Polymath Wiki - User contributions [en]

Linear norm grant acknowledgments

2019-03-10T06:25:29Z

Khare: /* Grant information */

Participants should be arranged in alphabetical order of surname.

== Participants and contact information ==

Caution: this list may be incomplete. Participants who have made significant contributions to the project (on par with a co-author on a traditional mathematical research paper) should add themselves to this list, or email tao@math.ucla.edu if they are unable to do so directly. Participants who have made auxiliary contributions to the project (on par with those mentioned in an Acknowledgments section in a traditional paper) should add themselves instead to the list at the bottom of the page.

* Tobias Fritz, MPI MIS, [http://personal-homepages.mis.mpg.de/fritz/]
* Siddhartha Gadgil, Indian Institute of Science, Bangalore [http://math.iisc.ac.in/~gadgil/]
* Apoorva Khare, Indian Institute of Science, Bangalore, and the Analysis and Probability Research Group, Bangalore [http://www.math.iisc.ac.in/~khare/]
* Pace Nielsen, BYU, [https://math.byu.edu/~pace/]
* Lior Silberman, UBC, [https://www.math.ubc.ca/~lior/]
* Terence Tao, UCLA, [http://www.math.ucla.edu/~tao]

=== Grant information ===

* Apoorva Khare was supported by Ramanujan Fellowship SB/S2/RJN-121/2017 and MATRICS grant MTR/2017/000295 from SERB (DST, Govt.~of India), by grant F.510/25/CAS-II/2018(SAP-I) from UGC (DST, Govt.~of India), and by a Young Investigator Award from the Infosys Foundation.
* Pace Nielsen was supported by NSA grant H98230-16-1-0048.
* Lior Silberman was supported by an NSERC Discovery grant.
* Terence Tao was supported by a Simons Investigator grant, the James and Carol Collins Chair, the Mathematical Analysis & Application Research Fund Endowment, and by NSF grant DMS-1266164.

=== Other acknowledgments ===

Other contributors to the project include Sean Eberhard, Tobias Hartnick, Chris Jerdonek, Antonio Machiavelo, Andy Putman, Will Sawin, Alexander Shamov, and David Speyer.

Thanks to Michael Nielsen for hosting the polymath wiki for this project.

Linear norm

2018-11-27T18:12:16Z

Khare: /* Writeup */

This is the wiki page for understanding ''seminorms of linear growth'' on a group <math>G</math> (such as the free group on two generators). These are functions <math>\| \|: G \to [0,+\infty)</math> that obey the triangle inequality

:<math>\|xy\| \leq \|x\| + \|y\| \quad (1)</math>

and the linear growth condition

:<math> \|x^n \| = |n| \|x\| \quad (2) </math>

for all <math>x,y \in G</math> and <math>n \in {\bf Z}</math>.

We use the usual group theory notations <math>x^y := yxy^{-1}</math> and <math>[x,y] := xyx^{-1}y^{-1}</math>.

== Threads ==

* [https://terrytao.wordpress.com/2017/12/16/bi-invariant-metrics-of-linear-growth-on-the-free-group/ Bi-invariant metrics of linear growth on the free group], Dec 16 2017.
* [https://terrytao.wordpress.com/2017/12/19/bi-invariant-metrics-of-linear-growth-on-the-free-group-ii/ Bi-invariant metrics of linear growth on the free group, II], Dec 19 2017.
* [https://terrytao.wordpress.com/2017/12/21/metrics-of-linear-growth-the-solution/ Metrics of linear growth – the solution], Dec 21 2017.
* [https://terrytao.wordpress.com/2018/01/11/homogeneous-length-functions-on-groups/ Homogeneous length functions on groups], Jan 11 2018.

== Key lemmas ==

Henceforth we assume we have a seminorm <math>\| \|</math> of linear growth. The letters <math>s,t,x,y,z,w</math> are always understood to be in <math>G</math>, and <math>i,j,n,m</math> are always understood to be integers.

From (2) we of course have
:<math> \|x^{-1} \| = \| x\| \quad (3)</math>

'''Lemma 1'''. If <math>x</math> is conjugate to <math>y</math>, then <math>\|x\| = \|y\|</math>.

'''Proof'''. By hypothesis, <math>x = zyz^{-1}</math> for some <math>z</math>, thus <math>x^n = z y^n z^{-1}</math>, hence by the triangle inequality

:<math> n \|x\| = \|x^n \| \leq \|z\| + n \|y\| + \|z^{-1} \|</math>

for any <math>n \geq 1</math>. Dividing by <math>n</math> and taking limits we conclude that <math>\|x\| \leq \|y\|</math>. Similarly <math>\|y\| \leq \|x\|</math>, giving the claim. <math>\Box</math>

An equivalent form of the lemma is that

:<math> \|xy\| = \|yx\| \quad (4).</math>

We can generalise Lemma 1:

'''Lemma 2'''. If <math>x^i</math> is conjugate to <math>wy</math> and <math>x^j</math> is conjugate to <math>zw^{-1}</math>, then <math> \|x\| \leq \frac{1}{|i+j|} ( \|w\| + \|z\| )</math>.

