As a followup to the earlier post of the evolution of the paper, here’s another piece of information: the word “conjecture”, in the sense currently used (and sometimes abused), doesn’t appear in the *Annales de l’ENS* before the 1950’s (searching for “conjectur*” in the whole text, and disregarding two early papers of Jacobi and Beltrami which are really discussions of how they were led to some result or other, and some chemistry/mineralogy papers).

# Category: Mathematics

## The evolution of the paper

One of the nice consequences of the current development of online archives is that reading the classics has really become much easier than before. Of course, “classics” refers here to those mathematical texts which were published in journals since the middle to late 19th century, and are in easily accessible languages (which, for me, means French or English, and I guess I can look a little bit at Italian papers without feeling too lost too quickly).

As a sometime “flâneur” along those roads, my favorite site is NUMDAM, which contains archives of many journals and seminars, mostly French (there are a few Italian journals and seminars included, as well as Compositio Mathematica). Compared with other sites like JSTOR, Project Euclid, or the Goettingen archive, NUMDAM seems much quicker and easier to browse. It is also freely available (except for the last few years of some journals, but since we speak of classics here, this is not an issue). Moreover, besides the standard PDF format, it has copies of the papers in the djvu format, which is much more compact.

During the last snowy Easter week-end, I’ve looked at NUMDAM to check some points of what might be called the natural evolution of the mathematical paper. I used the Annales Scientifiques de l’École Normale Supérieure as a source, because it has been published continously since 1864.

So one finds, in no particular order (nor guarantee of correctness of the dates mentioned…):

- That proper separate bibliographies did not occur until around 1948, with footnotes then indicating that “Numbers between brackets refer to the bibliography located at the end of the paper”.
- Except that this was mostly in French; in fact, the first papers not in French in this journal appear in 1968 only (series 4, volume 1; it may be, and this would be easy to check, that this new series was the first where languages other than French were permitted). The second such paper is quite famous: it is John Tate’s Residues of differentials on curves.
- What about the first
*joint*paper? The honor goes to Castelnuovo and Enriques, in 1906. But this is an outlier: the next two only occur in 1934 (one is also famous, due to Leray and Schauder), and papers with more than one author can be counted on two hands until 1954. (I am disregarding, here, two earlier papers by Pasteur and Raulin, in 1872, which have to do with the fight against silk-worm diseases, and another one on the construction of the official “mètre étalon”, or yardstick; such non-mathematical papers disappeared around that time, although in 1896, the journal included the discourse given by the renowned Désiré Gernez on the occasion of the inauguration of a statue of Pasteur.) - Then, what about the first joint paper in English? There’s Tate again, with F. Oort in 1970.
- OK, and what about the first paper written by a French person in English? Here, there can be some ambiguity, since someone may well be French without having a name that claims it to the world (…), but Alain Connes may be the first one.
- Another information that could be interesting would be the first article written by a woman, but since first names are typically missing from much of the early tables of contents (and can be ambiguous), this is even harder to decide. The first unambiguous example I saw is due to Jacqueline Ferrand, in 1942.

In an idle moment today, I also looked more quickly at the Bulletin de la Société mathématique de France, which goes back to 1872. The various dates are somewhat similar; there is a single paper not in French before 1952, by Wiener (in 1922). I can envision a vicious fight among the editors to decide if it could be published; at least the first few paragraphs mention that the work on which the paper is based was done in France, and thank profusely Fréchet for his insights… (There are also a few earlier papers *translated in French* from their original language, for instance Heegaard’s thesis appears in 1916, translated, presumably, from the Norwegian). The first joint paper appears in the same year, 1916, and the next one only in 1930.

Reading the titles of article before 1950 or so can be quite amusing; mixed with terminology that still seems very modern and recognizable, there are certain gems like the anallagmatic metric (“métrique anallagmatique”) of R. Lagrange, in 1942. Strangely enough, this is a word recognized by the OED: “Not changed in form by inversion: applied to the surfaces of certain solids, as the sphere”, with quotations from Clifford, in 1869, and Salmon, in 1874. This is too bad, since I was hoping that the date indicated that this paper was an elaborate coded transmission to the Free French… There is also in most titles an element of high and formal seriousness that can be a bit tiring; all those papers starting with “Sur une équation…”, or “Sur une propriété…”, or “Sur *quelque chose*…” (“On *something*…”) do not give a great impression of the fun of doing mathematics.

