Today, as it has been for a long time, it can only be a fantastic dream to know and understand all of mathematics, and virtuous mathematicians must perforce look for alternatives. One of the best is to find some analogy between different areas — a brilliant instance, being Vojta’s rapprochement between questions of diophantine approximation (e.g, Roth’s Theorem) and questions of Nevanlinna Theory. Another great satisfaction is to see surprising direct connections between two such areas: I still remember my surprise and delight on learning in a probability class how complex Brownian motion can be used to solve Partial Differential Equations (such as the Dirichlet problem, as shown by Kakutani).

A very new joint work with J. Ellenberg and C. Hall brought (at least to me!) some of these emotions. The barest summary would be as follows: we describe very strong connections between the combinatorial notion of expander graphs (or, more properly, expander families) and certain types of finiteness statements in arithmetic geometry. There is already a bit of magic here, but the result is even nicer in that the proof depends crucially on another unexpected connection with a result of differential geometry of Li and Yau.

I won’t describe the arithmetic geometry now (partly because Jordan has already written a very good summary…). Rather I want to explain what are the esperantist graphs that we introduce in the paper, and discuss some vague but enticing “philosophical” questions that this paper suggests.

Let us start with expanders (a very good place to start); it might be that, with the possible exception of root systems, this is the most amazingly ubiquitous notion of 20th Century Mathematics — amazing in the sense that the definition can look hyper-specialized, until its influence extends (pun intended), and one day you realize that what looked like a practical network-communication question is needed, for instance, to give counterexamples to some form of the Baum-Connes conjecture (about which I don’t know anything, except for what a superb talk by N. Higson taught me about ten years ago). I highly recommend downloading the long survey paper of Hoory, Linial and Wigderson to get an idea of the breadth and importance of this notion.

Now, there are different equivalent definitions of expanders, and I’ll use the least intuitive, for the simple reason that this is the one that comes in most naturally for our applications: given a sequence of graphs

,

which we assume — for simplicity — to be connected and k-regular for a fixed k, one says that this sequence is an expander if (1) the number of vertices goes to infinity

and (2) there is a uniform spectral gap δ>0 for all n:

for all n, where λ1 is the first non-zero eigenvalue of the square matrix of size |Γn| given by

in terms of the adjacency matrices of the graphs (this is known as the combinatorial Laplace operator; it is a non-negative symmetric matrix, with first eigenvalue equal to 0 with the constant eigenvector, which is unique up to scalar since the graph is assumed to be connected.)

As it turns out, for our application, one does not need such a strong condition as uniform spectral gap. Precisely, we say that the family above is an esperantist family if it satisfies

for all n and some constants

(Note: The factor 2 simply avoids ever talking of 1/0.) Thus expanders correspond to esperantist families where one can take A=0.

Now, the obvious question: why did we select this name? Mostly, it is a question of alliteration; and we feel it just sounds nice (like étale topology sounds nice, or barreled spaces, or adèles, etc…)

Then, scientifically, what does it mean, and why is it interesting? For us, the meaning is given in practice by a theorem of Diaconis and Saloff-Coste: if the family is a family of Cayley graphs for some finite groups Gn, with respect to systems of generators with constant size k, then they will form an esperantist family as soon as the diameter of the Cayley graphs grows polylogarithmically in the size of the groups, i.e., if there exists

such that

And the fact is that showing this type of diameter growth is, at the moment, an indispensable staging point in all the recent developpments concerning growth and expansion in linear groups over finite fields, starting with the breakthrough by Helfgott (for SL(2,Fp). (For our purpose, the results of Pyber-Szabó are the most directly useful, because sometimes we do not control the groups well enough to claim, for instance, that they are given by G(Fp), where p varies, for a fixed algebraic group G; however, other papers with important results in this direction are one of Gill and Helfgott and one of Breuillard, Green and Tao.)

