath project] Terence Tao, July 27, 2009.
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* [http://blog.jonudell.net/2009/07/31/polymath-equals-user-innovatio/ Polymath equals user innovation] Jon Udell, July 31, 2009.
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* [http://scienceblogs.com/christinaslisrant/2009/08/an_overview_of_the_polymath_pr.php An overview of the polymath project] Christina Pikas, August 1, 2009
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* [http://whatisresearch.wordpress.com/2009/08/09/collaborative-mathematics-etc/ Collaborative mathematics etc.] Vipulniak, August 9, 2009
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* [http://www.nature.com/nature/journal/v461/n7266/full/461879a.html Massively collaborative mathematics] Tim Gowers, Michael Nielsen, Nature, October 15, 2009
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* [http://portonmath.wordpress.com/2009/10/25/collaborative-research-of-filters/ Collaborative math research – a real example] Victor Porton, October 24, 2009
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* [http://whatisresearch.wordpress.com/2009/10/26/polymath-again/ Polymath again] Vipulniak, October 26, 2009
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* [http://www.kennislink.nl/publicaties/wiskunde-met-zijn-allen Wiskunde met zijn allen] (Dutch), Alex van den Brandhof, Kennislink, November 12, 2009
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* [http://www.sciencenews.org/view/generic/id/50532/title/Mathematics_by_collaboration Mathematics by collaboration], Julie Rehmeyer, ScienceNews, December 8, 2009
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* [http://www.nytimes.com/projects/magazine/ideas/2009/#m Massively Collaborative Mathematics], Jordan Ellenberg, The Ninth Annual Year in Ideas, New York Times, 2009.
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* [http://www.hypios.com/thinking/2010/01/13/massively-collaborative-mathematics-lessons-from-polymath1/ Massively Collaborative Mathematics: lessons from polymath1], Hypios, Jan 13 2010
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* [http://ths1104.wordpress.com/2010/02/13/open-reflexions-sur-fond-de-polymaths/ Open réflexions sur fond de Polymaths] (French), ths1104, Feb 13 2010
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* [http://www.javiertordable.com/blog/2010/02/25/collaborative-mathematics-future-of-science Collaborative Mathematics and The Future of Science] Javier Tordable, February 26 2010
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* [http://www.scientificamerican.com/article.cfm?id=problem-solved-tic-tac-toe-blog Problem Solved, LOL: A Complex Tic-Tac-Toe Puzzle Falls Thanks to Blog Comments] Davide Castelvecchi, Scientific American, March 17 2010
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* [http://www.thebigquestions.com/2010/04/08/blogging-tic-tac-toe-and-the-future-of-math/ Blogging, Tic Tac Toe, and the Future of Math] Steve Landsburg, The Big Questions, April 4 2010
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* [http://www.siam.org/news/news.php?issue=0043.03 Massively Collaborative Mathematics] Julie Rehmeyer, SIAM News, Volume 43(3), April 2010 (to appear)
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* [http://mbarany.com/publications.html#WikiSymPolymath  `But this is blog maths and we're free to make up conventions as we go along': Polymath1 and the Modalities of `Massively Collaborative Mathematics.'] Michael Barany,  Proceedings of the 6th International Symposium on Wikis and Open Collaboration, Gdansk, Poland, 2010.
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* [http://www.cs.cmu.edu/~jcransh/papers/cranshaw_kittur.pdf J. Cranshaw and A. Kittur. The Polymath Project: Lessons from a successful online collaboration in mathematics]. In Proceedings of the Conference on Human Factors in Computing Systems, Vancouver, BC, Canada, May 2011.
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* [http://www.newscientist.com/article/mg21028113.900-how-to-build-the-global-mathematics-brain.html How to build the global mathematics brain], Jacob Aron, New Scientist, 4 May 2011.
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* [http://www.newscientist.com/article/mg21028112.900-mathematics-becomes-more-sociable.html Mathematics becomes more sociable], New Scientist, 5 May 2011.
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* [http://polymathprojects.files.wordpress.com/2011/03/polymathias.jpg Mathematical Advances: Lone or Massively Collaborative Endeavors?] from IAS Institute Letter for fall 2010 based on a discussion organized by IAS fall 2010.
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* [http://today.uconn.edu/blog/2010/10/will-crowdsourcing-revolutionize-scholarship Will ‘Crowdsourcing’ Revolutionize Scholarship?] An article in UConn Today by Jeremy Teitelbaum, Fall 2010.
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* [http://online.wsj.com/article/SB10001424052970204644504576653573191370088.html The New Einsteins Will Be Scientists Who Share] The Wall street journal, October 2011.
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* [http://www.nature.com/news/parallel-lines-1.14759?WT.ec_id=NATURE-20140227 Parallel lines], editorial, Nature 506, 407–408 (27 February 2014).
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Additional links are very welcome.
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== Other links ==
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* [http://polymathprojects.org/ The polymath blog]
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* [http://polymathprojects.org/general-polymath-rules/ General polymath rules]
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== Note on anonymous editing ==
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To help combat spam, anonymous editing has been disabled, and a captcha system added to hinder automated account creation. If this is causing problems, please email mn@michaelnielsen.org.
 +
 
