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More things of the day(s)

(1) Today’s Word of The Day in the OED: afanc, which we learn is

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

Gaston

I’ve heard that Gaston

Billard

Billard

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

Contrats

Contrats

or to the interjection Rogntudju!.

Rogntudju

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

The many ways of affineness

Last Saturday, the OED Word of the Day was affineur. Now, I know very well what an affineur is (my favorite is Jean d’Alos, and I especially like his renowned Tome de Bordeaux, the excellence of which can probably be confirmed by Mr. Quomodocumque), but for a few seconds I had in mind the picture of a fearsome algebraic geometer busily transforming all projective varieties into affine ones.

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)…

Conferences

Here are two forthcoming conferences that I am co-organizing with Philippe Michel this year:

(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.

This will be counting but

For the first time ever, I have temporarily reached the top of the culture pecking order in France, since I was able to go see “Einstein on the Beach” during its run at the Théâtre du Châtelet in January. This was clearly “the” évènement to attend (even two months or so before, buying two good contiguous seats was almost impossible), and the reactions of the audience suggested that many people were there because of the “the” instead of their desire to see the évènement, and were correspondingly a bit nonplussed by the work. Thus the two people sitting on my right left after a bit more than a third of the opera. (It was clearly indicated that, during the performance, which is a bit longer than four hours and has no intermission, it was allowed to leave and come back as desired, but they did not reappear).

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…

Three little things I learnt recently

In no particular order, and with no relevance whatsoever to the beginning of the year, here are three mathematical facts I learnt in recent months which might belong to the “I should have known this” category:

(1) Which finite fields k have the property that there is a “square root” homomorphism
s\ :\ (k^{\times})^2\rightarrow k^{\times},
i.e., a group homomorphism such that s(x)^2=x for all non-zero squares x in k?

The answer is that such an s exists if and only if either p=2 or -1 is not a square in k (so, for k=\mathbf{Z}/p\mathbf{Z}, this means that p\equiv 3\pmod 4).

The proof of this is an elementary exercise. In particular, the necessity of the condition, for p odd, is just the same argument that works for the complex numbers: if s exists and -1 is a square, then we have
1=s(1)=s((-1)\times (-1))=s(-1)^2=-1,
which is a contradiction (note that s(-1) only exists because of the assumption that -1 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 K is a field of characteristic different from 2, then K admits a homomorphism
s\ :\ (K^{\times})^2\rightarrow K^{\times},
with s(x)^2=x, if and only if -1 is not a square in K.

(The argument for sufficiency is not very hard: one first checks that it is enough to find a subgroup R of K^{\times} such that the homomorphism
t\, :\, R\times \{\pm 1\}\rightarrow K^{\times}
given by t(x,\varepsilon)=\varepsilon x is an isomorphism; viewing K^{\times}/(K^{\times})^2) as a vector space over \mathbf{Z}/2\mathbf{Z}, such a subgroup R is obtained as the pre-image in K^{\times} of a complementary subspace to the line generated by (-1)(K^\times)^2, 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:
\int_0^u tJ_{\nu}(t)^2dt=\frac{1}{2}u^2\Bigl(J_{\nu}(u)^2-J_{\nu-1}(u)J_{\nu+1}(u)\Bigr)
where J_{\mu} 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 GL(n) 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
 \sum_{k\geq 0}\binom{N-1+k}{k}^2 T^k=\frac{P_N(T)}{(1-T)^{2N-1}}
where N\geq 1 is a fixed integer and
P_N(T)=\sum_{k=0}^{N-1}\binom{N-1}{k}^2T^k.

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 N 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 P_N 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
{}_2F_1(-(N-1),-(N-1);1;T)=(1-T)^{2N-1}{}_2F_1(N,N;1;T),
where
{}_2F_1(\alpha,\beta;1;z)=\sum_{k\geq 0}\frac{\alpha (\alpha+1)\cdots   (\alpha+k-1)\beta(\beta+1)\cdots \beta+k-1)} {(k!)^2}z^k
is (a special case of) the Gauss hypergeometric function.