'''Proof'''. By hypothesis, <math>x^i = s wy s^{-1}</math> and <math>x^j = t zw^{-1} t^{-1}</math> for some <math>s,t</math>. For any natural number <math>n</math>, we then have

:<math> x^{in} x^{jn} = s wy \dots wy s^{-1} t zw^{-1} \dots zw^{-1} t^{-1}</math>

where the terms <math>wy, zw</math> are each repeated <math>n</math> times. By Lemma 1, conjugation by <math>w</math> does not change the norm. From many applications of this and the triangle inequality, we conclude that

:<math> |i+j| n \|x\| = \| x^{in} x^{jn} \| \leq \|s\| + n \|y\| + \|s^{-1} t\| + n \|z\| + \|t^{-1}\|.</math>

Dividing by <math>n</math> and sending <math>n \to \infty</math>, we obtain the claim. <math>\Box</math>

== Corollaries ==

'''Corollary 0'''. The eight commutators <math>[x^{\pm 1}, y^{\pm 1}], [y^{\pm 1}, x^{\pm 1}]</math> all have the same norm.

'''Proof'''. Each of these commutators is conjugate to either <math>[x,y]</math> or its inverse. <math>\Box</math>

'''Corollary 1'''. The function <math>n \mapsto \|x^n y\|</math> is convex in <math>n</math>.

'''Proof'''. <math>x^n y</math> is conjugate to <math>x (x^{n-1} y)</math> and to <math>(x^{n+1} y) x^{-1}</math>, hence by Lemma 2
:<math>\| x^n y \| \leq \frac{1}{2} (\| x^{n-1} y \| + \| x^{n+1} y \|),</math>
giving the claim. <math>\Box</math>

'''Corollary 2'''. For any <math>k \geq 1</math>, one has
:<math>\| [x,y] \| \leq \frac{1}{2k+2} (\| [x^{-1},y^{-1}]^k x^{-1} \| + \| [x,y]^k x \|).</math>
Thus for instance
:<math>\| [x,y] \| \leq \frac{1}{4} (\| [x^{-1},y^{-1}] x^{-1} \| + \| [x,y] x \|).</math>

'''Proof'''. <math>[x,y]^{k+1}</math> is conjugate both to <math>x(y[x^{-1},y^{-1}]^k x^{-1}y^{-1})</math> and to <math>(y^{-1} [x,y]^k xy)x^{-1}</math>, hence by Lemma 2
:<math> \| [x,y] \| \leq \frac{1}{2k+2} ( \| y[x^{-1},y^{-1}]^k x^{-1} \| + \| (y^{-1} [x,y]^k xy)x^{-1}\|)</math>
giving the claim by Lemma 1. <math>\Box</math>

'''Corollary 3'''. One has
:<math> \|[x,y]^2 x\| \leq \frac{1}{2} ( \| x y^{-1} [x,y] \| + \| xy [x,y] \| ).</math>

'''Proof'''. <math>[x,y]^2 x</math> is conjugate both to <math>y (x^{-1} y^{-1} [x,y] x^2)</math> and to <math>(x[x,y]xyx^{-1}) y^{-1}</math>, hence by Lemma 2
:<math> \displaystyle \|[x,y]^2 x\| \leq \frac{1}{2} ( \|x^{-1} y^{-1} [x,y] x^2\| + \|x[x,y]xyx^{-1}\|)</math>
giving the claim by Lemma 1. <math>\Box</math>

'''Corollary 4'''. One has
:<math>\| [x,y] x\| \leq \frac{1}{4} ( \| x^2 y [x,y] \| + \| xy^{-1} x [x,y] \| ).</math>

'''Proof'''. <math>([x,y] x)^2</math> is conjugate both to <math>y^{-1} (x [x,y] x^2 y x^{-1})</math> and to <math>(x^{-1} y^{-1} x [x,y] x^2) y</math>, hence Lemma 2
:<math>\| [x,y] x\| \leq \frac{1}{4} ( \| x [x,y] x^2 y x^{-1} \| + \| x^{-1} y^{-1} x [x,y] x^2 \| ),</math>
giving the claim by Lemma 1. <math>\Box</math>

'''Corollary 5'''. One has
:<math> \|[x,y] x\| \leq \|x\| + \frac{1}{2} \| [x^2, y] \|</math>.

'''Proof'''. <math>[x,y]x</math> is conjugate to both <math>x [x^{-2},y^{-1}]</math> and to <math>(y^{-1} x^2 y) x^{-1}</math>, hence by Lemma 2
:<math>\| [x,y] x\| \leq \frac{1}{2} ( \| [x^{-2}, y^{-1}] \| + \| y^{-1} x^2 y \| ),</math>
giving the claim by Lemma 1 and Corollary 0. <math>\Box</math>

'''Corollary 6'''. One has
:<math> \| [x,y]\| \leq \frac{1}{4} ( \| x\| + \| [x^2,y] \| + \| [x,y] x\| ) </math>.

'''Proof'''. From Lemma 2 we have
:<math> \| [x,y] \| \leq \frac{1}{4} ( \| x^{-1} [x,y]^2 \| + \| [x,y] x \| ).</math>
Since <math>x^{-1} [x,y]^2</math> is conjugate to <math>(yx^{-1} y^{-1}) (xyx^{-2} y^{-1} x)</math>, we have
:<math> \| x^{-1} [x,y]^2 \| \leq \| yx^{-1} y^{-1} \| + \|xyx^{-2} y^{-1} x\|</math>
and the claim follows from Lemma 1 and (3). <math>\Box</math>

'''Corollary 7'''. For any <math>m,k</math>, one has
:<math> \| x^m [x,y]^k \| \leq \frac{1}{2} ( \| x^{m-1} [x,y]^k \| + \|x^{m+1} [x,y]^{k-1} \| )</math>.

