Mathematics – Page 68 – E. Kowalski's blog

Science and mathematics

Quite by chance, I’ve stumbled in the archive of Nature (alas, not freely available) on a paper by J. Sylvester (dated December 30, 1869) concerning, roughly, the status of mathematics among sciences. He says his text was a reaction to earlier talks and articles by Huxley (the biologist, not the limericks writer…). His esteemed opponent having stated

Mathematics “is that study which knows nothing of observation, nothing of induction, nothing of experiment, nothing of causation”, but knows only deduction,

Sylvester argues strongly in the opposite direction:

I think no statement could have been made more opposite to the fact,

and goes on to give examples from his own work, in particular, where conclusions were reached, and entire theories were constructed, based on simple apparently accidental remarks, by processes of observation, induction and imagination.

Besides this discussion, reading this paper is quite fascinating. Mostly, it must be said, because it is rather incredibly hard to read. Not only physically (the font size is small, and the footnotes even smaller, and printed 2-up, it really exercises your eyesight), but also because of the language, which I believe should cause many a lover of the English language to either faint or burst out laughing; the Gothic Victorian style (cleverly ridiculed by Jane Austen in “Northanger abbey”) is here put into overdrive for the purpose of scientific discussion. There are rather frightening mathematical terms which, presumably, a few living readers can still interpret,

canonisant, octodecadic skew invariant, invariantive criteria, amphigenous surface, a catena of morphological processes

and there are Latin, Greek and French quotations, untranslated, and a German one (which, strangely, Sylvester feels to be in need of translation). The following passage is quite typical:

Now this gigantic outcome of modern analytical thought, itself, only the precursor and progenitor of a future still more heaven-reaching theory, which will comprise a complete study of the interoperation, the actions and reactions, of algebraic forms (Analytical Morphology in its absolute sense), how did this originate? In the accidental observation by Eisenstein, some twenty or more years ago, of a single invariant (the Quadrinvariant of a Binary Quartic) which he met with in the course of certain researches just as accidentally and unexpectedly as M. Du Chaillu might meet a Gorilla in the country of the Fantees, or any one of us in London a White Polar Bear escaped from the Zoological Gardens. Fortunately, he pounced upon his prey and preserved it for the contemplation and study of future mathematicians…

But there are also interesting things, like a discussion of the status of higher-dimensional geometry, and indeed a forecast of Flatland (the book of that title was only published 15 years later):

for as we can conceive beings (like infinitely attenuated book-worms in an infinitely thin sheet of paper) which possess only the notion of space of two dimensions…

The follow-up paper is much in the same style (with beauties such as “the Eikosi-heptagram“, and flights of fancy like “my own latest researches in a field where Geometry, Algebra and the Theory of Numbers melt in a surprising manner into one another, like sunset tints or the colours of the dying dolphin” – this theory is that of “the Reducible Cyclodes”), and also quite insightful sometimes. For instance, there is en passant, the following very convincing footnote:

Is it not the same disregard of principles, the indifference to truth for its own sake, which prompts the question “Where’s the good of it?” in reference to speculative science, and “Where’s the harm of it?” in reference to white lies and pious frauds? In my own experience I have found that the very same people who delight to put the first question are in the habit of acting upon the denial implied in the second. Abit in mores incuria.

(Sylvester writes the “i” in the word “in” in the last quotation as a dotless i; I doubt it’s a typographical error, but I can’t find an indication that this is proper Latin grammar; does any reader here have an insight on this?)

An introductory course in integration and probability

A while ago (in 2002 to be precise), I taught an introductory course of integration, Fourier analysis, and probability in Bordeaux (for third-year university students). While giving the lectures during the first year, I typed them more or less in parallel (in fact, sufficiently close to the course that I could probably have done it also on a weblog, as T. Tao has done this year with his course on Ergodic Theory, and his course on Perelman’s proof of the Poincaré Conjecture…).

Except for using those notes two more years in the same course (or a very similar one the third year), I did not work on them anymore. But since I had spent a fair amount of time to bring the text to a reasonable state of polish, and since I’ve found myself using it a few times as a convenient reference for basic facts that I wanted to show to other students, it seems more than reasonable to put this course on the web somewhere. And so, here it is: Un cours d’intégration.

The text is in French, which diminishes its current interest, but as it is likely that I will teach this topic again in English at ETH, I’ll probably use it as a basis for a translation and/or adaptation.

As the introduction indicates, the only noticeable feature of this integration course (though I don’t think it is at all unique) is that I have treated probability theory in parallel with measure theory, instead of treating probabilistic language and its basic results after the main development, as is often done. I find this is better for a first course, because even students who never go on to another probability course can get, if they are attentive, a good immersion in the special langage and frame of mind of probabilists. (And this was a practical concern in Bordeaux because, typically, there wasn’t another probability course following this one, at that time at least).

