E. Kowalski’s blog

In the spirit of fairness and balance, after my ode to an American magazine, I would like now to mention my admiration for one of the great achievements of the French publishing world: La Bibliothèque de la Pléiade. This is one of the collections edited by Gallimard, maybe the greatest French publishing house, which is dedicated to producing definitive editions of the best of the world’s literature. There is a strong emphasis on French-language writers, of course (including thirteen volumes of Voltaire’s correspondance), but by no means an exclusivity (as can be seen in the catalogue: note Spanish-language writers, such as Borgès, Russian masters like Dostoievski, Boulgakov or Tolstoy, Italian writers like Machiavelli, and of course many English-speaking ones, such as Faulkner, Melville or the Brontë family). This is one of the great differences with the natural reference point in the American world, the Library of America. The other main difference is that, besides the text itself, the Pléiade aims to provide extensive (sometimes exhaustive) editorial information on the author and the work, with notes, introductions and discussions, bibliographies, sometimes early versions or other relevant sources, etc. The books themselves (like those of the Library of America) are beautifully produced, on the thinnest paper (papier bible), so each volume is routinely longer than 1000 pages without being much bigger or heavier than a (fairly fat) paperback. The font is the elegant Garamond, with its intricate ligatures.

Being in Paris earlier this week, I visited one of the many bookstores, and noticed that the second part of the new complete Pléiade edition of Shakespeare’s works, the Histories, had just appeared; I therefore snatched the two volumes without more ado, to add to the Tragedies which were published a few years ago.

Now, it might seem slightly ridiculous to spend a lot of money on a French edition of Shakespeare (however beautiful the italic font in the scenic indications), and this was a valid criticism of the earlier edition (dating to the 1950′s), but the new one is in fact bilingual. And I will venture the opinion that reading Shakespeare in a bilingual version makes very good sense: one can try to read the “original” version as much as possible, but in case the syntax or grammar becomes decidedly perplexing on the page, the translation gives a backup. If the translation is written from the point of view of actual theatrical experience, then the solutions which are offered to the many ambiguities in the texts (which can most often not be fully translated) are likely to make more sense and to flow more smoothly than isolated glosses or paraphrases in footnotes, even if they can not convey all the possible meanings. In the new Pléiade edition, the main translator is Jean-Michel Déprats, and most of the translations were indeed used for actual representations in France before they appeared; so even if one can not always be sure of reading Shakespeare’s intended meaning, at least one gets something which may be the next best thing: some well-defined meaning, coming from a writer with enormous theatrical experience. And I’m sure that anyone who has seen a few plays of Shakespeare on the stage knows how different the experience may be from reading them. (My personal favorite memory is a magical version of The Tempest in the Théâtre des Bouffes du Nord, in Paris, directed by Peter Brook in 1990, in a translation of J-C. Carrière).

Now, lest any scholar of the Elizabethan theatre jump on my word “original” in the previous paragraphs, I emphasize the quotation marks: just as in any modern English edition, there has, very often, been a real choice of which text to use (Good Quarto, Bad Quarto, First Folio, and what you will). The whole history behind those various versions can be quite fascinating, and the very detailed notes explain which was used, what principles were applied in terms of localized corrections, etc: again, very solid scholarship comparable to those detailed editions one can find in English. There is also a separate genealogical tree of the relevant Kings, Queens, Princes, Princesses, Dukes, and other divers Noblemen and Noblewomen, included in the first volume of the Histories, which is certainly quite useful…

Here’s a picture of the two-volume Histories:

and here’s one of the text of Richard III:

and the genealogical tree:

I can’t say that I really have anything like a “teaching philosophy” — maybe because I was lucky and never had to write a teaching statement. However, experimentally, I realize that there are two prominent features in my teaching that I end up feeling fairly strongly about, partly from an impression that they were not emphasized enough when I was myself a student: one is that the development of the course and the underlying theory should be emphasized and motivated enough that it does not feel like the teaching of some disembodied Voice Of God or Deus Ex Machina (even if the motivation is not historical, and indeed the motivations I give are often based on a fair amoung of hindsight); and two, that whenever a theorem of some importance is proved, it should be clear where and how its assumptions are used.

I’ve been thinking a bit about this second point because I’m currently teaching weak topologies in my Functional Analysis class, and in particular I just finished proving the Banach-Alaoglu Theorem that states that the closed unit ball in the dual V’ of a Banach space is compact for the weak-star topology. Here one basic point I wanted to make in class was that the analogue statement is not true (in general) for the closed unit ball of V itself, with the weak topology. I was then surprised to see, in browsing through various textbooks, that none of those I saw explained explicitly why the argument doesn’t work in this situation (it is implicit, of course, in all those that stated, e.g., that this second statement is equivalent with the reflexivity of V, but I think that for a student beginning in Functional Analysis, this may not be sufficient, because the proof of this equivalence is not really obvious). Sometimes, the “broken” step was just glossed over, but sometimes it was simply left as an exercise. In the worst case, the written proof did not even spell out which of the steps of the proof requires that the weak-star topology be used, leaving this also as an exercise (the beginning of the standard proof works just as well with the norm topology)!

