Sometimes proofs of long-outstanding problems emerge without prior warning. In the 1990s, Andrew Wiles proved Fermat’s Last Theorem. More recently, Yitang Zhang announced a key result on bounded gaps in the prime numbers. Both Wiles and Zhang had worked for years in isolation, keeping abreast of developments but carrying out intensive research programs unaided by others. This ensured that they did not have to share the glory of discovery, but it may not be an optimal way of making progress in mathematics.
Polymath
Timothy Gowers in 2012 [image Wikimedia Commons].
Is massively collaborative mathematics possible? This was the question posed in a 2009 blog post by Timothy Gowers, a Cambridge mathematician and Fields Medal winner. Gowers suggested completely new ways in which mathematicians might work together to accelerate progress in solving some really difficult problems in maths. He envisaged a forum for the online discussion of problems. Anybody interested could contribute to the discussion. Contributions would be short, and could include false routes and failures; these are normally hidden from view so that different workers repeat the same mistakes.
Some problems lend themselves to huge collaborations. For example, the classification of finite simple groups resulted from collaboration between hundreds of researchers, with different mathematicians or groups working on different, but related, topics. In the field of computation, the search for ever-larger prime numbers is carried out under the GIMPS Project: GIMPS stands for Grand Internet Mersenne Prime Search, and has broken the record many times for the largest prime.
It is more difficult, but possible, to distribute the effort to solve problems that do not easily split up into a large number of sub-tasks. Gowers discussed the way in which this could be done. He gave the name Polymath to the programme, and he also proposed some ground-rules for how the collaboration would operate.
Ongoing Success
The first problem suggested by Gowers – Polymath1 – was to find a new proof of the so-called (Density) Hales–Jewett Theorem (DJH). This result in Ramsey Theory had already been proved, but Gowers asked for a combinatorial proof. Within two months, as a result of many contributions, the proof had emerged, and was later published under the pseudonym DJH Polymath.
Several other problems followed. The work of Yitang Zhang inspired Polymath8. This had the aim of improving the bounds for small gaps between primes. Polymath8 resulted in several papers. The project was arranged into two components: Polymath8a, called “Bounded gaps between primes”, was a project to reduce the least gap between consecutive primes that is attained infinitely often, aiming to refine the techniques introduced by Zhang. The best estimate of the bounding gap was 4,680. Polymath8b, called “Bounded intervals with many primes”, aimed to combine Zhang’s methods with the techniques used by John Maynard. This project concluded with a bound on the gap of 246.
A recent review of the Polymath (https://polymathprojects.org/) indicates that the successful first project in 2009 has been followed by 15 other formal polymath projects together with a few other projects of a similar nature. It will be interesting to see how the collaboration develops in the future.
Sources
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Gowers original 2009 blog post: Is massively collaborative mathematics possible?
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The current central hub of the Polymath project is HERE
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Wikipedia article: The Polymath Project.
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Home page of the Polymath8 Project: Bounded Gaps between Primes.
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A 10-year Review of the Polymath Project.
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