La cascade d’homologie

It can be very rewarding to read old mathematical papers, in terms of accessing insights and ideas that may have been filtered out in later transformations of the results they contain. In my modest experience, this does not extend to notation and terminology, and it is much easier to appreciate the insights in question after translating them into modern language and formalism. This is an area where, maybe, progress is usually fairly steady. But still, there can be exceptions. I was recently rather struck, while reading the recently published collection of letters between Henri Cartan and André Weil, to discover that when they were exchanging many letters on algebraic topology just after 1945, they used the charming name cascade for what is now known as a “long exact sequence” (in homology or cohomology). I think it is too bad this didn’t become the standard name; one could have imagined that triangles
A\rightarrow B\rightarrow C\rightarrow A[1]\rightarrow
would be called “Escherian cascades”…

Incidentally, this book of letters is very interesting to read, in no small part because of the extensive notes and comments by Michèle Audin. It is published by the SMF in the same series where letters between Grothendieck and Serre were also published a few years ago.


One of the nicest things about Linux (and Open Source software in general) is that new versions often offer clear measurable improvements on the previous ones. And another is that this does not usually require abandoning whatever might have been worth keeping from other computer-ages. In particular, if one has very old software, there’s a good chance that one can still keep them working, even if they are written for a completely different operating system, through the wonders of emulation. In my case, this applies to Windows 3.1-era dictionary cdroms, and to Motorola 68000-era Mac software.
Recently, I had somewhat lapsed in performing the necessary tweaks to make these old programs work on my laptop (a decidedly modern 4-core Lenovo), but on upgrading Fedora, I decided to try again. It’s quite amazing that, through the wonders of Wine, I can enjoy again the Grand Robert de la Langue Française


(originally available for MS-DOS and Windows 3.1) as well as the American Heritage Dictionary

(though I use the O.E.D instead when I’m connected to the ETH network). The Grand Robert is the best anti-pedant tool I know against so-called défenseurs de la langue française; it usually reveals that their favorite anglicisms are perfectly French (e.g., opportunité, in the sense of “occasion, circumstance”, which goes back to 1355 in French, and is at least as French as Baudelaire…)

I’m even more impressed to be able to boot the equivalent of my old Mac SE30,

and thereby play with, or recover, the old files I used to work with during my PhD thesis and before. (In fact, the emulator boots in something like 1.5 seconds on my laptop, which is about a hundred times faster than it ever did in real life…) Afficionados will note the realistic 512 x 384 resolution of the screen.

And after Fermat…

… there came Jorge Luis Borges

as Google doodle.

By the way, people who have encountered many French mathematicians (say, in a conference) of a certain sharply defined age may have got the impression of finding themselves in a confusing self-referential Borgesian circle. The reason is that his book of short stories “Fictions” (Ficciones in Spanish) was assigned as one of the two texts during one year of the famous French classes préparatoires.

Strangely, the effect of the second book, a poetry collection of Francis Ponge, was much less obvious, though some highly refined friends of mine enjoyed it a lot; my own personal memory is restricted to the sad remark that it is rather a shame that the title of his poem La crevette dans tout ses états does not translate exactly to The startled shrimp, the (former) name of the night-club in which B. Wooster gets entangled with the awful majesty of the law in Jeeves and the Feudal Spirit.

On the other hand, the two hard drives of my computer at the time were called “Tlön” and “Uqbar”, and I dabbled in imitative short stories; I might as well put here a link to my favorite

Subtile finesse de la notation…

For a long time (since I first heard of it around 1990), I thought that the terminology “Property (T)” was a completely arbitrary name, and no better than the thousand notions of “admissible thingummy” or “good khraboute” which sprinkle too many mathematics papers. Then I learnt a few years ago that the “T” was supposed to refer to the Ttrivial representation, since — in a suitable language — the property is about the trivial representation of a group G.

This was already better. But much more recently, I learnt from S. Mozes that the typography “(T)” itself was not some random choice, but was meant to express the fact that the trivial representation T is supposed to be alone in some open set, incarnated as an open interval (so one should read () as in (a,b))…

A direct corollary is that the right translation in French is, of course, Property ]T[.

By the way, I personally much prefer the ]a,b[, [a,b], ]a,b], etc, notation for intervals, but I’m told that many find this ugly beyond belief and much prefer the (a,b), (a,b], etc, style…