  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.