As a matter of fact (this has been confirmed to us by many people), it is almost certain that all the families we consider in our paper are, really, expanders, and very likely that this will be formally proved and published in the near future. But as long as the proofs of this require showing the esperanto property first, and require additional non-trivial steps (which is the case for the moment), our own applications seem to be more transparent when phrased in terms of esperantism. And there might of course be applications where the graphs do not form expanders (though we have not found one yet).

Now for the philosophy: the very barest summary of our main result is that, provided a family of finite (unramified) coverings

of a fixed algebraic curve over a number field has the property that the associated Cayley-Schreier graphs associated to the sets of cosets

form an esperantist family, then there are very strong diophantine restrictions on the points of the coverings defined over extensions of the base field of a fixed degree. (There is more information in Jordan’s post.) There is an unsuspected deus ex machina hidden, which makes the proof quite surprising: we use an inequality from global analysis of Li and Yau (already used by Abramovich in the setting of classical modular curves), which seems to come completely out of the blue.

This seems to suggest that the following general analogue problem might deserve attention: suppose you have some “objects” for which it makes sense to speak of finite coverings, and of Galois groups, etc, and you have a sequence of these (say ?n) with finite coset spaces

and some finite generating set of the base Galois group (or something similar). Can one say anything interesting on the “geometry” if one assumes that this family of graphs is esperantist (or expanding)?

Even the most natural analogues of our setting seem very interesting and quite tricky to think about:

(1) what about coverings of a base curve defined over a function field of positive characteristic (say, with tame ramification to avoid unpleasantness)? Here one would think that the direct analogue of our statements might hold, but the Li-Yau inequality evaporates, and we are left scratching our heads (though one might hope, maybe, that some p-adic analogue could be true?)

(2) what about coverings of higher-dimensional base varieties over a number field? Here, we do not even know yet what a reasonable consequence could look like…

[Note: our results also depend on the comparison between hyperbolic and discrete laplace eigenvalues, specifically on the results of Burger in this direction, which are somewhat sharper than those of Brooks; since Burger's method is described only sketchily in his papers, and his thesis -- which contains the full details -- is hard to find, we included the proof of the comparison we require as an Appendix to our paper, in the case of surfaces with finite hyperbolic area. This might be of some interest to some readers.]

Every mathematician who has ever exchanged (La)TeX files by email must have noticed lines starting

¿From…

appearing in the resulting dvi or pdf file.

These charmingly infuriating lines are due (if I understand things right) to TeX’s transforming the character “>” into the inverted question mark, and to the tendency of email programs to consider that a line starting with “From” means that an included email is starting, which must be quoted with “>”. Since mathematical papers tend to have sentences like “From this, it follows that…”, this is what we end up with, unless one is careful to regularly search the document for the telltale “>From” in order to remove the offending symbol (or one gets the reflex of cleverly writing “{}From” instead of “From”, something I just learnt from a coauthor.)

But I find it ironic that computers, which can apparently spell-check documents, correct their grammar, or attempt translating them into Esperanto, are unable to understand that sentences starting with “From” might be legitimate…

I know about Japanese rings (commutative rings A which are integral domains and such that the integral closure of A in any finite extension of the ring of fraction is finite over A), and about Polish spaces (separable complete metric spaces). Are there any other mathematical concepts named after locations on Earth (or elsewhere)?

The only vaguely similar cases I can think of are K3 surfaces, which A. Weil mentions somewhere being named partly as a reference to the K2 mountain; and the recent innovation of esperantist graphs, which are defined in a new preprint of J. Ellenberg, C. Hall and myself (I’ll write about the latter in more detail soonish; the point of the name is that it alliterates with “expanders”, and it is indeed a condition related to, but weaker, than being an expander graph…)

What Jeeves calls an unfortunate concatenation of circumstances led me and mia sposa to go on rather short notice to Italy in early August for vacation. A priori, this is rather un-optimal timing; tourists flock to Tuscany while Tuscans escape. However, things turned out quite nicely. In case anyone gets in the same situation, here are a few random remarks on the topic…

