I saw Alain Resnais’ *Muriel* yesterday. After the film, an old man next to me addressed me out of the blue and said: Finally, they show something in French, and I do not understand a thing…

One of the most striking things at the beginning was the use of (extradiegetic) music. There a brief snippets in the style of atonal expressionism. (They seemed to refer to Schoenberg, via Berg; they turn out to be due to this guy.) It is a kind of music that calls forth associations of difficulty and meaning, or at least intellectualised emotion. Towards the beginning of the film, these phrases or fragments are played precisely at moments of perfect banality.

(Another reviewer points out that one of the uses of music here is to pinpoint deceit, or anxiety felt but not shown by a character. This was not clear to me at the time; if anything the filmmaker seemed to have avoided any very obvious move in this direction. This is a movie that will reward a second viewing.)

There are different ways in which to take this. It does signal that we are in the kind of film where characters and setting will be taken as aesthetic objects. At the same time, it quickly becomes clear that we are not in *Last Year at Marienbad*; we are dealing with a realistic setting and, what is more important, with a real issue of great literal importance.

It is tempting – and it makes specific sense in this film – to take this use of music and sound as a case of inappropriate affect, acquired by the narrative from the characters that likely suffer from it – certainly Bernard, who is just back from the Algerian war, but possibly also Hélène and even Alphonse, whose manifold deceit may or may not have an element of mythomania.

(This is, by the way, a film on the Algerian war, and specifically on the use of torture. It is also a film on the second world war. It thus makes perfect sense that the action takes place entirely in a coastal city in the north of France around 1962.)

It is also touching that it is banality – and the reflections and misgivings of two old lovers with whom it is difficult to emphatise – that is being emphasised here. To some small extent, bourgeois ritual may be being used by the characters to keep their own experiences at bay. (More broadly, we may be speaking not only of personal experiences, but of traumatic historical events that seem to have neither involved them as fully as possible nor spared them completely.) What this viewer, at least, is induced to think is that part of the tragedy of some of these individuals is that they can no longer lead fully banal lives – lives that seem banal in contrast with the events that made them partly impossible.

There is an odd bit of dialogue in a restaurant. I turn out to have remember it correctly:

Claudie: C’est comme pour l’anguille, ça se cuit vivant.

Hélène (lowering her voice): Le cuisinier d’ici a été déporté. S’il était mort, on aurait perdu sa recette.

There is something clearly askew here – but it is arguably not so much Hélène as the world both of the generations in the film were forced to live in.

A final remark. There may be something objectionable or at least very frustrating about movies that center on the emotional journey or moral dilemmas faced by a protagonist who is a young man on the better-equipped side of a war; surely we ought to be (and are) less interested in him than in his and his friends’ victims? This is averted here. The very title states clearly who the centre of this film is; the fact that it is an absent centre makes it more poignant, while raising valid issues of representability.

(Incidentally, one extra point for using “martingale” in its original non-mathematical setting.)

An issue in early higher mathematics education is that one is teaching people who are still far from being mathematical mature, but may have reached a certain level of philosophical maturity, at least as far as questions and interests are concerned. It is also true that part of a university education has to be a reeducation, in that any mathematical education up to that point has areas that need to be deemphasised (i.e. high-school algebra, ending at best with Cardano; plane Euclidean geometry). At the same time, there are also interests that arise often and could be channelled constructively, instead of being ignored. This would also help in smoothing the oddly disjointed nature of a typical mathematical formation, with strange caesurae before university and graduate school.

(Warning: some of the statements below may rest in part on personal experiences and draw from some very different systems I have studied and taught in.)

When I first studied any mathematics with a certain amount of rigour – that is, in my early teens – I was particularly interested in non-Euclidean geometry. (I can hardly be alone in this.) I recall with singular pleasure understanding the point that a model – such as the surface of a sphere in three-dimensional Euclidean space – showed right away that the consistency of Euclidean geometry implied the consistency of non-Euclidean geometry. (This may seem obvious to us now, but it is a point that is never made in basic popular explanations of the subject.)

My younger self was very interested in issues of limits, convergence and continuity. Besides some very basic stuff on transfinite numbers (Cantor’s diagonal proof), I learned about Dedekind cuts, how to prove things in geometry by (what amounts to) exhaustion, how to go about properly with s and s, why one cannot handle series as fecklessly as in the eighteenth century – and not much else.

