Il a publié il y a deux ans (…) un ouvrage relatif au sentiment de l’Infini sur la rive occidentale du lac Victoria-Nyanza et cette année un opuscule moins important, mais conduit d’une plume alerte, parfois même acérée, sur le fusil à répétition dans l’armée bulgare, qui l’ont mis tout à fait hors de pair.

or, in translation:

He has published two years ago (…) a book concerning the feeling of Infinity on the occidental shore of the Victoria-Nyanza lake, and this year another booklet, less important but written with a lively, and even piercing, pen, on the repeating rifle in the bulgarian army, which have made him rather peerless.

This reminds of the curiously little-known hilarious “Antrobus stories” of diplomatic mishaps:

It was during one of those long unaccountable huffs between ourselves and the Italians. You know the obscure vendettas which break out between Missions? Often they linger on long after the people who threw the first knife have been posted away. I have no idea how this huff arose. I simply inherited it from bygone dips whose bones were now dust. It was in full swing when I arrived — everyone applying freezing-mixture to the Italians and getting the Retort Direct in exchange. (…) So while bows were still exchanged for protocol reasons they were only, so to speak, from above the waist. A mere contortion of the dickey, if you take me, as a tribute to manners. A slight Inclination, accompanied by a

moue. Savage work, old lad, savage work!

(from “The game’s the thing”, where a soccer game between the English and Italian embassies rather degenerates.)

]]>In Welsh mythology: an aquatic monster. Also: an otter or beaver identified as such a monster.

Maybe the Welsh otters, like their rugbymen, are particularly fierce?

(2) Yesterday’s Google doodle, in Switzerland at least, celebrated the 57th birthday of Gaston Lagaffe

I’ve heard that Gaston

is mostly unknown to the US or English, leaving many people with no reaction to the mention of the *contrats de Mesmaeker*

or to the interjection *Rogntudju!*.

This lack of enlightenment is a clear illustration of the superiority of the continental mind.

]]>I looked at the adjacent words in the OED; there is quite a list of them involving affine-ness in some way (listed here with dates of first use, as recorded in the OED); actually, affineness is not in the list:

- affinage 1656
- affinal 1609
- affine 1509
- affine v 1473
- affined 1586, 1907
- affineur 1976
- affining 1606
- affinitative 1855
- affinitatively 1825
- affinition 1824
- affinitive 1579
- affinity 1325

It is interesting to think of an algebraico-geometric meaning for each of them (especially the tongue twister affinitatively, and affinition)…

]]>(1) Quite soon, the traditional Number Theory Days (the eleventh edition of this yearly two-day meeting that alternates between EPF Zürich and ETH Lausanne), will be held in Zürich on March 7 and 8; the web page is available, with the schedule and the titles of the talks; the speakers this year are Raf Cluckers (who is also giving a Nachdiplomvorlesung at FIM on the topic of motivic integration and applications), Lillian Pierce, Trevor Wooley, Tamar Ziegler and (Tamar is probably happily surprised not to come last in alphabetical order!) David Zywina.

Anyone interested in participating should send an email to Mrs Waldburger as soon as possible (see the web page for the address).

(2) In July, intersecting neatly the last stages of the soccer world cup, and beginning in the middle of a week to avoid (thanks to some fancy footwork) starting on the 14th of July, we organize a summer school on analytic number theory at I.H.É.S; people interested in participating should follow the instructions on the web site. The detailed programme will soon be available.

]]>Personally, I knew the music extremely well and it was a great pleasure to finally see the full spectacle. I had already attended two other Robert Wilson stagings, and I like his style (i.e., I have no objection to watching a light pillar move from horizontal to vertical in the space of twenty minutes or so), but it was the first time I saw a full-length Glass opera live, and I’d have paid gladly quite a bit more than I did.

The full staging certainly answers a few of the puzzled questions one might get from the music alone. In particular, the scenes entitled “Dance 1″ and “Dance 2″ felt more alive when seen as, well, dances. In fact, I arbitrarily decided that the first represents electromagnetism (it begins with a voice saying “Bern, Switzerland, 1905″), and the second nuclear forces (a sinister character crosses the stage and I see it as representing the potential for evil arising from nuclear physics). Einstein would have appreciated that, despite the random-looking evolutions of the dancers, these were certainly not the result of dice throws, since collisions would certainly have been unavoidable otherwise.

One of the Paris representations was filmed, and shown on French television, so that it can be found on the internet. However, I watched it a bit, and the fact that there are many cameras giving different angles of view seems to diminish the full immersive effect of the live show. But that’s certainly better than nothing…

]]>(1) Which finite fields have the property that there is a “square root” homomorphism

i.e., a group homomorphism such that for all non-zero squares in ?

The answer is that such an exists if and only if either or is not a square in (so, for , this means that ).

The proof of this is an elementary exercise. In particular, the necessity of the condition, for odd, is just the same argument that works for the complex numbers: if exists and is a square, then we have

which is a contradiction (note that only exists because of the assumption that is a square).

The question, and the similarity with the real and complex cases, immediately suggests the question of determining (if possible) which other fields admit a square-root homomorphism. And, lo and behold, the first Google search reveals a nice 2012 paper by Waterhouse in the American Math. Monthly that shows that the answer is the same: if is a field of characteristic different from , then admits a homomorphism

with , if and only if is not a square in .

(The argument for sufficiency is not very hard: one first checks that it is enough to find a subgroup of such that the homomorphism

given by is an isomorphism; viewing as a vector space over , such a subgroup is obtained as the pre-image in of a complementary subspace to the line generated by , which is a one-dimensional space because $-1$ is assumed to not be a square.)

It seems unlikely that such a basic facts would not have been stated before 2012, but Waterhouse gives no previous reference (and I don’t know any myself!)

(2) While reviewing the Polymath8 paper, I learnt the following identity of Lommel for Bessel functions (see page 135 of Watson’s treatise:

where is the Bessel function of the first kind. This is used to find the optimal weight in the original Goldston-Pintz-Yıldırım argument (a computation first done by B. Conrey, though it was apparently unpublished until a recent paper of Farkas, Pintz and Révész.)

There are rather few “exact” indefinite integrals of functions obtained from Bessel functions or related functions which are known, and again I should probably have heard of this result before. What could be an analogue for Kloosterman sums?

(3) In my recent paper with G. Ricotta (extending to automorphic forms on all the type of central limit theorem found previously in a joint paper with É. Fouvry, S. Ganguly and Ph. Michel for Hecke eigenvalues of classical modular forms in arithmetic progressions), we use the identity

where is a fixed integer and

This is probably well-known, but we didn’t know it before. Our process in finding and checking this formula is certainly rather typical: small values of were computed by hand (or using a computer algebra system), leading quickly to a general conjecture, namely the identity above. At least **Mathematica** can in fact check that this is correct (in the sense of evaluating the left-hand side to a form obviously equivalent to the right-hand side), but as usual it gives no clue as to why this is true (and in particular, how difficult or deep the result is!) However, a bit of looking around and guessing that this had to do with hypergeometric functions (because is close to a Legendre polynomial, which is a special case of a hypergeometric function) reveal that, in fact, we have to deal with about the simplest identity for hypergeometric functions, going back to Euler: precisely, the formula is identical with the transformation

where

is (a special case of) the Gauss hypergeometric function.

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.

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.

(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 , not sparse, not weird, that satisfies the Siegel-Walfisz property, but has unbounded gaps, i.e.,

(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...]

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