'''Proof'''. <math>x^m[x,y]^k</math> is trivially conjugate to <math>x(x^{m-1}[x,y]^k)</math> and conjugate to <math>(y^{-1}x^m[x,y]^{k-1}xy)x^{-1}</math>. Hence by Lemma 2,
:<math>\| x^m[x,y]^k \| \leq \frac{1}{2} ( \| x^{m-1}[x,y]^k \| + \| y^{-1}x^m[x,y]^{k-1}xy \| ) = \frac{1}{2} ( \| x^{m-1}[x,y]^k \| + \| x^{m+1}[x,y]^{k-1} \|),</math>
where the final equation is by conjugation invariance (Lemma 1). <math>\Box</math>

'''Corollary 8'''. One has <math>\|x\| \leq \| [x,y] x \|</math>.

'''Proof'''. <math>x</math> is equal to both <math> (x^2 y x y^{-1} x^{-2}) (x^2 y x^{-1} y^{-1} x^{-1})</math> and to <math>(x^2 y x^{-1} y^{-1} x^{-1})^{-1} (x^2 y x^{-1} y^{-1})</math>, hence by Lemma 2
:<math> \|x\| \leq \frac{1}{2} ( \| x^2 y x y^{-1} x^{-2} \| + \|x^2 y x^{-1} y^{-1}\| ).</math>
By Lemma 1, the RHS is <math>\frac{1}{2} \|x\| + \frac{1}{2} \| [x,y] x \|</math>, and the claim follows. <math>\Box</math>

== Iterations ==

Call a pair of real numbers <math>(\alpha,\beta)</math> '''admissible''' if one has the inequality

:<math> \| [x,y] \| \leq \alpha \|x\| + \beta \|y \|</math>

for all <math>x,y</math>. Clearly the set of admissible pairs is closed and convex, and if <math>(\alpha,\beta)</math> is admissible then so is <math>(\alpha',\beta')</math> for any <math>\alpha' \geq \alpha, \beta' \geq \beta</math>. From Corollary 0 we also see that the set is symmetric: <math>(\alpha,\beta)</math> is admissible if and only if <math>(\beta,\alpha)</math> is.

Writing <math>[x,y] = y^x y^{-1}</math> we see that <math>(0,2)</math> is admissible, and similarly so is <math>(0,2)</math>.

'''Proposition 1'''. If <math>(\alpha,\beta)</math> is admissible, then so is <math>(\frac{\alpha+1}{2}, \frac{\beta}{4})</math>.

'''Proof'''. From Corollary 5 and hypothesis one has
:<math>\| [x,y] x\| \leq \|x\| + \frac{1}{2} ( \alpha \|x^2\| + \beta \|y\| ) = (\alpha+1) \|x\| + \frac{\beta}{2} \|y\|</math>
and hence also
:<math>\| [x^{-1},y^{-1}] x^{-1}\| \leq (\alpha+1) \|x\| + \frac{\beta}{2} \|y\|.</math>
From Corollary 2 we thus have
:<math>\| [x,y]\| \leq \frac{\alpha+1}{2} \|x\| + \frac{\beta}{4} \|y\|.</math>

The map <math>(\alpha,\beta) \mapsto (\frac{\alpha+1}{2}, \frac{\beta}{4})</math> is a contraction with fixed point <math>(1,0)</math>. Thus

:<math> \|[x,y]\| \leq \|x\| \quad (4)</math>.

From symmetry we also see that if <math>(\alpha,\beta)</math> is admissible, then so is <math>(\frac{\beta+1}{2}, \frac{\alpha}{4})</math>. The map <math>(\alpha,\beta) \mapsto (\frac{\beta+1}{2}, \frac{\alpha}{4})</math> is a contraction with fixed point (4/7,1/7), thus

:<math> \|[x,y]\| \leq \frac{4}{7} \|x\| + \frac{1}{7} \|y\| </math>.

== Solution ==

''Note: this argument only requires Lemma 1, Lemma 2, and Corollary 7 from the preceding sections.''

'''Theorem 1''' <math>\|[x,y]\| = 0</math>.

'''Proof''' Let <math>n</math> be a large natural number. Write <math>f(m,k) := \| x^m [x,y]^k \|</math>. Let <math>X_1,\dots,X_{2n}</math> be iid random variables, each taking a value of <math>(-1,0)</math> or <math>(1,-1)</math> with equal probability <math>1/2</math>. From Corollary 7 one has

:<math>f(m,k) \leq {\bf E} f( (m,k) + X_j)</math>

for any <math>(m,k), j</math>, and in particular on iterating

:<math>f(0, n) \leq {\bf E} f( (0,n) + X_1 + \dots + X_{2n} ).</math>

By the triangle inequality, we conclude that

:<math>f(0, n) \leq (\|x\|+\|y\|) {\bf E} | (0,n) + X_1 + \dots + X_{2n} |.</math>

But the random variable <math>(0,n) + X_1 + \dots + X_{2n}</math> has mean zero and variance <math>O(n)</math>, hence by Cauchy-Schwarz

:<math>f(0, n) \ll n^{1/2} (\|x\|+\|y\|).</math>

But the left-hand side is <math>n \|[x,y]\|</math>, so on dividing by <math>n </math> and taking limits we obtain the claim.<math>\Box</math>

As a consequence of this theorem and the triangle inequality, any seminorm on a group will factor through to its abelianisation.