All of these are, broadly speaking, issues on foundations, and thus are adolescent concerns in the best sense of the word. Some of the territories that these questions lead to – axiomatic systems implying the consistency of some other axiomatic systems – have become specialist fields now that mathematics has passed through some of its own adolescence(s). Nevertheless, these questions can be made to lead to matters of everyday mathematical practice outside the specialist subject of foundations.

This became clear to me only much later in the game. Take, again, non-Euclidean geometry. Before university, it seemed to be one of the main things to learn about. Then, in my college years, it never appeared as such in what I was taught. I was taught some of the right *language* for it, viz., the basic concepts of differential geometry, but somehow this was never tied in with Riemann’s original concerns. (To be completely fair – this may in part be an artifact of the fact that the main text I read on the matter centered on Nash’s embedding theorem and related matters rather than on Gauss-Bonnet.)

Then, in my first year of graduate school, automorphic forms appeared out of nowhere and took centre stage. What amounted to harmonic analysis in the hyperbolic plane was now supposed to be one of my main topics of study. Something that was a premium was a kind of very concrete familiarity that I could have acquired a long time before.

Incidentally, how much basic harmonic analysis did you learn before graduate school? I knew from physics that Fourier series were important, and had probably seen Fourier inversion over once – and that was about it. One of my friends at the first-year barracks was shocked at my ignorance, which I tried to amend rapidly – but I now realise I do not have any clear idea of what the others knew, meaning other first- and second-year students (excluding the handful who were actually specialising in harmonic analysis, obviously).

I was recently teaching some analytic number theory in different places. I gave a Tauberian proof of the prime number theorem – the one based on Wiener-Ikehara. I did this because I was assuming that, while not everybody would feel comfortable with complex-analytic magic – which is at any rate not very easy to motivate in this instance – everybody would surely have seen Fourier transforms, and would feel both comfortable and interested in a close look at them.

Somewhat to my amazement, I was informed that, while most advanced undergraduates at X and Y had already taken complex analysis, Fourier analysis was largely unknown to them, and in fact wasn’t a standard part of their curriculum. Perhaps I should have been less amazed; I had somehow assumed that the very small role that Fourier transforms played in my undergraduate education was completely outside the norm, and that most people nearly everywhere would have seen much more of them, partly due to their applications.

(Also – I am under the impression that statistics students do learn about the Fourier transform early on – curiously, both over and over , due to computational issues. Or am I getting also this wrong?)

Incidentally, did it also take you — assuming you are a mathematical reader, possibly of a particular kind — a long time till you saw a clear explanation of how Tauberian theorems (as currently used in analytic number theory, say) truly relate to Abel’s and Tauber’s work on convergence? This again is an example of something important that could be introduced early on as a response to a natural early interest in a basic question – namely, what it means for a series to converge.

Does anybody know whether Jean Ferrat’s

*Potemkine*was actually composed so as to synchronise with the sequence below? I cannot decide either way.

> Incidentally, how much basic harmonic analysis did you learn before graduate

> school?

Like most French mathematicians who passed through the

classes préparatoires, I knew about Fourier series during the second year after high-school, though without proofs of anything interesting (Parseval, Dirichlet’s theorem being just stated) in my case — I suspect a number of teachers do actually prove these things. In the first year afterwards (three years after high-school), we had a strong integration course with basic facts about Fourier transforms and Fourier series, this time with complete proofs. In principle this is before the (nearest) equivalent to graduate school, but my impression is that such a course would be typically a first-year graduate course at a place like Rutgers.Do you think it would be a good idea if, together with the basics of Fourier series (given without proofs of the main statements, as you said), students were taught about Fourier analysis in the to context – with proofs? That way the main ideas would be presented – with convergence and integration issues removed.

That would certainly make a lot of sense; whether the glacially slow and reactionary syllabus committes have/will/would do it is another question…

Where I teach at the moment (ETHZ), integration/measure theory is a third-semester class, and(although there is no fixed syllabus) I think most of the time the course would contain an introduction to either Fourier series or Fourier integrals.