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== Note on image uploads ==
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Image uploads have been disabled, as they were causing problems with spam.  If you'd like to upload an image, please email mn@michaelnielsen.org

Latest revision as of 11:52, 14 April 2018

This is the wiki for polymath projects - massively collaborative online mathematical projects. The idea of such projects originated in Tim Gowers' blog post Is massively collaborative mathematics possible?

Many polymath projects will be proposed, planned, and run at This Blog.

A Polymath logo is being trialled. If you have more suggestions, please add them to the logo page, or add to the discussion at Talk:logo.

The wiki is currently locked down due to a major influx of spam (July 29, 2013). Please email mn@michaelnielsen.org if you'd like an account set up, and I'll do my best to reply quickly.

Existing polymath projects

  • Polymath1: New proofs and bounds for the density Hales-Jewett theorem. Initiated Feb 1, 2009; research results have now been published.
  • Polymath2: Must an “explicitly defined” Banach space contain [math]c_0[/math] or [math]l_p[/math]? Initiated Feb 17, 2009; attempts to relaunch via wiki, June 9 2010.
  • Mini-polymath1: Solving Problem 6 of the 2009 International Mathematical Olympiad. Initiated July 20, 2009; five proofs obtained so far.
  • Polymath3. The polynomial Hirsch conjecture. Proposed July 17, 2009; launched, September 30, 2010.
  • Polymath4: A deterministic way to find primes. Proposed July 27, 2009; launched Aug 9, 2009. Research results have now been published.
  • Polymath5: The Erdős discrepancy problem. Proposed Jan 10, 2010; launched Jan 19, 2010. Activity ceased by the end of 2012, but results from the project were used to solve the problem in 2015.
  • Mini-polymath2: Solving Problem 5 the 2010 International Mathematical Olympiad. Proposed Jun 12, 2010; launched and solved, Jul 8 2010.
  • Polymath6: Improving the bounds for Roth's theorem. Proposed Feb 5, 2011.
  • Mini-polymath3: Solving a problem from the 2011 International Mathematical Olympiad. Proposed Jun 9, 2011; launched and solved, Jul 19, 2011.
  • Mini-polymath4: Solving a problem from the 2012 International Mathematical Olympiad. Proposed, Jun 3, 2012; launched, July 12 2012.
  • Polymath7: Establishing the Hot Spots conjecture for acute-angled triangles. Proposed, May 31st, 2012; launched, Jun 8, 2012.
  • Polymath8: Improving the bounds for small gaps between primes. Proposed, June 4, 2013; launched, June 4, 2013. Research results have now been published.
  • Polymath9: exploring Borel determinacy-based methods for giving complexity bounds. Proposed, Oct 24, 2013; launched, Nov 3, 2013.
  • Polymath10: improving the bounds for the Erdos-Rado sunflower lemma. Launched, Nov 2, 2015.
  • Polymath11: proving Frankl's union-closed conjecture. Proposed Jan 21, 2016; launched Jan 29, 2016. Concluded, Jan 17, 2017.
  • Polymath12: proving Rota's conjecture. Proposed Feb 28, 2017.
  • Polymath13: Intransitive dice. Proposed Apr 28, 2017.
  • Polymath14: Classifying homogeneous norms on groups. Initiated Dec 16, 2017. Results submitted, Jan 11, 2018.
  • Polymath15: Upper bounding the de Bruin-Newman constant. Proposed, Jan 24 2018; launched Jan 27 2018.
  • Polymath16: Simplifying the lower bound proof for the Hadwiger-Nelson problem. Proposed, Apr 10, 2018; launched, Apr 14, 2018.

Polymath-like projects

Proposed polymath projects


A (partial) list of proposed projects can be found here.

If you have a tentative proposal for a polymath project, you can either make a post on it on your own blog, or place it here.

Discussions about polymath

Additional links are very welcome.

Other links

Note on anonymous editing

To help combat spam, anonymous editing has been disabled, and a captcha system added to hinder automated account creation. If this is causing problems, please email mn@michaelnielsen.org.

Note on image uploads

Image uploads have been disabled, as they were causing problems with spam. If you'd like to upload an image, please email mn@michaelnielsen.org