Big Project Proves Many Hands Do Not Spoil Math Broth

October 28, 2009

Many minds working in tandem can collaborate and solve a difficult mathematical problem.

So claimed Timothy Gowers (University of Cambridge) and Toronto-based author-physicist Michael Nielsen about a Polymath project: an effort to find an elementary proof of a special case of the density Hales–Jewett theorem (DHJ), which is a central outcome in combinatorics. Although a long and complex proof exists, the two were in search of a solution involving basic techniques, which might spur mathematical breakthroughs.

After Gowers posted a description of the problem, pointers to background material, and rules for online collaboration, the first replies rolled in, on Feb. 1, 2009, from Jozsef Solymosi (University of British Columbia); Arizona high school teacher Jason Dyer; and Terence Tao (University of California, Los Angeles). In the course of several weeks, more than two dozen people contributed 800 "substantive comments, containing 170,000 words," according to the team leaders. Nielsen set up a wiki of "notable insights from the blog discussions."

By early March 2009, Gowers indicated that not only had the continent-wide collaboration produced an elementary proof of the special case of DHJ, but that "the argument could be straightforwardly generalized" to prove the complete theorem. The project had attracted notice by 16 blogs, made the front page of the Slashdot technology-news aggregator—and spawned related mathematics on Tao's blog.

Results: Papers describing the proof and related mathematics are in process; a record of the project offers insight into ideas and methods behind mathematical discovery; and Tim Austin (University of California, Los Angeles) produced a non-elementary proof of DHJ based on ideas from this Polymath project.

Conclusion: "Who would have guessed that the working record of a mathematical project would read like a thriller?" Gower and Nielsen write in a commentary published in the Oct. 15 Nature.

Source: Nature, Oct. 15, 2009.