== Writeup ==

* Files for the writeup may be found in [https://www.dropbox.com/sh/wg4y7ptahwq3xo1/AABreDLrXH3hniz1jiFTtvska?dl=0 this directory].
* The arXiv preprint may be found [https://arxiv.org/abs/1801.03908 here].
* The paper has been published in [http://dx.doi.org/10.2140/ant.2018.12.1773 Algebra & Number Theory]].

Here are the [[linear norm grant acknowledgments]].

Main Page

2018-11-27T18:10:24Z

Khare: /* Existing polymath projects */

{{RightTOC}}
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?]

Many polymath projects will be proposed, planned, and run at [http://polymathprojects.org/ 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.
* [[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.
* [[imo 2009 q6|Mini-polymath1]]: Solving Problem 6 of the 2009 International Mathematical Olympiad. Initiated July 20, 2009; five proofs obtained so far.
* [[The polynomial Hirsch conjecture|Polymath3]]. The polynomial Hirsch conjecture. Proposed July 17, 2009; launched, September 30, 2010.
* [[finding primes|Polymath4]]: A deterministic way to find primes. Proposed July 27, 2009; launched Aug 9, 2009. Research results have now been published.
* [[The Erdős discrepancy problem|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.
* [[imo 2010|Mini-polymath2]]: Solving Problem 5 the 2010 International Mathematical Olympiad. Proposed Jun 12, 2010; launched and solved, Jul 8 2010.
* [[Improving the bounds for Roth's theorem|Polymath6]]: Improving the bounds for Roth's theorem. Proposed Feb 5, 2011.
* [[imo 2011|Mini-polymath3]]: Solving a problem from the 2011 International Mathematical Olympiad. Proposed Jun 9, 2011; launched and solved, Jul 19, 2011.
* [[imo 2012|Mini-polymath4]]: Solving a problem from the 2012 International Mathematical Olympiad. Proposed, Jun 3, 2012; launched, July 12 2012.
* [[The hot spots conjecture|Polymath7]]: Establishing the Hot Spots conjecture for acute-angled triangles. Proposed, May 31st, 2012; launched, Jun 8, 2012.
* [[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.
* [[Discretized Borel Determinacy and P=NP|Polymath9]]: exploring Borel determinacy-based methods for giving complexity bounds. Proposed, Oct 24, 2013; launched, Nov 3, 2013.
* [[The Erdos-Rado sunflower lemma|Polymath10]]: improving the bounds for the Erdos-Rado sunflower lemma. Launched, Nov 2, 2015.
* [[Frankl's union-closed conjecture|Polymath11]]: proving Frankl's union-closed conjecture. Proposed Jan 21, 2016; launched Jan 29, 2016. Concluded, Jan 17, 2017.
* [[Rota's conjecture|Polymath12]]: proving Rota's conjecture. Proposed Feb 28, 2017.
* [[Intransitive dice|Polymath13]]: Intransitive dice. Proposed Apr 28, 2017.
* [[linear norm|Polymath14]]: Classifying homogeneous norms on groups. Initiated Dec 16, 2017; solved Dec 21, 2017. Research results have now been published.
* [[De_Bruijn-Newman constant|Polymath15]]: Upper bounding the de Bruin-Newman constant. Proposed, Jan 24 2018; launched Jan 27 2018.
* [[Hadwiger-Nelson problem|Polymath16]]: Simplifying the lower bound proof for the Hadwiger-Nelson problem. Proposed, Apr 10, 2018; launched, Apr 14, 2018.

== Polymath-like projects ==

* 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.
* A wiki page clearinghouse for the [[Deolalikar P vs NP paper]]. Launched Aug 10, 2010.
* <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 [https://conference.portonvictor.org/wiki/Research_in_the_middle Research in the middle project] at [https://conference.portonvictor.org Virtual scientific conference].
* The page for the [[ABC conjecture]] contains links and information about Mochizuki's claimed proof of this conjecture.

== Proposed polymath projects ==

* [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).
* [[Boshernitzan’s problem]]. Proposed July 27, 2009.
* [http://gowers.wordpress.com/2009/09/16/possible-future-polymath-projects/ Possible future polymath projects]. Discussion opened September 16, 2009.
* [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.

A (partial) list of proposed projects can be found [http://polymathprojects.org/category/polymath-proposals/ 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 [[other proposed projects|here]].

