The "bounded gaps between primes" Polymath project - a retrospective

Abstract: For any $m \geq 1$, let $H_m$ denote the quantity $H_m := \liminf_{n \to \infty} (p_{n+m}-p_n)$, where $p_n$ denotes the $n^{\operatorname{th}}$ prime; thus for instance the twin prime conjecture is equivalent to the assertion that $H_1$ is equal to two. In a recent breakthrough paper of Zhang, a finite upper bound was obtained for the first time on $H_1$; more specifically, Zhang showed that $H_1 \leq 70000000$. Almost immediately after the appearance of Zhang's paper, improvements to the upper bound on $H_1$ were made. In order to pool together these various efforts, a \emph{Polymath project} was formed to collectively examine all aspects of Zhang's arguments, and to optimize the resulting bound on $H_1$ as much as possible. After several months of intensive activity, conducted online in blogs and wiki pages, the upper bound was improved to $H_1 \leq 4680$. As these results were being written up, a further breakthrough was introduced by Maynard, who found a simpler sieve-theoretic argument that gave the improved bound $H_1 \leq 600$, and also showed for the first time that $H_m$ was finite for all $m$. The polymath project, now with Maynard's assistance, then began work on improving these bounds, eventually obtaining the bound $H_1 \leq 246$, as well as a number of additional results, both conditional and unconditional, on $H_m$. In this article, we collect the perspectives of several of the participants to these Polymath projects, in order to form a case study of online collaborative mathematical activity, and to speculate on the suitability of such an online model for other mathematical research projects.