Page 30 – E. Kowalski's blog

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.

Random CIRM happenings

I was last week at the conference on “Number theory and its applications” which was excellently organized by C. Delaunay and F.X. Roblot at the CIRM conference center, close to Marseille. Although I don’t have last year’s excuse at the end of the Joint Math Meetings, my remarks will be just as incoherent…

M. Watkins showed a book he recently bought in the Canary Islands, which proves that G. Perelman is on his way to becoming a pop-culture figure:

A cursory look at the content (though not by native Spanish speakers!) does not seem to suggest that this a serious work of mathematical scholarship…
For the future writer of the definitive history of analytic number theory, I offer this remark from É. Fouvry, who said one could quote him:

… et Chebychev arrive avec une astuce de voleur de mobylette… (…and then Chebychev comes around with a trick worthy of a bicycle thief)
Charles Boyd, an enterprising soul worthy of homeric epithets, has ported Pari/GP to android

The package can be downloaded here. There’s something confortable in having your phone factor the 8-th Fermat number during a post-dinner round-wine discussion… One may object that, at the moment, any syntax error causes the program to exit unceremoniously, but certainly this will soon improve. (Note: the broken screen was not caused by Paridroid…)

Interlude

I presume that a number of readers are getting tired of my stories of growth and expansion, especially when it seems I can’t keep a value straight for two days in a row. There will be at least one more post about this, but for a relaxing change, here are some recent animal pictures…

First, a raptor looking at me straight in the eyes,

and then a shingleback lizard from the Zürich Zoo being handed his lunch on chopsticks:

he (or she) was very lazy about actually starting eating his (or her) cricket, which makes you wonder how things would go in the wild…

And finally a leaf-fish, still from the Zürich Zoo, which is a species I had never seen before:

Self-improvements

I’m currently profiting from the vacations to clean-up various aspects of my tentative to obtain explicit expansion bounds for subgroups of $\mathrm{SL}_2(\mathbf{Z})$ which are Zariski dense in $\mathrm{SL}_2$ . To review the previous episodes, there are three basic ingredients that needed to be made explicit:

basic multiplicative combinatorics;
Helfgott’s growth theorem;
the Bourgain-Gamburd expansion criterion (especially the so-called “ $L^2$ -flattening” lemma).

I’ve already talked about the first and second. For the growth theorem, after a few more changes, my current result is that any generating subset $H$ of $G=\mathrm{SL}_2(\mathbf{F}_p)$ satisfies either $H\cdot H\cdot H=G$ or
$|H\cdot H\cdot H|\geq |H|^{1+1/1344},$
(with no condition on $p$ ).

For multiplicative combinatorics, I have reworked the argument after noticing the fact (certainly obvious to all cognoscenti) that, for the purpose I need it, one can work mainly with symmetric sets, for which the basic relation between tripling constant and growth of $n$ -fold product sets is quite a bit better (in explicit terms) than the corresponding one of “mixed” $n$ -fold products involving either a set or its inverse. This gives much better exponents.

The most important gain comes, however, from a second look at the Bourgain-Gamburd inequality. The point is that they argue from a “dyadic” viewpoint, considering (in effect) the steps $X_{2^jk}$ of a random walk (for a suitable starting point $k$ where the walk has started to expand non-trivially). Each step from $j$ to $j+1$ gives a fixed improvement of the counting of closed geodesics of the corresponding length, and the number of steps which is required is directly related to the spectral gap one obtains.

From a qualitative point of view, there is nothing to argue with this. But if one wants to get explicit constants, one notices (this is also certainly known to the aforementioned cognoscenti, as shown by Helfgott’s comment on my previous post…) that the argument of Bourgain-Gamburd works essentially just as well for steps $2jk$ of the random walk: what is needed is a small adaptation of their main inequality to bound the spread of products $X_1X_2$ where $X_1$ and $X_2$ are independent random variables with $X_2$ at least “as well spread out” as $X_1$ .

Apart from this last part, which I will include soon since I just wrote down the details, I’ve incorporated these improvements in my notes. The last point, rather satisfyingly, improves exponentially the estimates for spectral gaps: for the Lubotzky group, it becomes now $2^{-38}$ , instead of $2^{-2^{36}}$ (for $p$ large enough, and I confess that I don’t yet know what this last condition means explicitly…)

We’re not yet in the realm of really macroscopic numbers, but this is certainly encouraging…

Emulation

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.