Polymath Wiki - User contributions [en]

Definable Banach Spaces

2010-06-10T08:32:34Z

Colin Tan: /* Fundamental Tension Resulting in This Problem */ correct typo

This is an attempt to summarize in the form of a wiki the proposals for formalizing the notion of a "definable" Banach space that has been brought in the comments to [https://gowers.wordpress.com/2009/02/17/must-an-explicitly-defined-banach-space-contain-c_0-or-ell_p/ this post] at Gower's Weblog.

== Fundamental Tension Resulting in This Problem ==

As with much of mathematics, Banach spaces developed from explicit examples, leading to an axiomatization and consequent theory. The explicit examples of Banach spaces have historically been function spaces.

Let <math>(X,{\mathcal{M}},\mu) </math> be a measure space. The <math>L_p</math>, Lorentz, Olicz, Schreier, Sobolev, Besov spaces are in general defined as the class of functions <math>f:X\to {\mathbb{R}}</math> with a finite norm <math> \| f\| < +\infty</math>. These spaces contain either <math>c_0</math> or <math>l_p</math> as a subspace for some <math>p\in [1,+\infty)</math>.

In the above paragraph, we make the choice to use function spaces of functions over '''an abstract measure space''' rather than over the natural numbers with counting measure. Choosing to focus on Banach spaces of sequences of real numbers obscures the dual role that the natural numbers <math>{\mathbb{N}}</math> plays in the notion of definablility:
# The natural numbers serve as the domain of these functions;
# The real numbers can be coded as subsets of the natural numbers, via a bijection between <math>{\mathcal{P}}({\mathbb{N}})</math> and <math>{\mathbb{R}}</math>.
Admittedly, using an abstract measure space entails that the notion of definability of a Banach space will be relative to this abstract measure space. By focusing on an abstract measure, the first role that the natural numbers play will be removed, and the recursion theory used when employing the second role will be stark and clear.

Let us now look at a space, Tsirelson space, which when contrasted with the spaces above lead to the fundamental tension resulting in this problem.

== Proposals To Formalize The Notion of Definability ==

These proposals are listed in approximate choronological order as they appear on [https://gowers.wordpress.com/2009/02/17/must-an-explicitly-defined-banach-space-contain-c_0-or-ell_p/ this post] at Gower's Weblog.

* The asymptotic complexity of functions which the norm gives rise to.
Although the Tsirelson space gives rise to fast-growing functions, Gowers defined a variant known as Schlumprecht’s space in an attempt to give an example of a norm that does not give rise to fast-growing functions.
* The recursive complexity of functions which the norm gives rise to.
* The extent to which the space is combinatorial.
* Placing the definition of norm in a hierarchy of recursive notions weaker than primitive recursion.
The definition of the norm of the Tsirelson space is primitive recursive. In order to distinguish the Tsirelson space from other spaces explicitly defined, it is necessary to have notions of recursion that are weaker than recursive notions.
* Impredicativity of the definition of norm.

Definable Banach Spaces

2010-06-09T06:32:14Z

Colin Tan: Problem motivation and Proposals of Definability

This is an attempt to summarize in the form of a wiki the proposals for formalizing the notion of a "definable" Banach space that has been brought in the comments to [https://gowers.wordpress.com/2009/02/17/must-an-explicitly-defined-banach-space-contain-c_0-or-ell_p/ this post] at Gower's Weblog.

== Fundamental Tension Resulting in This Problem ==

As with much of mathematics, Banach spaces developed from explicit examples, leading to an axiomatization and consequent theory. The explicit examples of Banach spaces have historically been function spaces.

Let <math>(A,{\mathcal{M}},\mu) </math> be a measure space. The <math>L_p</math>, Lorentz, Olicz, Schreier, Sobolev, Besov spaces are in general defined as the class of functions <math>f:X\to {\mathbb{R}}</math> with a finite norm <math> \| f\| < +\infty</math>. These spaces contain either <math>c_0</math> or <math>l_p</math> as a subspace for some <math>p\in [1,+\infty)</math>.

In the above paragraph, we make the choice to use function spaces of functions over '''an abstract measure space''' rather than over the natural numbers with counting measure. Choosing to focus on Banach spaces of sequences of real numbers obscures the dual role that the natural numbers <math>{\mathbb{N}}</math> plays in the notion of definablility:
# The natural numbers serve as the domain of these functions;
# The real numbers can be coded as subsets of the natural numbers, via a bijection between <math>{\mathcal{P}}({\mathbb{N}})</math> and <math>{\mathbb{R}}</math>.
Admittedly, using an abstract measure space entails that the notion of definability of a Banach space will be relative to this abstract measure space. By focusing on an abstract measure, the first role that the natural numbers play will be removed, and the recursion theory used when employing the second role will be stark and clear.

Let us now look at a space, Tsirelson space, which when contrasted with the spaces above lead to the fundamental tension resulting in this problem.

== Proposals To Formalize The Notion of Definability ==

These proposals are listed in approximate choronological order as they appear on [https://gowers.wordpress.com/2009/02/17/must-an-explicitly-defined-banach-space-contain-c_0-or-ell_p/ this post] at Gower's Weblog.

* The asymptotic complexity of functions which the norm gives rise to.
Although the Tsirelson space gives rise to fast-growing functions, Gowers defined a variant known as Schlumprecht’s space in an attempt to give an example of a norm that does not give rise to fast-growing functions.
* The recursive complexity of functions which the norm gives rise to.
* The extent to which the space is combinatorial.
* Placing the definition of norm in a hierarchy of recursive notions weaker than primitive recursion.
The definition of the norm of the Tsirelson space is primitive recursive. In order to distinguish the Tsirelson space from other spaces explicitly defined, it is necessary to have notions of recursion that are weaker than recursive notions.
* Impredicativity of the definition of norm.

Main Page

2010-06-09T05:47:25Z

Colin Tan: /* Existing polymath projects */ linked to wiki on polymath2

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].

== Existing polymath projects ==

* [[Polymath1]]: New proofs and bounds for the density Hales-Jewett theorem. Initiated Feb 1, 2009; research results have now been submitted for publication.
* [[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.
* [[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.
* [[finding primes|Polymath4]]: A deterministic way to find primes. Proposed July 27, 2009; launched Aug 9, 2009.
* [[The Erdős discrepancy problem|Polymath5]]. The Erdős discrepancy problem. Proposed Jan 10, 2010; launched Jan 19, 2010.

== 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.

A (partial) list of proposed projects can be found [http://en.wordpress.com/tag/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)

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 ==

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Definable Banach Spaces

2010-06-09T05:37:14Z

Colin Tan: Set up main page for polymath2