[Note: The standard proof of Banach-Alaoglu proceeds by embedding the closed unit ball B in the compact product space

by

then one shows that i is injective, continuous (this works even for the norm topology on B), that the inverse of i, defined on i(B), is also continuous (this requires that the weak-star topology be used, and fails for the norm topology if V is not finite-dimensional); then, finally, one shows that i(B) is closed in C, and this works because we deal with B in V’ instead of the closed ball of V itself — the basic issue is that knowing that (for some sequence in V) we have

for all continuous linear functionals λ on V does not always imply that we can find some w such that

A simple example where this fails (i.e., w does not exist), is provided by taking

(the space of sequences converging to 0) and

where there are n ones at the beginning: from the fact that the dual of V can then be identified with the space l₁ of absolutely convergent series, one checks that

for all linear functionals λ represented by the sequence (u_n) in l₁. But

is not of the form

for any sequence w in c₀.]

When the conversation turns to anti-américanisme primaire, as it will every once in a while in France, the first argument I use if I intend to display a contrary argument is the New Yorker. Indeed, this magazine is so much above the level of the available French weeklies that (to use a cliché), it’s not even funny. Not only is the content much better — more art, more poetry, more humor, more fiction, less French politics –, but the difference is even stronger from the purely visual point of view (typography, art, design: no need to be able to read French or English to see which editors/writers/readers have better taste). This is especially shameful, considering that the French penchant for style over substance would seem to guarantee that we would (at least) do much better in this respect. However, it is not for any of the French magazines that J.J. Sempé draws covers, but for The New Yorker. (Sempé is known in France in particular for inventing the second most important fictional Nicolas in history).

I think I first read about The New Yorker in the introduction to a French translation of Woody Allen’s short humor pieces for the magazine (if you’ve never read any of them, I suggest googling for “Gossage Vardebedian”), in which (the introduction) it was identified as “the most snobbish magazine in the world”, which immediately piqued my curiosity. However, I think I read an issue for the first time when I went for a month to the US to work with Henryk Iwaniec in 1992. Then in late 1993, I decided to start subscribing from France. At that time, issues arrived there about one month after publication, so that reading the jazz programme at the Café Carlyle, for instance, was a somewhat quixotic thing to do, but most of the articles were of lasting enough interest that this delay was not much a problem.

My interest for this magazine has been considered somewhat obsessive at times. It is not true that I brought my fledgling (three years old) collection to the US when I went there for Graduate School, but I must admit that I did ship back to France all issues accumulated during that period (and the resulting post-doc), and then had them also sent to Switzerland, together with the issues of the last eight years or so (they are now in storage somewhere in Zürich).

Frankly, my justification for this accumulation was not quite convincing: it is not really useful to have physical issues of The New Yorker somewhere in the basement in a random order, since (until recently) it did not really help to remember vaguely that, say, there was a hilarious story about a mathematics class by some Irish author sometime during the first (or was it second?) Clinton administration — the time to locate it would still be discouraging to consider. Moreover, I couldn’t help feeling terribly jealous of older subscribers who could reach (if they knew where they were located) for issues containing stories by I.B. Singer, for instance, and read them whenever they wanted.

In principle, this two problems were solved a few years ago when The New Yorker published a set of eight DVD’s containing all issues of the magazine (until that date, of course; it has been updated regularly). I bought it immediately, but the fact that the DVD’s were encrypted, and the reader program did not work under Linux was something of a problem. Because we had a Mac in the house, it was still theoretically possible to take advantage of the archive, but in practice it was very inconvenient (except for the fact that the search database was a standard SQLite database, and could thus very well be queried from my Linux computers; so I could say very quickly when the Gossage-Vardebedian papers were published — January 22, 1966 –, but actually reading it involved complicated manipulations and printing to PDF from a very slow Mac whose DVD reader was broken, etc.)

But, at last, this is old history: just recently, The New Yorker started making available both a digital edition (which is convenient, but not so important for me), and the complete archive online, available more or less as in the DVD set, as exact reproductions of the actual magazine (so even the ads, etc, are exactly as in the printed edition, which is quite wonderful actually). Better yet: both services are available free to subscribers.

[Note: I am aware that many older subscribers believe the magazine went downhill starting about 1990; but I can't really be held responsible for not reading it before, and (1) now I can; (2) it is still much better than the French weeklies...]