(1) Between the Forum and the Colosseum, no need to make a choice, since the ticket is the same and is valid all day; however, unless queuing is a délicatesse in your mind, it is much better to get the tickets at the Forum office (via dei Rori Imperiali), visit the Forum (and the Palatine Hill) first in the morning, have a good lunch, and then visit the Colosseum when the sun is fierce. The queue there will be enormous, but the tourist au courant, holding his daily ticket, scorns the long line, and there is surprisingly much more shade inside the standing circular Colosseum than on the mostly ruined Forum…

(2) Speaking of lunch, part of the difficulty was that not only are quite a few ristoranti closed in August, but those that remain within a certain radius of the centers of interest tend to be, shall we say, not particularly interesting. Clearly, there is life is smaller streets, and we were rather lucky in terms of finding very decent places. For the lunch break between the Forum and the Colosseum, we found La Taverna dei Fori Imperiali, located 9, via della Madonna dei Monti, which was very nice.

(3) Firenze, or Florence as we say in France, is absolutely amazing if you have any interest in anything related to the Renaissance, be it painting, sculpting, science, politics or heretic-burning. (Not an original observation.) It is moreover quite compact and one can happily walk all over the place without needing or caring for any other mode of transportation. And although the touristic density is higher than in Rome, there are more small streets. We had a wonderful dinner about two to three hundred meters from Santa Maria dei Fiore in a very nice restaurant which was entirely empty apart from us (Cantina Barbagianni, located 13r, via Sant’Egidio). And another dinner at Ristorante Pensavo Peggio, at 51r, Via del Moro was also very good; the place — more traditional — was a bit more crowded, but not much.

(4) Because Firenze is small, it is easy to have a very good look at the doors of the Baptistery of the Duomo, simply by starting the day early enough before the crowd arrives. On the other hand, to visit the Galleria degli Uffizi, it is clear that the only reasonable thing to do is to buy a ticket in advance; otherwise, even arriving 15 minutes before opening (as we did…), you’re in for a good hour of standing in line. (In the shade, but that’s not so important in the morning…) And then of course the place will be packed all along your visit…

(5) A good option to escape the crowd is always to look for the science museum. The one in Milano (named after Leonardo) is fairly big and had an exhibition of early electric machinery including a 19th century Fax Machine, invented by a rather cunning Abbé (if I remember right his ecclesiastic position). The one in Firenze (named after Galileo) is smaller and has mostly older apparatus on display.

(6) And if you ever wondered what Galileo, Machiavelli, Michelangelo and Rossini have in common, you can visit the Basilica di Santa Croce in Firenze; it is just a bit outside of the most busy center, and hence a bit less crowded.

(7) Last, but not least: if you enjoy a nice digestif after a long day walking around, a good choice if you don’t feel equal to a Grappa is to order Limoncello (something I picked up in Trieste three years ago). Apparently this makes a good impression, since — in two different places — our glasses were liberally refilled.

Earlier today, while hacking my way with A. Saha through Gradshteyn-Rizhik in search of some clue to an integral he wanted to compute, we found a close enough approximation where, on the right-hand side, a function Dp appeared. I had no idea what it could possibly be; the lack of an index meant we had to go through the whole back section on special functions to locate it (it turns out to be a “parabolic cylinder function”, a close relative of Whittaker functions and of confluent hypergeometric functions).

This led me to wonder if one could make a whole alphabet song of special functions: is there a letter, poor thing, such that no well-known special function is named after it? (I’m allowing multi-letter names, so that “A” goes with the Airy function Ai(z)). I’m not even sure about B, though some Bessel function should fit…

As I’ve just lent my copy of G-R, I can’t look right away. But aspiring song-writers can start looking and suggesting catchy rhymes and couplets to go with the long overdue “Song of special functions”…

During the Spring Semester 2011, Philippe Michel and myself will be co-organizing a semester-long programme at the Centre Interfacultaire Bernoulli of EPF Lausanne, dedicated to the topic of

Group Actions in Number Theory

or GANT, for short — a name that, to make a bad pun, fits like a glove.

