Crowdsourcing Mathematics

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By Alan Scrivner

Sir Andrew Wiles, Image copyright C. J. Mozzochi, Princeton N.J, Freely reused.

Mathematicians love problem solving and there is nothing like a good problem to bring them out to toss ideas around. This year, as he has for the last three years, Terence Tao is hosting a mini-polymath.  This a online group discussion, run at the same time the International Math Olympiad (IMO) to solve one of the IMO’s more challenging problems. Mini-polymath4 will start at Thu July 12 2012 UTC 22:00.  As usual, the main research thread will be held at the polymath blog, with the discussion thread hosted separately on Terrence Tao’s blog. Everyone is welcome to come and toss their ideas into the ring or just stand on the sidelines and see how mathematicians think and solve problems.

 

Mathematics is typically thought of as a solitary affair - a lone mathematician toiling away for days, weeks, or years on end with the singular goal of solving an abstract problem in mathematics. Indeed, this is exactly how Andrew Wiles spent seven years of his life before cracking, what is arguably the most famous problem in mathematics, Fermat’s Last Theorem [1].

He did this not by solving the immediate problem, that , but by proving a theorem on modular forms called the Taniyama-Shumura conjecture which states that all elliptic curves are modular forms and, if true, implied Fermat’s Last Theorem.

However, this solitary approach to problem solving is starting to change. In a 2009 post on his blog, Timothy Gowers, a Cambridge University mathematician and a Fields Medalist (1998), poised the provocative question - “Is massively collaborative mathematics possible? [2].” His post began mathematical crowdsourcing and led to the creation of the Polymath Project [3] where blog comments are used to solve serious mathematical problems collaboratively and the mini-polymath project where mathematicians go to have some fun.

The initial problem proposed by Gowers for “Polymath 1,” as is called, was to find a combinatorial proof for the density version of the Hales-Jewett theorem. This theorem is stated below.

The density Hales-Jewett theorem (DHJ) had already been solved but the proof was extremely complex and Gowers believed a simpler one was possible.

The H-J Theorem is usually explained with reference to a tic-tac-toe game (played in multiple dimensions). To get the gist of it, visualize such a game played in multiple dimensions with multiple sets of squares. Then ask yourself how many squares you would have to block off to prevent the other player from winning? Using this image of the game, DHJ states that the more dimensions you have the more squares you would have to block off. Fascinating in its own right, the theorem rests in a pretty active mathematical space, so it was believed that work on the problem would likely have a larger-than-average effect.

Well, the first experiment in this direction succeeded way beyond Gowers’ expectations. More than two dozen mathematicians came together on the blog and found a direct, combinatorial proof of the density Hales–Jewett theorem over the course of a mere six weeks.

Polymth1 has spawned additional polymath projects and the community is currently attempting to solve Polymath7 [4] on “The Hotspot Conjecture” which can be thought of as follows:

Suppose a flat piece of metal, represented by a two-dimensional bounded connected domain, is given an initial heat distribution which then flows throughout the metal. Assuming the metal is insulated (i.e. no heat escapes from the piece of metal), then given enough time, the hottest point on the metal will lie on its boundary.

Or, in more mathematical terms - a (generic) eigenfunction of the first nontrivial eigenvalue of the Laplacian (with Neumann boundary data) attains its maximum on the boundary.

As described in a Nature article that appeared shortly following the DHJ proof.

“For the first time one can see on full display a complete account of how a serious mathematical result was discovered. It shows vividly how ideas grow, change, improve and are discarded, and how advances in understanding may come not in a single giant leap, but through the aggregation and refinement of many smaller insights [5].”

In addition to the polymath projects, other crowd sourced mathematical forums have appeared. Below are brief descriptions of some of the better known:

TRIKI: A website with a large store of useful mathematical problem-solving techniques. Some of these techniques are very general, but all of them are used regularly by mathematical problem-solvers (http://www.tricki.org/).

MathOverflow: MathOverflow's primary goal is for users to ask and answer research level math questions, the sorts of questions you come across when you're writing or reading articles or graduate level books (http://mathoverflow.net/)

Mathematics - Stack Exchange: This is for people studying mathematics at any level and professionals in related fields. (http://math.stackexchange.com/) Common questions one will see on this site are:

  • Understanding mathematical concepts and theorems
  • Hints on mathematical problems (but see the FAQ about homework questions first)
  • The history and development of mathematics
  • Solving mathematical puzzles
  • Software that mathematicians use

Footnotes:

[1] http://math.stanford.edu/~lekheng/flt/wiles.pdf Last accessed 24 June 2012.

[2] http://gowers.wordpress.com/2009/01/27/is-massively-collaborative-mathematics-possible/
Last accessed 24 June 2012.

[3] http://polymathprojects.org/ Last accesses 24 June 2012.

[4] http://polymathprojects.org/2012/06/09/polymath7-discussion-thread/ Last accessed 24 June 2012.

[5]T. Gowers and M. Nielsen, “Massively collaborative mathematics,” Nature, vol. 461, no. 7266, pp. 879–881, Oct. 2009.

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