The authors of the early papers are also divided pretty sharply between names we all know (or have heard about, not only French, but from most countries we think of as having a strong mathematical culture in that period), and completely obscure characters (at least to me). The most magnificent I found is the (probably) redoubtable Gaston Gohierre de Longchamps who published three papers in the *Annales de l’ENS* between 1866 and 1880 (the second of which concerns Bernoulli numbers, and quotes the equally remarkable M. Haton de la Goupillère).

Finally, one observes with interest that the poor reputation of rigor of these older mathematicians is an unwarranted slander; no erratum is needed for the entire corpus of the *Annales de l’ENS* until 1953, with the single exceptions of one in 1907 (a paper by Émile Merlin), and of a remonstrance by Brouwer pointing out a few mistakes in a paper of Zoretti in 1910, which the latter rather grudgingly accepts (by claiming that another mathematician had priority in finding those)…

## Deligne’s proof of the Weil conjectures

Today, I will jump on the coat-tails of Terry Tao’s link to Brian Osserman’s article on the Weil Conjectures for the (upcoming) Princeton Companion to Mathematics (edited by Tim Gowers and June Barrow-Green), and use this as an occasion to mention an old text of mine which was meant to be an introduction to Deligne’s (first) proof of the Riemann Hypothesis over finite fields. [The date of 2008 on the PDF file simply reflects that I recompiled it recently to incorporate a PDF version of the single figure in it, which I may as well include here to break the textual monotony…

this is supposed to illustrate the famous “cycle évanescent”, a terminology which I find somewhat more poetic in French, where “évanescent” is rather old-fashioned and formal, than in the English “vanishing cycle”, although the etymology is in fact the same].

I say “text” because it’s a bit hard to define in what category this could/should be put. I wrote it as a Graduate Student at Rutgers in 1997, as the text for two lectures I gave at the end of a graduate course by my advisor, Henryk Iwaniec, on the general topic of solutions of equations over finite fields. His lectures had ended with a detailed proof of the Riemann Hypothesis for curves (and their all-important consequence, for analytic number theorists, the Weil bound for Kloosterman sums), following Bombieri’s adaptation of Stepanov’s method.

I took this background as starting point to give as complete a sketch as I could (given my understanding of the algebraic geometry at the time…). This means the text is not really self-contained: the basic theory of algebraic curves is assumed to be known, at least informally. However, the somewhat original feature (which is why I am mentioning this text here) was that I tried to explain the origin, the formal definition and some of the properties of étale cohomology “from scratch”. This culminates, at the end of the first part, with an almost complete and rigorous computation of the étale cohomology of an elliptic curve over an algebraically closed field (based on the basic theory of isogenies of elliptic curves). This is in fact not difficult at all, but I had not seen it done elsewhere in a self-contained way, and it certainly seemed enlightening to me.

As implied in the previous paragraph, there are two parts to the paper. The first is the preparation (i.e., supposedly leading to where things stood before Deligne’s breakthrough), and it still seems to have some value for people interested in starting to learn this important theory. The second tries to follow Deligne’s proof very closely, and is in fact probably hard to read, and therefore of less interest. (Moreover, Deligne’s first proof was largely subsumed in his *second* proof, which leads to much more general statements, and in particular to his Equidistribution Theorem; this second proof is really quite a bit more involved than the first one, in terms of algebraic geometry).

## Averages of singular series, or: when Poisson is everywhere

Here is another post where the mediocre mathematical abilities of HTML will require inserting images with some TeX-produced text…

I have recently posted on my web page a preprint concerning some averages of “singular series” (another example of pretty bad mathematical terminology…) arising in the prime *k*-tuple conjecture, and its generalization the Bateman-Horn conjecture. The reason for looking at this is a result of Gallagher which is important in the original version of the proof by Goldston-Pintz-Yildirim that there are infinitely many primes *p* for which the gap *q-p* between *p* and the next prime *q* is smaller than *ε* times the average gap, for arbitrary small *ε>0*.

This result refers to the behavior, on average over * h=(h_1,…,h_k)*, of the constant

*S(*which is supposed to be the leading coefficient in the conjecture

**h**) |{*n*<*X* | *n*+*h_i* is prime for *i*=1,…,*k*}|~*S( h)*

*X*(log

*X*)

^{-k}

Gallagher showed that the average value of *S( h)* is equal to

*1*, and I’ve extended this in two ways…

## Schlomo Cohen

Quite a few years back, when I was finishing the (in)famous *Classes préparatoires*, I started writing a series of stories entitled *Les fabuleuses aventures de Schlomo Cohen le Mathématicien détective* (“The Wonderlous Adventures of Schlomo Cohen, Detective-Mathematician”); after finishing four texts ranging in length from a short story to a modestly-sized novel, this ended around 1994 when I really didn’t have time anymore for the type of imaginative concentration needed for even my level of fiction-writing.

