## One shouldn’t disturb binomial identities too much

When teaching the most elementary courses at the university, a little bit of combinatorics enters, and the relation between the binomial coefficients and the expansion of (x+y)k, for non-negative integers k. An often appreciated trick for the students is to prove some identities among binomial coefficients using “analytic” properties of polynomials, and interpret them combinatorially, or conversely. The most basic of these identities is probably

In the spirit of fun, assume we think of rewriting this as

And now, maybe while whistling idly to pretend that we are not doing anything, let’s jolt the denominators with a quick flick of the finger:

Since it seems that no one has noticed anything, let’s do it again:

and again, and again… but stop! a red-faced policeman comes, and says that he has nothing against some good clean fun, especially on Boat Race Night, but enough is enough, and what horror have we done with poor Newt’s lovely identity:

Well, of course, what is behind this is an undoubtedly well-known identity, which can be expressed in terms of hypergeometric functions with very general parameters. But the sliding denominators might be a nice thing to show students, and the (or at least, one elementary one) actual proof of the actual formula is a good exercise in exploiting “polynomiality” in various forms, so it can be used for that purpose…

## Zazie count

So after re-reading carefully Zazie au pot de thèse, I have counted 74 references to mathematical terminology, including names of actual mathematicians (a few of which are actually hidden in puns which can be considered as Joycean or atrocious, depending on the point of view), and words which are used for their mathematical meaning in the text; this seems fair enough since, after all, the story is supposed to happen during (or mostly after) a mathematical PhD defence.

The best-hidden name (the construction being utterly untranslatable) is in the following sentence:

Il faut préserver les plus infimes de nos coutumes obsolètes, car nous, Gaulois irréductibles, sans us, perdrions notre esprit de corps, notre unité fondamentale.

(Litterally: We must preserve the most minute of our obsolete habits, since we, irreducible Gauls, without customs, would lose our esprit de corps, our fundamental unity.)

The mathematician here is Galois, seen as “Gaulois-without-us”; the surroundings of field and Galois-theoretic wordings (irreducible, fundamental unit, field – which is the translated “corps” in French) were supposed to make this noticeable…

## Zazie

My favorite French novel is “Zazie dans le métro”, by Raymond Queneau. I won’t go into detailed literary criticism to explain why, though the amazing inventivity of the language is one reason, but mention only that the fact that Queneau is well-known to have had a lively interest in mathematics is certainly another factor (there are not many mathematical traces in this book, though there is a very nice sentence, which I can’t locate at the moment, which mixes elliptic, parabolic and hyperbolic…).

Parenthetically, Queneau’s interest was shared by all the writers of the OULIPO group, the best known of whom is probably G. Perec. Perec is the author of the book-without-e “La disparition”, but he also wrote an absolutely hilarious pastiche of a scientific paper (not a mathematical one) entitled Cantratrix Sopranica L.” (which you really should read if you have never seen it; it’s in English). Even the bibliography is a jewel, as shown by the citation of a paper by “Einstein, Zweistein, Dreistein, Vierstein et Saint Pierre”. Note that this mathematical interest actually makes plausible the claim (which I have just seen on the web) that Grothendieck played a small role in L. Malle’s film version of “Zazie…”.

Coming back to “Zazie…”, when, to celebrate a friend’s PhD defense in Orsay in 1998, I decided to write a short story with the intention of cramming it with as many mathematical terms as I could in a non-professional context, I chose to imitate Queneau’s masterpiece, and make use of his characters Zazie, of the famously free vocabulary, and her uncle Gabriel, amateur of “sirop de grenadine” and, in the evening, dancer in a transvestite cabaret in Paris.

The resulting text is here; I will leave it as an exercise to find all the hidden mathematical terminology. The counts you may reach might not be the same as mine (which I will include in a follow-up after re-reading the text carefully, since I don’t remember it…). For instance, I don’t think I knew that “net” was a terme de métier when I wrote this text, and other instances yet unknown to me might lurk in it. As a teaser, I will mention “Eh quoi, si on élit P. Tique…”, which really must be read “Équation elliptique” (elliptic equation), and “Et les uns poussaient en avant, et les autres tiraient en arrière” (and some pushed forward, and some pulled back), which should be self-explanatory.

## Singular poles

By a lucky coincidence, we went today on an outing for lunch at the restaurant of the “Schloss Raperswil”, about half an hour from Zürich. On the way to the castle, I noticed a poster for the current exhibition at the Polish Museum there, and here is the interesting part of it:

I’m quite happy, in particular, to see here my mathematical great-grand-father…

I didn’t have time to go see the exhibition itself today, but I hope to go back soon to check its actual contents. In particular, I’d like to know the date this drawing was made, and why Denjoy and Choquet, both French mathematicians, are included.

## Jeeves and the PhD

Since the topic of graduate schools and the choice thereof seems suddenly popular, discussions of the outcome of graduate school, the dreaded PhD thesis, should also start soon. So it seems a good time to link to an old text of mine, entitled Jeeves and the PhD, which describes fairly and realistically some universal aspects of thesis-writing.

Compared with the Adventures of Schlomo Cohen, this text has the advantage of being much shorter (and in English). It also contains no mathematics whatsoever (this blog is about mathematics, but only mostly).

From the literary point of view, just as in the Scholomo Cohen stories, one can not claim that I try to hide my influences. I was, and still am, a great fan of P.G. Wodehouse (indeed, a positive proportion of my English vocabulary derives directly from reading his books, from “flabbergasted” to “flummoxed”, with “tidly om pom pom” in between).

This story was written, over one night in 1996 or 1997, to cheer up a friend of mine (the author of a very nice book of Mathematics for physicists) when he was, like many a graduate student, pulling all-nighters in order to finish typing his thesis (which he did defend brilliantly not long afterwards). It may still be considered as funny by similarly shackled graduate students (in any field), and can be read as cautionary tale by those hesitating to pursue graduate studies…