== Discussions about polymath ==

* [http://gowers.wordpress.com/2009/01/27/is-massively-collaborative-mathematics-possible/ Is massively collaborative mathematics possible?] Tim Gowers, January 27, 2009.
* [http://lucatrevisan.wordpress.com/2009/02/01/a-peoples-history-of-mathematics/ A people's history of mathematics] Luca Trevisan, February 1, 2009.
* [http://michaelnielsen.org/blog/?p=553 The polymath project] Michael Nielsen, February 3, 2009.
* [http://www.neverendingbooks.org/index.php/yet-another-math20-proposal.html Yet another math 2.0 proposal] Lieven le Bruyn, February 11, 2009.
* [http://gowers.wordpress.com/2009/03/10/polymath1-and-open-collaborative-mathematics/ Polymath1 and open collaborative mathematics] Tim Gowers, March 10, 2009.
* [http://maxwelldemon.com/2009/03/14/polymath/ Polymath] Edmund Harriss, March 14, 2009.
* [http://science.slashdot.org/article.pl?sid=09/03/18/194228 Massive open collaboration in mathematics declared a success] Slashdot, March 18, 2009.
* [http://michaelnielsen.org/blog/?p=581 How changing the technology of collaboration can change the nature of collaboration] Michael Nielsen, March 18, 2009.
* [http://michaelnielsen.org/blog/?p=584 The polymath project: scope of participation] Michael Nielsen, March 20, 2009.
* [http://gowers.wordpress.com/2009/03/24/can-polymath-be-scaled-up/ Can polymath be scaled up?] Tim Gowers, March 24, 2009.
* [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.
* [http://michaelnielsen.org/blog/on-scaling-up-the-polymath-project/ On scaling up the polymath project] Michael Nielsen, March 25, 2009.
* [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.
* [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
* [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.
* [http://polymathprojects.org/2009/07/27/selecting-the-next-polymath-project/ Selecting the next polymath project] Terence Tao, July 27, 2009.
* [http://blog.jonudell.net/2009/07/31/polymath-equals-user-innovatio/ Polymath equals user innovation] Jon Udell, July 31, 2009.
* [http://scienceblogs.com/christinaslisrant/2009/08/an_overview_of_the_polymath_pr.php An overview of the polymath project] Christina Pikas, August 1, 2009
* [http://whatisresearch.wordpress.com/2009/08/09/collaborative-mathematics-etc/ Collaborative mathematics etc.] Vipulniak, August 9, 2009
* [http://www.nature.com/nature/journal/v461/n7266/full/461879a.html Massively collaborative mathematics] Tim Gowers, Michael Nielsen, Nature, October 15, 2009
* [http://portonmath.wordpress.com/2009/10/25/collaborative-research-of-filters/ Collaborative math research – a real example] Victor Porton, October 24, 2009
* [http://whatisresearch.wordpress.com/2009/10/26/polymath-again/ Polymath again] Vipulniak, October 26, 2009
* [http://www.kennislink.nl/publicaties/wiskunde-met-zijn-allen Wiskunde met zijn allen] (Dutch), Alex van den Brandhof, Kennislink, November 12, 2009
* [http://www.sciencenews.org/view/generic/id/50532/title/Mathematics_by_collaboration Mathematics by collaboration], Julie Rehmeyer, ScienceNews, December 8, 2009
* [http://www.nytimes.com/projects/magazine/ideas/2009/#m Massively Collaborative Mathematics], Jordan Ellenberg, The Ninth Annual Year in Ideas, New York Times, 2009.
* [http://www.hypios.com/thinking/2010/01/13/massively-collaborative-mathematics-lessons-from-polymath1/ Massively Collaborative Mathematics: lessons from polymath1], Hypios, Jan 13 2010
* [http://ths1104.wordpress.com/2010/02/13/open-reflexions-sur-fond-de-polymaths/ Open réflexions sur fond de Polymaths] (French), ths1104, Feb 13 2010
* [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
* [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
* [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
* [http://www.siam.org/news/news.php?issue=0043.03 Massively Collaborative Mathematics] Julie Rehmeyer, SIAM News, Volume 43(3), April 2010 (to appear)
* [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.
* [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.
* [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.
* [http://www.newscientist.com/article/mg21028112.900-mathematics-becomes-more-sociable.html Mathematics becomes more sociable], New Scientist, 5 May 2011.
* [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.
* [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.
* [http://online.wsj.com/article/SB10001424052970204644504576653573191370088.html The New Einsteins Will Be Scientists Who Share] The Wall street journal, October 2011.
* [http://www.nature.com/news/parallel-lines-1.14759?WT.ec_id=NATURE-20140227 Parallel lines], editorial, Nature 506, 407–408 (27 February 2014).
Additional links are very welcome.

== Other links ==

* [http://polymathprojects.org/ The polymath blog]
* [http://polymathprojects.org/general-polymath-rules/ General polymath rules]

== 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

Main Page

2018-11-27T18:09:57Z

Khare: Undo revision 10944 by Khare (talk)

{{RightTOC}}
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?]