The course itself is roughly 140 pages long, and is fairly standard. In measure theory, most of the basic results are treated, though not the Radon-Nykodim theorem. In probability, things go as far as the Central Limit Theorem, but there are no martingales.

The last 40 pages contain the exams given the first two years I gave the course, with corrections. Some of the problems are dedicated to proofs of quite nice results, including the Radon-Nykodim theorem on [0,1] and Lebesgue’s differentiation theorem, and some are more standard. (Those who have never seen a French-style three/four hours long exam can have a look to get an idea of how this type of things are done in this strange country; with hindsight, it’s clear the exams were quite a bit too difficult for the students I had…)

Tidbits of terminology and other folklore

Two small and independent interrogative remarks:

(1) Nowadays, an extension of a group G by a group K is a group H fitting in a short exact sequence

$1\rightarrow K\rightarrow H\rightarrow G\rightarrow 1$

in other words, and rather counterintuitively, the group G is a quotient of the group which is the extension. When did this terminology originate? A paper of Alan Turing (entitled, rather directly, “The extensions of a group”) defines “extension”, in the very first paragraph, exactly in the opposite (naïve) way, quoting Schreier and Baer who, presumably, had the same convention.

(2) There’s a whole lot of discussion here and there about the mystical “field with one element”; usually, papers of Tits from around 1954 are mentioned as being the source of the whole “idea”; however, the following earlier quote from a 1951 paper of R. Steinberg (“A geometric approach to the representations of the full linear group over a Galois field”, 1951, p. 279, TAMS 71, 274–282) seems to also contain a germ of the often mentioned analogy between the formulas for the order of the Weyl groups and those of groups of Lie type over a field with q elements, the former being obtained by specializing the latter for q=1:

In closing this section, a remark on the analogy between G and H seems to be in order. Instead of considering G as a group of linear transformations of a vector space, we could consider G as a collineation group of a finite (n-1)-dimensional geometry. If q=1, the vector space fails to exist but the finite geometry does exist and, in fact, reduces to the n vertices of a simplex with a collineation group isomorphic to H. “

In this citation, G is GL(n,F_q), and H is the symmetric group on n letters.

(3) Here’s a third question: when did the terminology “Galois field” become more or less obsolete within the pure mathematics community?

Yet another property of quadratic fields with extra units

Here is an amusing exercise that is suitable for a course on basic algebraic number theory: let p be a prime number. Consider integral solutions (a,f) to

$4p=a^2+3f^2$

with a and f positive. The claim is that, if p is congruent to one mod 3, there are three distinct solutions (a,f), (b,g), (c,h), and if they are ordered

$1\leq a<b<c$

then we have

$c=a+b$

For example, in the case p=541, we find a=17, b=29, c=46:

$4p=2164=17^2+3\times 5^5=29^2+3^3\times 7^2=46^2+3\times 2^4$

The pedagogic value of the exercise is that while it looks like something that one could prove by a simple brute force computation, this is not so easy to do, while it becomes elementary knowing the basic facts about factorizations of primes in quadratic fields (and units of imaginary quadratic fields, in this case Q(√-3).

Indeed, the equation means that p is the norm of the integral ideal generated by

$\frac{a}{2}+ \frac{f\sqrt{-3}}{2}$

It is known that only primes congruent to 1 modulo 3 are norms in this field, hence the first condition on p. Then, it is known that the ideal above is unique up to conjugation. So the only possible extra solutions, given one of them, are obtained by multiplying by a unit of the field, and isolating the “coefficient of 1”, or in other words taking the trace to Q. Since the units are

$\pm 1,\, \pm j=\frac{\mp 1\pm \sqrt{-3}}{2},\, \pm j^2$

simply multiplying shows there are three positive solutions:

$a,\,\frac{a+3f}{2},\,\frac{|a-3f|}{2}$

Depending on the sign of a-3f, the conclusion follows by considering two cases.

[This minor property of quadratic fields was motivated by the question of finding interesting examples of relations between the zeros of zeta functions of algebraic curves over finite fields; for quite a bit more about this – both results of independence and examples of relations -, see my preprint on the subject, in particular Section 6 for the examples.]

Équidistribution, or équirépartition ?

For years, I have been convinced that the proper French translation of “equidistribution” was not the faux ami (false friend) “équidistribution”, but rather the word “équirépartition”. The latter is for instance used by Serre (and Bourbaki).

But then I realized recently that Deligne uses “équidistribution” in his great paper containing his second proof of the Riemann Hypothesis over finite fields, which contains in particular his famous equidistribution theorem (see Section 3.5, entitled “Application: théorèmes d’équidistribution”).

Since, in fact, neither word appears in the French dictionaries I have available (unsurprisingly: “equidistribution” is not in the OED), and since moreover “distribution” and “répartition” do appear and are identified as synonyms, it seems now that in fact both words should be acceptable…