Reflections on reading the Polymath8(a) paper

Earlier today, I finished reviewing the Polymath8 paper (except for the last section, which is a discussion of possible improvements to some of the results in the paper; this is now a bit obsolete, and will probably be deleted).

Since the discussion on the project (and on the paper in particular) is almost entirely available online on Terry Tao’s blog, one can review the reviewing process, which is somewhat unusual in current mathematics, where reading/reviewing/refereeing a paper is mostly done in the dark.

Looking back, it seems that I began on September 30 (I thought it had been earlier…) At that time, the paper was about 164 pages long, and the current version is 176 pages long, so it is not so surprising that it took a bit more than two months to read through it, when combined with the semester teaching and other duties. (In fact, I changed some line-spacing settings, and the final version would have been probably at least five pages longer otherwise.)

At first, I actually intended to read the paper as if refereeing it, in some sense, with just corrections and minor changes, and things would certainly have gone faster in that case. But it seems that I have some genetic disorder and that I can’t read a paper through without wanting to change all the notation and quite a bit more (for good or ill…) Thus, where quite a few sections are concerned (especially towards the end), I basically re-typed much of the text with many many changes of notation. This was of course slower, but there is one big advantage in proceeding this way: I was much more likely to catch and correct typos and minor slips, and moreover I couldn’t just decide to go over a section with glassy eyes, since all the variables, functions, names of auxiliary quantities, and so on needed to be changed…

As far as mathematical issues in the first draft are concerned, I found only one which required any work to correct. I’d be very surprised if there were any still lurking, since I basically re-did every single computation, including checking numerical constants. And even this issue was really minor in comparison with the complexity of the whole argument, and didn’t affect the “generic” cases of the bounds where it was present.

So I think one can say that this paper has been reviewed in full basically at the thoroughest possible technical level (i.e., excluding philosophical or high-level comments). I actually wonder, in a hypothetical way, if I would ever have accepted to referee this paper, and if I had, whether I would have worked as carefully as I did…

And I am not quite done with bounded gaps between primes, since the Association des collaborateurs de Nicolas Bourbaki has asked me to give a Bourbaki seminar on the work of Zhang (and Maynard); this will be on March 29, 2014, in Paris, and I will now start preparing the text that will accompany the lecture…
But before I begin, I will offer myself a nice whisky this evening.

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I am a professor of mathematics at ETH Zürich since 2008.

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