Many polymath projects will be proposed, planned, and run at [http://polymathprojects.org/ 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.
* [[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.
* [[imo 2009 q6|Mini-polymath1]]: Solving Problem 6 of the 2009 International Mathematical Olympiad. Initiated July 20, 2009; five proofs obtained so far.
* [[The polynomial Hirsch conjecture|Polymath3]]. The polynomial Hirsch conjecture. Proposed July 17, 2009; launched, September 30, 2010.
* [[finding primes|Polymath4]]: A deterministic way to find primes. Proposed July 27, 2009; launched Aug 9, 2009. Research results have now been published.
* [[The Erdős discrepancy problem|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.
* [[imo 2010|Mini-polymath2]]: Solving Problem 5 the 2010 International Mathematical Olympiad. Proposed Jun 12, 2010; launched and solved, Jul 8 2010.
* [[Improving the bounds for Roth's theorem|Polymath6]]: Improving the bounds for Roth's theorem. Proposed Feb 5, 2011.
* [[imo 2011|Mini-polymath3]]: Solving a problem from the 2011 International Mathematical Olympiad. Proposed Jun 9, 2011; launched and solved, Jul 19, 2011.
* [[imo 2012|Mini-polymath4]]: Solving a problem from the 2012 International Mathematical Olympiad. Proposed, Jun 3, 2012; launched, July 12 2012.
* [[The hot spots conjecture|Polymath7]]: Establishing the Hot Spots conjecture for acute-angled triangles. Proposed, May 31st, 2012; launched, Jun 8, 2012.
* [[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.
* [[Discretized Borel Determinacy and P=NP|Polymath9]]: exploring Borel determinacy-based methods for giving complexity bounds. Proposed, Oct 24, 2013; launched, Nov 3, 2013.
* [[The Erdos-Rado sunflower lemma|Polymath10]]: improving the bounds for the Erdos-Rado sunflower lemma. Launched, Nov 2, 2015.
* [[Frankl's union-closed conjecture|Polymath11]]: proving Frankl's union-closed conjecture. Proposed Jan 21, 2016; launched Jan 29, 2016. Concluded, Jan 17, 2017.
* [[Rota's conjecture|Polymath12]]: proving Rota's conjecture. Proposed Feb 28, 2017.
* [[Intransitive dice|Polymath13]]: Intransitive dice. Proposed Apr 28, 2017.
* [[linear norm|Polymath14]]: Classifying homogeneous norms on groups. Initiated Dec 16, 2017. Results submitted, Jan 11, 2018.
* [[De_Bruijn-Newman constant|Polymath15]]: Upper bounding the de Bruin-Newman constant. Proposed, Jan 24 2018; launched Jan 27 2018.
* [[Hadwiger-Nelson problem|Polymath16]]: Simplifying the lower bound proof for the Hadwiger-Nelson problem. Proposed, Apr 10, 2018; launched, Apr 14, 2018.

== Polymath-like projects ==

* 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.
* A wiki page clearinghouse for the [[Deolalikar P vs NP paper]]. Launched Aug 10, 2010.
* <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 [https://conference.portonvictor.org/wiki/Research_in_the_middle Research in the middle project] at [https://conference.portonvictor.org Virtual scientific conference].
* The page for the [[ABC conjecture]] contains links and information about Mochizuki's claimed proof of this conjecture.

== Proposed polymath projects ==

* [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).
* [[Boshernitzan’s problem]]. Proposed July 27, 2009.
* [http://gowers.wordpress.com/2009/09/16/possible-future-polymath-projects/ Possible future polymath projects]. Discussion opened September 16, 2009.
* [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.

A (partial) list of proposed projects can be found [http://polymathprojects.org/category/polymath-proposals/ 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 [[other proposed projects|here]].

== Discussions about polymath ==

* [http://gowers.wordpress.com/2009/01/27/is-massively-collaborative-mathematics-possible/ Is massively collaborative mathematics possible?] Tim Gowers, January 27, 2009.
* [http://lucatrevisan.wordpress.com/2009/02/01/a-peoples-history-of-mathematics/ A people's history of mathematics] Luca Trevisan, February 1, 2009.
* [http://michaelnielsen.org/blog/?p=553 The polymath project] Michael Nielsen, February 3, 2009.
* [http://www.neverendingbooks.org/index.php/yet-another-math20-proposal.html Yet another math 2.0 proposal] Lieven le Bruyn, February 11, 2009.
* [http://gowers.wordpress.com/2009/03/10/polymath1-and-open-collaborative-mathematics/ Polymath1 and open collaborative mathematics] Tim Gowers, March 10, 2009.
* [http://maxwelldemon.com/2009/03/14/polymath/ Polymath] Edmund Harriss, March 14, 2009.
* [http://science.slashdot.org/article.pl?sid=09/03/18/194228 Massive open collaboration in mathematics declared a success] Slashdot, March 18, 2009.
* [http://michaelnielsen.org/blog/?p=581 How changing the technology of collaboration can change the nature of collaboration] Michael Nielsen, March 18, 2009.
* [http://michaelnielsen.org/blog/?p=584 The polymath project: scope of participation] Michael Nielsen, March 20, 2009.
* [http://gowers.wordpress.com/2009/03/24/can-polymath-be-scaled-up/ Can polymath be scaled up?] Tim Gowers, March 24, 2009.
* [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.
* [http://michaelnielsen.org/blog/on-scaling-up-the-polymath-project/ On scaling up the polymath project] Michael Nielsen, March 25, 2009.
* [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.
* [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
* [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.
* [http://polymathprojects.org/2009/07/27/selecting-the-next-polymath-project/ Selecting the next polymath project] Terence Tao, July 27, 2009.
* [http://blog.jonudell.net/2009/07/31/polymath-equals-user-innovatio/ Polymath equals user innovation] Jon Udell, July 31, 2009.
* [http://scienceblogs.com/christinaslisrant/2009/08/an_overview_of_the_polymath_pr.php An overview of the polymath project] Christina Pikas, August 1, 2009
* [http://whatisresearch.wordpress.com/2009/08/09/collaborative-mathematics-etc/ Collaborative mathematics etc.] Vipulniak, August 9, 2009
* [http://www.nature.com/nature/journal/v461/n7266/full/461879a.html Massively collaborative mathematics] Tim Gowers, Michael Nielsen, Nature, October 15, 2009
* [http://portonmath.wordpress.com/2009/10/25/collaborative-research-of-filters/ Collaborative math research – a real example] Victor Porton, October 24, 2009
* [http://whatisresearch.wordpress.com/2009/10/26/polymath-again/ Polymath again] Vipulniak, October 26, 2009
* [http://www.kennislink.nl/publicaties/wiskunde-met-zijn-allen Wiskunde met zijn allen] (Dutch), Alex van den Brandhof, Kennislink, November 12, 2009
* [http://www.sciencenews.org/view/generic/id/50532/title/Mathematics_by_collaboration Mathematics by collaboration], Julie Rehmeyer, ScienceNews, December 8, 2009
* [http://www.nytimes.com/projects/magazine/ideas/2009/#m Massively Collaborative Mathematics], Jordan Ellenberg, The Ninth Annual Year in Ideas, New York Times, 2009.
* [http://www.hypios.com/thinking/2010/01/13/massively-collaborative-mathematics-lessons-from-polymath1/ Massively Collaborative Mathematics: lessons from polymath1], Hypios, Jan 13 2010
* [http://ths1104.wordpress.com/2010/02/13/open-reflexions-sur-fond-de-polymaths/ Open réflexions sur fond de Polymaths] (French), ths1104, Feb 13 2010
* [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
* [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
* [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
* [http://www.siam.org/news/news.php?issue=0043.03 Massively Collaborative Mathematics] Julie Rehmeyer, SIAM News, Volume 43(3), April 2010 (to appear)
* [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.
* [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.
* [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.
* [http://www.newscientist.com/article/mg21028112.900-mathematics-becomes-more-sociable.html Mathematics becomes more sociable], New Scientist, 5 May 2011.
* [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.
* [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.
* [http://online.wsj.com/article/SB10001424052970204644504576653573191370088.html The New Einsteins Will Be Scientists Who Share] The Wall street journal, October 2011.
* [http://www.nature.com/news/parallel-lines-1.14759?WT.ec_id=NATURE-20140227 Parallel lines], editorial, Nature 506, 407–408 (27 February 2014).
Additional links are very welcome.

