Polymath

I told here the story of W. E. Opencomb, a nom de plume for a set of ten mathematicians (including me) at a research week at the Open University.

This and other examples show that large collaborations in mathematics, though not yet common, are not a new thing. But it is true that technology, and the leadership of people like Tim Gowers and Terry Tao, have changed the character of such collaborations.

Gowers proposed an experiment in collaboration on his blog in 2007, and receiving an enthusiastic response, set up the first Polymath project. The aim was to find an “elementary” combinatorial proof of the density version of the Hales–Jewett theorem. Success was declared in six weeks, though the write-up took a bit longer. The collaborators chose the name D. H. J. Polymath to publish the result: the initials for the theorem they had proved, and Polymath for the generic name of the collaboration, which used technological facilities such as blogs and wikis to enhance the collaboration.

There have been several Polymath projects since, not all of them so successful. But Polymath 8 has achieved much mathematics, and much notoriety, and is described by an article by D. H. J. Polymath in the current Newsletter of the European Mathematical Society, which I recommend.

The impetus for the project was the astonishing result by Yitang Zhang, that there are infinitely many prime pairs differing by at most 70000000. Tao fairly soon decided that improving the bound would make a good Polymath project, since there are many places where Zhang’s estimates could be refined, and these would involve a variety of mathematical and programming skills. Many people contributed, and eventually the bound was reduced to 4680.

Then James Maynard came up with a different argument which reduced Zhang’s bound to 600. (He has just been awarded the 2014 Ramanujan Prize for this and other work.) So it was decided to enlist him in the second phase of Polymath 8, and see how far this bound could be improved. They have got it down to 246, and there are various generalisations and conditional results as well.

One of the reasons for the wide interest created is that there was a “headline” figure, namely the best value so far achieved, and mathematicians on the sideline could watch this figure decreasing as the project continued, and read and understand the arguments used.

The EMS Newsletter article collects the perspectives of ten of the participants (Tao, Andrew Gibson, Pace Nielsen, Maynard, Gergely Harcos, David Roberts, Andrew Sutherland, Wouter Castryck, Emmanuel Kowalski, and Philippe Michel) on their involvement in the project. Some common themes run through their accounts:

• One of the hardest things was to get used to making mistakes in public, with no chance to suppress them; but all the participants who commented on this felt that it was crucial for the success of the project that everyone was prepared to contribute and risk looking foolish.
• Several participants, especially the younger ones, remarked on the danger for a non-tenured mathematician taking part in a Polymath collaboration, since no proper academic credit can be given; but none of them regretted their own involvement.
• There was a lot of praise for the way that Tao as “ringmaster” handled the project, which was felt to have contributed to its success.
• Everyone spoke of the excitement of the collaboration, and the exhaustion which it produced!

(In the Opencomb collaboration, the first point was not an issue, since we could stick our necks out and only appear foolish to the other nine participants; in Polymath 8, if not all the world, then a good crowd of spectators was watching.)