2014-12-94

Featuresure that comes from deriving an elegant computational result.Another reason I joined is that I wanted to understandYitang Zhang’s breakthrough as much as possible. His resultwas quickly followed by James Maynard’s (and independentlyTerry Tao’s) multi-dimensional, probabilistically motivatedsieve, which is also incredibly interesting to me. Inmy opinion, this improved sieve deserves a spot as one of thetop advancements in analytic number theory of the past halfcentury.My initial incursions into the project consisted mainly ofpointing out minor corrections to the Polymath8a paper andasking questions about some of Tao’s blog posts. Hence, Idon’t consider myself a full participant in the 8a portion ofthe work; it was only during the 8b half of the work that Ibecame a contributor.There were a few things that surprised me about the wholeexperience. First was the friendliness of the other participants,particularly our host Terry Tao. I want to publicly thank everyoneinvolved in the project for the positive experience. Aspecial thanks goes to James Maynard, whose kindness insending me some of his original Mathematica code was whatfinally pushed me into full activity in the project.Second, a large number of mathematicians I know commented(in personal communications to me) on the fact thatthey were “impressed with my bravery” in participating. Itcaught me off guard that so many people had been followingthe project online and that all of my comments (including mymistakes) were open to such a wide readership. I believe it isimportant to consider this issue before deciding to participatein a public project. Some of the mistakes I made would neverhave seen the light of day in a standard mathematical partnership.However, any collaboration relies on the ability for theparticipants to share ideas freely, even the “dumb” ones.The third surprise was how much time I devoted to thisproject. I think that came from really getting excited aboutthe work. This is also the point at which I want to mention a(mild) concern on my part. As a pre-tenure professor, I madethe conscious choice going into this project that any time Ispent was not necessarily going to be reflected in my tenurefile. I knew that even if I contributed enough work to be notedas an official “participant” in the online acknowledgements,it wasn’t clear whether this would count as co-authorship inthe eyes of those on my tenure committee or even of my colleaguesgenerally. Indeed, there are aspects of this type of cooperativemathematics which make it difficult to decide howparticipation in this venture should be treated by the mathematicalcommunity at large. There are so many levels of effortthat it can be somewhat confusing where the line is drawn betweenproviding comments vs. being a full co-author vs. everythingin-between. Also, the Polymath projects don’t followthe convention used in other sciences of having the projectmanager decide who is a co-author on the final paper.On the one hand, I support the idea of collaborative mathematicswithout an eye towards recognition. With respect tothe Polymath8b project in particular, I’m happy to give D. H.J. Polymath all the credit. On the other hand, I do considermyself a co-author on the Polymath8b portion of the projectand thus want to take ownership for my part of the work.The fourth and final surprise was that I did contributesomething meaningful! Sometimes this contribution happenedby making simple comments on the blog. For instance, I rememberwaking up one morning with the realisation that aconstruction we were attempting would contradict the paritybarrier in sieves. This idea then led the experts to write someinteresting formal mathematics on this issue. Sometimes mycontribution was made in the time-consuming, busy work ofwriting, running and debugging computer code. And sometimesit was just in contributing my experience after thinkingabout the geometric picture long enough. While it is intimidatingworking with Fields Medallists and other experts, itis also a once in a lifetime opportunity to rub shoulders withsuch a wide array of mathematicians.While I rate my involvement as extremely positive, otherswho are contemplating joining a massive mathematical onlinecollaboration should keep in mind the costs in time, publicembarrassment and potential lack of control of your workbefore fully committing to the experience.5 James MaynardJames Maynard is a fellow at Magdalen College, Oxford.As a graduate student who had been looking at closelyrelated problems, it was thrilling to hear of Zhang’s initialbreakthrough. I didn’t participate in the subsequent Polymath8a project, although I found myself reading several of theposts whilst studying Zhang’s work for myself. I intended toavoid working on anything too close to the Polymath project(to avoid any competition) but one day I was going backthrough some ideas I’d had several months earlier on modificationsof the ‘GPY sieve’ and I realised I could overcomethe obstacles I had faced in my original attempt. This modification(also discovered by Tao) gave an alternative, strongerapproach to gaps between primes, although it didn’t producethe equidistribution results which lie at the heart of Zhang’swork. With some small numerical calculations, this allowedme to show that there were infinitely many pairs of primeswhich differ by at most 600, and allowed one to show the existenceof many primes in bounded length intervals.I knew the numerical bounds in my work hadn’t been fullyoptimised – there was some slack in my approach and therewere also several opportunities to extend the method (suchas incorporating ideas from Zhang and Polymath8a, or usingmore careful arguments). I was therefore pleased and excitedwhen I learnt there was the intention for a Polymath8b projectwhich I could be part of – it was very exciting for me that therewas such an interest in my work!The style of a large collaborative project was very new tome. I had relatively little experience of research collaborationand I was used to mainly trying out ideas alone. I certainlyhadn’t fully appreciated quite how public the posts were (andhow many mathematicians who were not active participantswould read the comments). In some ways this was quite fortunate;not realising the attention posts might receive mademe more willing to contribute openly. I posted several ideaswhich were not fully thought through, some of which wererather stupid in hindsight, but some of which I believe wereuseful to the project (and probably more useful than posting afully thought out idea a few days later). The atmosphere of theproject certainly helped encourage such partial contributions,which I feel was a large factor in the project’s success.18 EMS Newsletter December 2014