== Other links ==

* [http://polymathprojects.org/ The polymath blog]
* [http://polymathprojects.org/general-polymath-rules/ General polymath rules]

== 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

Main Page

2018-11-27T18:08:51Z

Khare: Undo revision 10945 by Khare (talk)

Main Page

2018-11-27T18:08:16Z

Khare:

{{RightTOC}}
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?]

Many polymath projects will be proposed, planned, and run at [http://polymathprojects.org/ 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.
* [[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.
* [[imo 2009 q6|Mini-polymath1]]: Solving Problem 6 of the 2009 International Mathematical Olympiad. Initiated July 20, 2009; five proofs obtained so far.
* [[The polynomial Hirsch conjecture|Polymath3]]. The polynomial Hirsch conjecture. Proposed July 17, 2009; launched, September 30, 2010.
* [[finding primes|Polymath4]]: A deterministic way to find primes. Proposed July 27, 2009; launched Aug 9, 2009. Research results have now been published.
* [[The Erdős discrepancy problem|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.
* [[imo 2010|Mini-polymath2]]: Solving Problem 5 the 2010 International Mathematical Olympiad. Proposed Jun 12, 2010; launched and solved, Jul 8 2010.
* [[Improving the bounds for Roth's theorem|Polymath6]]: Improving the bounds for Roth's theorem. Proposed Feb 5, 2011.
* [[imo 2011|Mini-polymath3]]: Solving a problem from the 2011 International Mathematical Olympiad. Proposed Jun 9, 2011; launched and solved, Jul 19, 2011.
* [[imo 2012|Mini-polymath4]]: Solving a problem from the 2012 International Mathematical Olympiad. Proposed, Jun 3, 2012; launched, July 12 2012.
* [[The hot spots conjecture|Polymath7]]: Establishing the Hot Spots conjecture for acute-angled triangles. Proposed, May 31st, 2012; launched, Jun 8, 2012.
* [[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.
* [[Discretized Borel Determinacy and P=NP|Polymath9]]: exploring Borel determinacy-based methods for giving complexity bounds. Proposed, Oct 24, 2013; launched, Nov 3, 2013.
* [[The Erdos-Rado sunflower lemma|Polymath10]]: improving the bounds for the Erdos-Rado sunflower lemma. Launched, Nov 2, 2015.
* [[Frankl's union-closed conjecture|Polymath11]]: proving Frankl's union-closed conjecture. Proposed Jan 21, 2016; launched Jan 29, 2016. Concluded, Jan 17, 2017.
* [[Rota's conjecture|Polymath12]]: proving Rota's conjecture. Proposed Feb 28, 2017.
* [[Intransitive dice|Polymath13]]: Intransitive dice. Proposed Apr 28, 2017.
* [[linear norm|Polymath14]]: Classifying homogeneous norms on groups. Initiated Dec 16, 2017, 2017; solved Dec 21, 2017. Research results have now been published.
* [[De_Bruijn-Newman constant|Polymath15]]: Upper bounding the de Bruin-Newman constant. Proposed, Jan 24 2018; launched Jan 27 2018.
* [[Hadwiger-Nelson problem|Polymath16]]: Simplifying the lower bound proof for the Hadwiger-Nelson problem. Proposed, Apr 10, 2018; launched, Apr 14, 2018.

== Polymath-like projects ==

* 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.
* A wiki page clearinghouse for the [[Deolalikar P vs NP paper]]. Launched Aug 10, 2010.
* <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 [https://conference.portonvictor.org/wiki/Research_in_the_middle Research in the middle project] at [https://conference.portonvictor.org Virtual scientific conference].
* The page for the [[ABC conjecture]] contains links and information about Mochizuki's claimed proof of this conjecture.

