I stayed two days after the conference and had time to do some visiting and shopping. My favorite browsing time was in the Yue Hwa Chinese Emporium (the address was suggested by Yuk-Kam Lau, who also suggested the restaurant where we had our best dinner). I couldn’t resist buying the torus-shaped box that I saw there.

According to my wife, this is certainly intended to hold necklaces and bracelets. It is decorated with a dragon and what I think is a phoenix (maybe representing two elements of a basis of the homology or cohomology of the torus). I haven’t yet measured it to check if it has complex multiplication.

I took a few animal pictures, although the rather oppressive weather made it a bit difficult to spend time walking leisurely outside.

And I saw a rather melancholy piano in a side street…

Musically, I think there is a minor modernist masterpiece to be written based on the rings, whistles and noises of the Octopus card system, recorded at one of the more important exits of the MTR subway at some of its busier hours of operation.

]]>In comparison with the 1987 edition (called more modestly, if apparently inaccurately, “The complete works”), the new version identifies more works where Shakespeare was involved, and (taking from the other hand) finds also more plays where other writers participated. This is all explained in fair detail in a companion book full of statistical studies of proportions of rhymes or of feminine endings, or other fine points of prosody. Maybe most interesting (to me) is the play “Arden of Fevershame” that is now attributed in part to Shakespeare at the very beginning of his career, since its theme (the story of a then fairly recent murder most foul committed among England commoners) is rather far from the themes of most of his other plays.

The impressive volumes also make for excellent book-ends.

And there is still apparently a further volume (or two) to come, of “alternative versions” of those plays that are known in two or more substantially different early texts (e.g., “King Lear”).

I am eagerly anticipating a similarly ambitious scholarly N.O.W (New Oxford Wodehouse); in fact, I am happy to volunteer for the exacting role of editor of the Jeeves & Wooster canon. Or, if objections are raised against the attribution of such a crucial part of the *oeuvre* to a Frenchman of Polish and Breton origins, I will gladly take responsibility for the volumes encompassing the acts of the fifth Earl of Ickenham, fewer in number but by no means in importance.

P.S. Ellman’s biography at least convincingly corrects the story I mentioned in an earlier post about Joyce moving frequently in Zürich because of his inability to pay the rent: his Zürich years during the first World War coincide with the time when he finally got a decent income (in principle) to not have to worry about such things. It seems however that he was very inventive in dealing with creditors earlier in Trieste…

P.P.S. The first name “Thelonious” apparently comes from Saint Tillo or Théau or Tillonius, who was active in the late 7th Century in Flanders and France; his feast day is January 7th, and he is noted for having cured the Bishop Hermenus of Limoges by informing him that he (Tillonius) was dying and requested him (the Bishop) to come and bury him.

]]>The Norwegian Academy of Science and Letters announces the 2017 Abel Prize is awarded to Yves Meyer, École normale supérieure Paris-Saclay, France…

(from the AMS home page, as of right now…)

I’m even prouder now than before of having bought Meyer’s “Wavelets and operators” in the Peoria, Illinois, science museum store…

]]>My son is currently reading an abridged version of *Les Misérables* for his French class. This is a text intended for schools and comes (among other things) with explanations of “hard words”. While glancing through it recently, I noticed the following striking instance:

Le hasard, c’est-à-dire la providence

^{(1)}…

where the footnote translates, in lapidary style:

1. Providence = chance

(in English: *Providence = luck*). I may not know a lot about Victor Hugo, but it’s as clear as day to me that nothing could be further away from his use of the word “providence” than the idea that this is mere luck.

This reminded me of another definition I have seen in the French Larousse Universel encyclopedic dictionary from 1922 concerning the German language (see here in the middle of the page):

Langue: … une langue laborieuse… de là un certain manque de rapidité et de précision dans l’expression de la pensée.

(Or: *… a clumsy language… from this comes a certain lack of speed and precision in the expression of one’s thoughts.*)

This is actually a very nice book overall, with wonderfully useful illustrations to understand what, say, a “face-à-main” is, or to remind yourself of the important classification of “chapeaux bicornes”

(the scan I am linking to does not do justice to the book; one can download the PDFs of the two volumes, but each is a huge file of at least 250 MB, and the quality is also not so great — but the books become searchable).

]]>A few months ago, I found *“Condorcet journaliste, 1790-1794”*,

which one cannot call a well-known book. It is the printed version of the 1929 thesis (at the École des Hautes Études Sociales) of Hélène Delsaux, and its main goal is to survey and discuss in detail all the journal articles that Condorcet, that particularly likable character of the French revolution (about the only one to be happily married, one of the very few in favor of a Republic from the outset, and — amid much ridicule — a supporter of vote for women), wrote during those years.

