Two biographies

Coincidentally, I finished reading two biographies in the last few days: R. Ellman’s “James Joyce” and R. D.G. Kelley’s “Thelonious Monk: The Life and Times of American Original”. Although I admire both Joyce and Monk, there is no question who is my favorite: on a desert island, I would take Monk’s records with me. And yet, I read through Joyce’s biography in barely more than a week, and only read Monk’s rather slowly over a few months — obviously, it is much easier to write about the life of a writer, quoting liberally from his letters and limericks, than to write about a musician without an accompanying CD or recording.

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.

De l’inconvénient des usines à gaz

This is maybe for the cognoscenti, but rather funny:

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…

Définition tendancieuse

(I guess the title of this post would translate as something like “Biased definition” in English; according to the OED, “tendencious” does exist, but is ascribed as coming from the German “tendenziös”)

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”

Bicornes
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).

Condorcet, Dedekind, Minkowski

One of my great pleasures in life is to walk leisurely down from my office about 30 minutes before the train (to Paris, or Göttingen, or Basel, or what you will) starts, browse a few minutes in one of the second-hand bookstores on the way, and get on the train with some wonderfully surprising book, known or not.

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

Condorcet
Condorcet

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)

Dedekind
Dedekind

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

Minkowski
Minkowski

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,

Minkowski
Minkowski
Minkowski
Minkowski

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

Minkowski
Minkowski

Update on a bijective challenge

A long time ago, I presented in a post a fun property of the family of curves given by the equations
x+1/x+y+1/y=t,
where t is the parameter: if we consider the number (say N_p(t)) of solutions over a finite field with p\geq 3 elements, then we have
N_p(t)=N_p(16/t),
provided t is not in the set \{0, 4, -4\}. This was a fact that Fouvry, Michel and myself found out when working on our paper on algebraic twists of modular forms.

At the time, I knew one proof, based on computations with Magma: the curves above “are” elliptic curves, and Magma found an isogeny between the curves with parameters t and 16/t, which implies that they have the same number of points by elementary properties of elliptic curves over finite fields.

By now, however, I am aware of three other proofs:

  • The most elementary, which motivates this post, was just recently put on arXiv by T. Budzinski and G. Lucchini Arteche; it is based on the methods of Chevalley’s proof of Warning’s theorem: it computes N_p(t) modulo p, proving the desired identity modulo p, and then concludes using upper bounds for the number of solutions, and for its parity, to show that this is sufficient to have the equality as integers.  Interestingly, this proof was found with the help of high-school students participating in the Tournoi Français des Jeunes Mathématiciennes et Mathématiciens. This is a French mathematical contest for high-school students, created by D. Zmiaikou in 2009, which is devised to look much more like a “real” research experience than the Olympiads: groups of students work together with mentors on quite “open-ended” questions, where sometimes the answer is not clear (see for instance the 2016 problems here).
  • A proof using modular functions was found by D. Zywina, who sent it to me shortly after the first post (I have to look for it in my archives…)
  • Maybe the most elegant argument comes by applying a more general result of Deligne and Flicker on local systems of rank 2 on the projective line minus four points with unipotent tame ramification at the four missing points (the first cohomology of the curves provide such a local system, the missing points being 0, 4, -4, \infty). Deligne and Flicker prove (Section 7 of their paper, esp. Prop. 7.6), using a very cute game with products of matrices and invariant theory, that if \mathcal{V} is such a local system and \sigma is any automorphism of the projective line that permutes the four points, then \sigma^*\mathcal{V} is isomorphic to \mathcal{V}.

Not too bad a track record for such a simple-looking question… Whether there is a bijective proof remains open, however!