As the title already suggests, there was a strong influence of British-style super-detective crime fiction when I started, based on reading too much Sherlock Holmes and Agatha Christie when I was a few years younger. Indeed, Schlomo Cohen is said to be “the best non-fictional private detective in London”. However, the third, and longest, story, concludes in Los Angeles with a clear debt to R. Chandler, since P. Marlowe makes his appearance, showing at least some improvement in stylistic debt over the years, and the last one is full of direct or indirect quotations of “The Waste Land”…

The other two obvious characteristics are that the hero, the superior detective S. Cohen, is (1) a mathematician; (2) Jewish. The second part may seem somewhat strange (I am not Jewish myself), and is due mostly to the twin influences of I. Bashevis Singer and W. Allen when I started writing the stories (in fact, S. Cohen is theoretically quite orthodox, quoting the Talmud and other sacred texts rather freely, and his mother, Masha Cohen, plays a big part in the last three stories).

The first characteristic (mathematics) is of course more understandable, and was for me the source of much of the fun in writing the stories. The first idea, as far as I remember, was to use sophisticated plots carefully designed so that insights from great theorems and their proofs would be genuinely useful to solve the mysteries S. Cohen was confronted with. This was quickly replaced with essentially completely random *associations d’idées*, which (quite obviously) make no sense whatsoever, but which nevertheless lead S. Cohen to the guily partie(s).

Here are the main mathematical results Schlomo Cohen claims led him to solve the outstanding problems of the age:

- Some theorems of Church on ultrapowers (which I don’t remember at all!) in
*Le vol du traité secret*(“The theft of the secret treaty”); - The Hahn-Banach theorem in
*De la banane dans le Bourgogne*(“Banana in Burgundy”); - The theory of Lefschetz pencils, van der Waerden’s theorem on finite colorings of the integers, Riemannian geometry, in
*Schlomo Cohen contre les maîtres criminels*(“Schlomo Cohen against the master criminals”); - Non-euclidean geometry, in
*Le traducteur subreptice*(“The surreptitious translator”).

Quite a few other results are discussed, and they mostly show what type of mathematics I was learning (and finding striking!) at the time.

I am mentioning this today mostly because I have just discovered that the second story (“Banana in Burgundy”) has become part of a (semi?) official bibliography of works about Bourbaki (look at “Anonymes” as author). The point is that, in this particular story, I envision Schlomo Cohen as a great friend of André Weil, so much so that he is invited to attend the first Bourbaki Congress, in Burgundy. There, terrible events occur, involving a conspiration against the wines of Burgundy, the French constabulary, and the transformation of all but one of the Bourbakists into amateur detectives…

People more knowledgeable about the history of Bourbaki than I was at the time will of course realize that the date, place and much else doesn’t make any sense at all, but still, even today, I find pleasure in re-reading the exchanges I wrote between mathematicians, and I think they are quite realistic in a way. Indeed, I feel flattered that the bibliography above says that the story is a *Récit plaisant et fantaisiste qui décrit tout de même le groupe Bourbaki de façon réaliste* (“A pleasant whimsical story which nevertheless describes realistically the Bourbaki group”).

The story, if you want to read it (it’s in French), is online here (search for “Humour”). Note that most of it does not involve Bourbaki at all; for the best parts, you can look around page 26, up to 44 or 45.

I should say that I had put the stories anonymously on the internet a few years ago (except the first one, which was really too embarrassing to my mind), and this is where the authors of the bibliography must have found it. The third story is therefore also available on this original site (if you look for it a bit)… But from a literary point of view, and indeed also plot-wise, the fourth and last one, Le traducteur subreptice is by far the best. There one finds all ingredients for a smashing hit: Masha Cohen’s theory of “Golems of the second kind”; Schlomo’s monograph on “The babylonian Talmud considered as a formal system”; the beautiful daughter of a mysterious Jesuit Father and her sulfurous thesis, “The judicial arsenal of the Spanish Inquisition”; the Vermont senator Philip P. Mark and his erstwhile English revolutionary friends; and the mysterious criminal of the title, whose devious deed is to replace the original manuscript of “The Waste Land” with a French translation… (One also learns in passing that S. Cohen wrote his thesis under the direction of J.E. Littlewood, on “A refinement of the circle method with applications to Waring’s problem”; in “Banana in Burgundy”, on the other hand, he is said to have interesting results concerning torsion points of elliptic curves).