Condorcet was also known at the time as a mathematician; hence this remarkable quote from the book in question:

Il est généralement admis que rien ne dessèche le coeur comme l’étude approfondie des mathématiques…

or in a rough translation

It is a truth universally acknowledged that nothing shrivels the heart more than the deep study of mathematics…

[Ed. Note: what about real estate?]

This book cost me seven Francs. More recently, my trip to the bookstore was crowned by the acquisition of a reprint of R. Dedekind’s *“Stetigkeit und irrationale Zahlen”* and *“Was sind und was sollen die Zahlen”* (five Francs)

and of a first edition (Teubner Verlag, Leipzig, 1907) of Minkowski’s *“Diophantische Approximationen”*

for the princely sum of thirty-eight Francs.

The content of Minkowski’s book is not at all what the title might suggest. There are roughly two parts, one concerned with the geometry of numbers, and the second with algebraic number theory. In both cases, the emphasis is on dimensions 2 and (indeed, especially) 3, so cubic fields are at the forefront of the discussion in the second part. This leads to a much greater number of pictures (there are 82) than a typical textbook of algebraic number theory would have today. Here are two examples,

and here is Minkowski’s description of the Minkowski functional (or gauge) of a convex set:

]]>I think that this is a great idea, and am very happy that, as the web site is now public, two of my own lecture notes can be found among the inaugural set! The highlight of the current selection is however undoubtedly “A singular mathematical promenade”, by Étienne Ghys, his beautiful book on graphs of polynomials, Newton’s method, Puiseux expansions, divergent series, and much much else that I have yet to see (I’m only one-third through looking at it…)

Hopefully, the Open Math Notes collection will grow to contain many further texts. The example of the book of Ghys is already an illustration of how useful this may be — although it is also available on his home page, one doesn’t necessarily visit it frequently enough to notice it…

Two final whimsical remarks to conclude: (1) among the six authors currently represented [Update (four hours later): this has already changed!], three [Update: four] (at least) are French; (2) one of my set of notes promises a randonnée, and Ghys’s book is a promenade — clearly, one can think of mathematics as a journey…

]]>Although I mentioned this briefly in my talk in the recent conference in honor of A. Lubotzky’s 60th birthday (where I was greatly honored to be invited!), it’s only last week or so that I somehow finally did the obvious thing, namely some experiments with **Magma** to see if the property of having proportion of -cycles is widespread.

When doing this, the first thing to realize (which I only did when P.P. Palfy pointed it out during that conference…) is that this condition, for a subgroup of , is equivalent with containing a unique conjugacy class of -cycles (simply because the centralizer of an -cycle, either in or in the bigger symmetric group, is the cyclic subgroup of order that it generates). So we can coin a solid decent name for these groups: we call them the unicyclic permutation groups.

**Magma** has a database of all transitive permutation groups of degree up to (even if one installs an extra specific database). Experimentation shows that there often exist unicyclic groups that are not the symmetric group. For instance, for degree , there exist (up to isomorphism) transitive permutation groups, and of them contain at least one -cycle, and of them are unicyclic.

However, the same experiments show that if we restrict our attention to primitive subgroups, then the situation is very different: either there is only the symmetric group , or there are two unicyclic groups. Amusingly the second case occurs if and only if is prime (an amusing primality test…)

One can indeed prove that these facts (which I did first experimentally notice) are correct, but it is not cheap. More precisely, from the Classification of Finite Simple Groups, Feit and later Jones (see for instance this paper which has a slightly more general result) deduced a full classification of primitive permutation groups that contain an -cycle. It is restrictive enough that one can then relatively easily exclude all groups except the symmetric groups and, when is prime, the group of transformations of the finite field .

That’s a nice result, but is it relevant for our original motivation? I think so, because (as Will Sawin pointed out), one can prove in considerable generality that the Galois group in (say) the Bunyakowski problem is primitive, before or without computing it exactly, because it suffices to prove that it is -transitive, and this is accessible to the diophantine interpretation using Chebotarev’s Density Theorem, as in Entin’s approach. (Indeed, Entin goes on to check that the Galois group is often -transitive or so, and from the Classification of Finite Simple Groups, deduces that the group contains the alternating group). Hence, if the degree is not prime, granted that one first applies Entin’s method to check primitivity, having Galois group is equivalent to the natural version of Bunyakowski’s conjecture in the large finite field limit…

If is prime, well one can say that if is not solvable, then it must be the symmetric group (assuming is at least …), although that is not as good — is there a better way to do it?