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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.

Trace functions, a survey

At last count, my series of works with Étienne Fouvry and Philippe Michel on trace functions and their applications consists of seven research papers or preprints, amounting to a bit more than 200 pages. To these are added a number of works-in-progress or partial notes (some with results we did not need or use and so took out of earlier drafts of our papers, some with worked-out examples or remarks, etc). We have a relatively firm project of writing a monograph account of the whole theory and applications, which we view in part as a way of making accessible some of the deep consequences of the Riemann Hypothesis over finite fields of Deligne, but this is clearly a long-term project. In the meantime, we have written a first short survey, the first draft of which can be found on my web page. This is in fact the written (and slightly expanded) account of a Colloquio de Giorgi that I gave at the

[Scuola Normale Superiore]

Scuola Normale Superiore

Scuola Normale Superiore in Pisa earlier this year, and we have included an unusual representation of the fundamental domain of the modular group acting on the upper half-plane as an homage and acknowledgement of this occasion.

[Pisa]

Pisa

James Maynard, auteur du théorème de l’année

How many times in a year is an analytic number theorist supposed to faint from admiration? We’ve learnt of the full three prime Vinogradov Theorem by Helfgott, then of Zhang’s proof of the bounded gap property for primes. Now, from Oberwolfach, comes the equally (or even more) amazing news that James Maynard has announced a proof of the bounded gap property that manages not only to ask merely for the Bombieri-Vinogradov theorem in terms of information concerning the distribution of primes in arithmetic progressions, but also obtains a gap smaller than 700 (in fact, even better when using optimal narrow k-tuples), where the efforts of the Polymath8 project only lead to 4680, using quite a bit of machinery.

(The preprint should be available soon, from what I understand, and thus a full independent verification of these results.)

Two remarks, one serious, one not (the reader can guess which is which):

(1) Again, from friends in Oberwolfach (teaching kept me, alas, from being able to attend the conference), I heard that Maynard’s method leads to the bounded gap property (with increasing bounds on the gaps) using as input any positive exponent of distribution for primes in arithmetic progressions (where Bombieri-Vinogradov means exponent 1/2; incidentally, this also means that the Generalized Riemann Hypothesis is strong enough to get bounded gaps, which did not follow from Zhang’s work). From the point of view of modern proofs, there is essentially no difference between positive exponent of distribution and exponent 1/2, since either property would be proved using the large sieve inequality and the Siegel-Walfisz theorem, and it makes little sense to prove a weaker large sieve inequality than the one that gives exponent 1/2. Question: could one conceivably even dispense with the large sieve inequality, i.e., prove the bounded gap property only using the Siegel-Walfisz theorem? This is a bit a rhetorical question, since the large sieve is nowadays rather easy, but maybe the following formulation is of some interest: do we know an example of an increasing sequence of integers n_k, not sparse, not weird, that satisfies the Siegel-Walfisz property, but has unbounded gaps, i.e., \liminf (n_{k+1}-n_k)=+\infty?

(2) There are still a bit more than two months to go before the end of the year; will a bright PhD student rise to the challenge, and prove the twin prime conjecture?

[P.S. Borgesian readers will understand the title of this post, although a spanish version might have been more appropriate...]

Alas, poor Yorick…

Je suis de passage, presque par hasard, ce soir à Paris, et je viens de lire que Patrice Chéreau est mort. Il y a peu d’occasions dont je me souvienne aussi vivement que d’avoir vu sa mise en scène de Hamlet, il y a longtemps, à Grenoble — “Good night, sweet prince // And flights of angels sing thee to thy rest!”

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