Matemáticos colaboram num projecto aberto de investigação conjunta através do blogue do Professor Timothy Gowers

Print screen do feed na barra lateral deste blogue, 11-3-2009

Concretizou-se um novo modelo de produzir Matemática usando como meio principal o blogue do Professor Timothy Gowers, que em 1-2-2009 nele lançou um projecto aberto de tipo colaborativo, escolhendo para o iniciar o que considerou um exemplo genuíno de um problema de investigação matemática.

No seu último post — Polymath1 and open collaborative mathematics –, passado seis semanas, faz um balanço da iniciativa. Um dos aspectos mais relevante é porventura o próprio processo de surgimento de ideias e de interacção que houve entre os participantes.

Quanto ao resultado obtido, anunciou-o em Problem solved (probably).

25.06.09: Em post de hoje (DHJ write-up and other matters) o Professor Gowers anuncia que

http://www.cs.cmu.edu/~odonnell/dhj-june-25.pdf

é  um rascunho completo de A new proof of the density Hales-Jewett theorem, D. H. J. Polymath*, June 25, 2009.

« Abstract

The Hales-Jewett theorem asserts that for every $r$ and every $k$ there exists $n$ such that every$r$-colouring of the $n$-dimensional grid $\{1,...,k\}^{n}$ contains a combinatorial line. This result is a generalization of van der Waerden’s theorem, and it is one of the fundamental results of Ramsey theory. The theorem of van derWaerden has a famous density version, conjectured by Erdös and Turán an in 1936, proved by Szemerédi in 1975 and given a different proof by Furstenberg in 1977. The Hales-Jewett theorem has a density version as well, proved by Furstenberg and Katznelson in 1991 by means of a signicant extension of the ergodic techniques that had been pioneered by Furstenberg in his proof of Szemerédi’s theorem. In this paper, we give the first elementary proof of the theorem of Furstenberg and Katznelson, and the first to provide a quantitative bound on how large $n$ needs to be. In particular, we show that a subset of $[3]^{n}$ of density contains a combinatorial line if $n\geq 2\uparrow\uparrow O(1/\delta ^{3})$. Our proof is surprisingly[["reasonably", maybe]] simple: indeed, it gives what is probably the simplest known proof of Szemerédi’s theorem. »

26.06.09: Por parte do Professor Terence Tao, sabe quem acompanha o seu blogue que está igualmente envolvido no projecto. O artigo  DHJ: Still writing the second paper, de 14.06.09, é o último sobre o assunto.

17.01.10: O draft quase final do artigo “Density Hales-Jewett and Moser numbers”, em versão pdf é este. A este respeito, o Professor Terence Tao escreveu no seu blogue:

« I’ve made an effort to advance the writing of the second Polymath1 paper, entitled “Density Hales-Jewett and Moser numbers”. This is in part due to a request from the Szemeredi 60th 70th birthday conference proceedings (which solicited the paper) to move the submission date up from April to February. »

No resumo do artigo lê-se:

« Abstract. For any $n\geq 0$ and $k\geq 1$, the density Hales-Jewett number $c_{n,k}$ is defined as the size of the largest subset of the cube $[k]^{n}:=\left\{ 1,\ldots ,k\right\} ^{n}$, which contains no combinatorial line; similarly, the Moser number $c_{n,k}^{\prime }$ is the largest subset of the cube $[k]^{n}$ which contains no geometric line. A deep theorem of Furstenberg and Katznelson [11], [12], [19] shows that $c_{n,k}=o\left( k^{n}\right)$ as $n\rightarrow \infty$ (which implies a similar claim for $c_{n,k}^{\prime }$; this is already non-trivial for $k=3$. Several new proofs of this result have also been recently established [23], [2].

Using both human and computer-assisted arguments, we compute several values of $c_{n,k}$ and $c_{n,k}^{\prime }$ for small $n,k$. For instance the sequence $c_{n,3}$ for $n=0,\ldots ,6$ is $1,2,6,18,52,150,450$, while the sequence $c_{n,k}^{\prime }$ for $n=0,\ldots,6$ is $1,2,6,16,43,124,353$.

We also establish some results for higher $k$, showing for instance that an analogue of the LYM inequality (which relates to the $k=2$ case) does not hold for higher $k$. »

16.01.10: dentro do mesmo espírito, está em curso o The Erdős discrepancy problem, no blogue do Professor Gowers, que já vai no seu quarto artigo sobre este problema.

18.01.10: continuando no quinto.

20.01.10: E, em 19.01.10, o artigo EDP1 — the official start of Polymath5, no Gowers’s Weblog,  lança oficialmente o ataque ao The Erdős discrepancy problem.

2.03.10:  Com o título sugestivo EDP — a new and very promising approach prossegue a tentativa.

6.03.10: Um rascunho (draft) de Scott Morrison sobre o Math Overflow, publicado ontem, termina assim:

It is hard to explain Math Overflow without showing it to you, so please visit http://mathoverflow.net and poke around! If you have a question that’s been bothering you that you are pretty sure someone  must able to answer, try asking it. If you find the main page overwhelming, go to http://mathoverflow.net/tags and click on a tag corresponding to your specialty. Currently some areas of math are disproportionately represented, but just as with the arXiv, it is gradually “filling out” as it matures. We’d love to have mathematicians from more areas actively involved, and already you’re sure to find something you’re interested in.

[1] John Baez, Notices of the AMS, Febuary 2010 Volume 57 Issue 03 http://www.ams.org/notices/201003/rtx100300333p.pdf

[2] Secret blogging seminar, http://sbseminar.wordpress.com/

[3] Terry Tao’s blog, http://terrytao.wordpress.com/

[4] Gowers’ blog, http://gowers.wordpress.com/

[5] The nLab, http://ncatlab.org/nlab/show/HomePage

[6] Polymath project, http://polymathprojects.org/

[7] Shtetl-Optimized, “Prove my lemma, get acknowledged in a paper!”, http://scottaaronson.com/blog/?p=432

[8] F. Calegari, S. Morrison, N. Snyder, “Cyclotomic integers, fusion categories, and subfactors”, to appear.

[9] N. Snyder, “Number theoretic spectral properties of random graphs”, http://mathoverflow.net/questions/5994/.

[10] Mathoverflow database dumps, http://dumps.mathoverflow.net/