Some of the discussions during the Random Matrix, L-functions and primes conference reminded me of an old combinatorial question I had been struggling with around the time of my PhD thesis, because of some potential (but highly hypothetical) applications to automorphic forms. After moving to Bordeaux, I had the chance of having Laurent Habsieger as a colleague and I told him about the question, which he then solved within a few days, around April 2001. (He is currently Directeur de Recherche for the CNRS in Lyon).

Here’s the problem in question, which involves an arbitrary positive integer m:

Is it true, or not, that if N denotes any integer divisible by all integers up to m, then for any collection

of N integers up to m, there is a subset I of the integers up to N such that

I see this as a kind of “intermediate value” property, since the sum over all integers of f(i) must be at least N. The condition that N be divisible by each integer up to m is necessary (because one can take each integer f(i) to be the same k in that range), and it is also clear that if N has the stated property, then any multiple of it also does (by splitting the range into subsets of size N).

I won’t explain the rather technical application I had in mind (which turned out to be so hypothetical as to vanish away anyway), but rather propose this as a challenge to combinatorialists (it might be already known, of course, though there would be a good chance Habsieger would have recognized it if it was standard). In fact, Habsieger’s proof shows that the answer is “Yes”, except possibly if m=5 or 6 (where the smallest N allowed, which is 60 here, might not work, and must be replaced, e.g., by 120 and 240, and their multiples, respectively). I believe these exceptions are just due to my ignorance in clearing up the corner cases in Habsieger’s argument, but in a sense I would be delighted if it were the case that the exceptions are genuine. (In fact, in my version, his argument works for m at least equal to 7, but the other potential exceptions can easily be treated by hand).

Since this is a challenge, I won’t reproduce the main ideas of the (quite elegant) argument, but the whole solution can be read in this write-up of his solution, which I have just put on the “Notes and unpublished results” section of my home page.

And now for the last day of the Conference… We started the morning with a talk of P.O. Dehaye, who explained his method (joint with Borodin) for computing averages of values of derivatives of characteristic polynomials of random unitary matrices. This approach generalizes to this case the method of Bump and Gamburd (as explained in Bump’s talk) and is based, besides partition and representation theory, on a theorem of Okounkov and Olshanski (involving shifted Schur functions). The talk ended with a suggestion that one should look closely at the Riemann zeta function (and other L-functions) from the point of view of symmetric functions of infinitely many variables, either from the Euler product over primes, or from the Hadamard product over zeros. This is quite a striking idea; certainly, from the analytic number theory point of view, primes are not really symmetric, because in most applications they do come up ordered with the information contained in their size. But maybe this ordering should be less important for certain problems — and the supply of problems in analytic number theory is not close to depletion…

The next talk, by H. Iwaniec, had been announced as Rational and p-adic zeros of ternary quadratic forms; however, as described in the previous instalment, the fall of a coin led him to change the topic and title to Asymptotic large sieve and moments of zeta and L-functions. It was a careful account of his recent work with B. Conrey and K. Soundararajan where, for the first time, an unconditional confirmation was obtained for one of the moment conjectures for families of L-functions beyond the fourth moment. More precisely, they have succeeded in proving the following asymptotic formula:

where χ runs over even primitive Dirichlet characters modulo q, and both f and g are bump functions, f being supported on [1,2] and the other analytic with rapid decay at infinity, and the constants which appear are

In fact, quite spectacularly, this is obtained as a special case of a more general theorem involving both shifts and with complete lower-order terms (a polynomial in log q) and power-saving. Iwaniec explained very carefully the statement of the general conjecture which is confirmed by this result, and discussed briefly the main new technical innovation: an asymptotic large sieve formula for certain specific coefficients, which breaks the limit imposed by the usual large sieve inequality for Dirichlet characters.

Iwaniec mentioned that the 8th-power moment analogue is barely outside of their reach, and it might also be accessible with more work. He also very briefly explained the need for the (aesthetically displeasing) smoothing in t which is involved in the statement (and should be unnecessary): it is used to obtain cancellation in expressions of the type

unless n and m are of comparable size.

After this impressive result, the main talks of the conference concluded with a lecture by H. Montgomery who reviewed very nicely various instances of subtle combinatorics involved in performing computations of moments of arithmetic quantities of interest. One of the basic examples is found in the derivation of the expected Poisson-distribution for the number of primes in intervals of the type

which Gallagher derived conditionally, based on a uniform Hardy-Littlewood conjecture for prime k-tuples (for some more on this, my paper on averages of singular series may be useful). Montgomery also discussed his work with Soundararajan on understanding the deviations between the naive probabilistic model for the number of primes in larger intervals and what is expected based on various conjectures (such as the Pair Correlation Conjecture for zeros of the Riemann zeta function).