== Proposed polymath projects ==

* [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).
* [[Boshernitzan’s problem]]. Proposed July 27, 2009.
* [http://gowers.wordpress.com/2009/09/16/possible-future-polymath-projects/ Possible future polymath projects]. Discussion opened September 16, 2009.
* [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.

A (partial) list of proposed projects can be found [http://polymathprojects.org/category/polymath-proposals/ 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 [[other proposed projects|here]].

== Discussions about polymath ==

* [http://gowers.wordpress.com/2009/01/27/is-massively-collaborative-mathematics-possible/ Is massively collaborative mathematics possible?] Tim Gowers, January 27, 2009.
* [http://lucatrevisan.wordpress.com/2009/02/01/a-peoples-history-of-mathematics/ A people's history of mathematics] Luca Trevisan, February 1, 2009.
* [http://michaelnielsen.org/blog/?p=553 The polymath project] Michael Nielsen, February 3, 2009.
* [http://www.neverendingbooks.org/index.php/yet-another-math20-proposal.html Yet another math 2.0 proposal] Lieven le Bruyn, February 11, 2009.
* [http://gowers.wordpress.com/2009/03/10/polymath1-and-open-collaborative-mathematics/ Polymath1 and open collaborative mathematics] Tim Gowers, March 10, 2009.
* [http://maxwelldemon.com/2009/03/14/polymath/ Polymath] Edmund Harriss, March 14, 2009.
* [http://science.slashdot.org/article.pl?sid=09/03/18/194228 Massive open collaboration in mathematics declared a success] Slashdot, March 18, 2009.
* [http://michaelnielsen.org/blog/?p=581 How changing the technology of collaboration can change the nature of collaboration] Michael Nielsen, March 18, 2009.
* [http://michaelnielsen.org/blog/?p=584 The polymath project: scope of participation] Michael Nielsen, March 20, 2009.
* [http://gowers.wordpress.com/2009/03/24/can-polymath-be-scaled-up/ Can polymath be scaled up?] Tim Gowers, March 24, 2009.
* [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.
* [http://michaelnielsen.org/blog/on-scaling-up-the-polymath-project/ On scaling up the polymath project] Michael Nielsen, March 25, 2009.
* [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.
* [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
* [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.
* [http://polymathprojects.org/2009/07/27/selecting-the-next-polymath-project/ Selecting the next polymath project] Terence Tao, July 27, 2009.
* [http://blog.jonudell.net/2009/07/31/polymath-equals-user-innovatio/ Polymath equals user innovation] Jon Udell, July 31, 2009.
* [http://scienceblogs.com/christinaslisrant/2009/08/an_overview_of_the_polymath_pr.php An overview of the polymath project] Christina Pikas, August 1, 2009
* [http://whatisresearch.wordpress.com/2009/08/09/collaborative-mathematics-etc/ Collaborative mathematics etc.] Vipulniak, August 9, 2009
* [http://www.nature.com/nature/journal/v461/n7266/full/461879a.html Massively collaborative mathematics] Tim Gowers, Michael Nielsen, Nature, October 15, 2009
* [http://portonmath.wordpress.com/2009/10/25/collaborative-research-of-filters/ Collaborative math research – a real example] Victor Porton, October 24, 2009
* [http://whatisresearch.wordpress.com/2009/10/26/polymath-again/ Polymath again] Vipulniak, October 26, 2009
* [http://www.kennislink.nl/publicaties/wiskunde-met-zijn-allen Wiskunde met zijn allen] (Dutch), Alex van den Brandhof, Kennislink, November 12, 2009
* [http://www.sciencenews.org/view/generic/id/50532/title/Mathematics_by_collaboration Mathematics by collaboration], Julie Rehmeyer, ScienceNews, December 8, 2009
* [http://www.nytimes.com/projects/magazine/ideas/2009/#m Massively Collaborative Mathematics], Jordan Ellenberg, The Ninth Annual Year in Ideas, New York Times, 2009.
* [http://www.hypios.com/thinking/2010/01/13/massively-collaborative-mathematics-lessons-from-polymath1/ Massively Collaborative Mathematics: lessons from polymath1], Hypios, Jan 13 2010
* [http://ths1104.wordpress.com/2010/02/13/open-reflexions-sur-fond-de-polymaths/ Open réflexions sur fond de Polymaths] (French), ths1104, Feb 13 2010
* [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
* [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
* [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
* [http://www.siam.org/news/news.php?issue=0043.03 Massively Collaborative Mathematics] Julie Rehmeyer, SIAM News, Volume 43(3), April 2010 (to appear)
* [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.
* [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.
* [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.
* [http://www.newscientist.com/article/mg21028112.900-mathematics-becomes-more-sociable.html Mathematics becomes more sociable], New Scientist, 5 May 2011.
* [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.
* [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.
* [http://online.wsj.com/article/SB10001424052970204644504576653573191370088.html The New Einsteins Will Be Scientists Who Share] The Wall street journal, October 2011.
* [http://www.nature.com/news/parallel-lines-1.14759?WT.ec_id=NATURE-20140227 Parallel lines], editorial, Nature 506, 407–408 (27 February 2014).
Additional links are very welcome.

== Other links ==

* [http://polymathprojects.org/ The polymath blog]
* [http://polymathprojects.org/general-polymath-rules/ General polymath rules]

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