We now have web pages online that describe the basic intent of the semester, with some indication of the activities that will happen (in particular, the instructional conference from January 18 to 28, 2011, and the final research conference, from June 6 to 10), and information concerning the application process for anyone interested in participating…

In arithmetic and geometry, there is a well-known split between right-wingers who write

and left-wingers who prefer

More insidious and mysterious is the deeper split between the inliners who write

and the redoubtable subscripters who will only write

I’ve used almost exclusively the first (almost because I don’t think a paper referring to E(8,Z) could possibly be accepted), but I have no particular memory of why I started. Does anyone have an argument (in bad faith or otherwise) for either choice?

I’ve just realized a rather obvious fact concerning the vague question at the end of my latest post on Kloostermania, which I’ll rephrase informally:

Can one guess with better than even chance that a graph like Kloostermania’s represents a Kloosterman or a Salié sum?

I had said that, for fixed p, just knowing the sum shouldn’t leave much room to make any choice except a random one. To make sense, the rule must be made more precise: you are given a bare real number, say

and you are told that it is either the value of a Kloosterman sum S(1,1;p) for some prime, or of a Salié sum T(1,-1;p) for some prime. You can win a drink of your choice by picking up the right one. What can you say? The reason it is hard to do better than heads-Kloosterman/tail-Salié is that you do not know p. If you knew the value of the prime, then you could compute the angle in [0,π] such that

and use the fact that these are supposed to be distributed differently for Kloosterman and Salié sums. For instance, the probability that θ is(conjecturally) larger than 3π/4 is bigger for Salié sums (it is 1/4) than for Kloosterman sums (about 0.09…), and hence, finding an angle in this range would give a big hint that it comes from a Salié sum (but no certainty, of course).

And now for the really obvious point: if you can, in addition to knowing S, actually see the graph of the partial sums, then — of course — you know the prime: you just have to count the steps on the graph.

It seems that similar ideas should lead to a slightly better than average guess even without knowing p: the size of S gives a lower bound on p, and for each possible guess of p, we have a guess of either Salié or Kloosterman. Presumably, combining these should be possible to get a small gain on pure chance…

(Note: readers are welcome to make a guess concerning the value above…)

The platonic solids are of course quintessentially Greek (although a claim to their discovery has apparently been staked on behalf of rugged Scots — who certainly play rugby better, not that this should influence priority disputes). I was therefore quite intrigued to see today, in the Roman Museum of the town of Avenches, a very beautiful Roman dodecahedron:

The decorations are quite interesting; note for instance that the holes in the faces are not all of the same size. The accompanying text mentioned that at least 60 such objects have been found in what was ancient Roman territories north of the Alps, and that their purpose (if any) is not known. This one was found in a private house (the approximate date is not mentioned, but the old Roman city of Aventicum apparently flourished mostly during the first Century).

One of the most charming category of words in any language is that of eponyms, nouns taken from the names of actual people (or places), when this origin is completely forgotten (not like euclidean…) Two favorites in French are

Poubelle (garbage container), from the name of the prefect Poubelle, who apparently made the use of such an implement mandatory in 1884;

Silhouette (silhouette), from a French finance minister of the middle 18th Century; here the etymology claims that the French people disliked his economy politics and attributed the name to drawings done equally economically… This word is even more remarkable in that it is now common in at least three languages (in English but also in German according to my dictionary). Are there others?

I’ve just learnt of a new one, a word I’ve used many times without ever wondering where it came from…

Barème: this may roughly be translated as “scale” or ‘table”; it’s commonly used in French for the distribution of points in an exam (e.g., four points for the first exercise, six for the second, and ten for the last problème). The name is from François Barrême (note the change of spelling; I’m using the Grand Robert as dictionary), a now obscure French mathematician from the late 17th century, who was once called ce fameux arithméticien (”this famous arithmetician”) for his works Tarifs et Comptes faits du grand commerce and Livre des comptes faits — see here; these books seem to have been simply conversion tables between units of measurements and money systems of various countries and provinces.

This is rather off-topic, but maybe not all owners of Wordpress-hosted blogs are aware of it: since a few days, the mobile version of their sites (which displays when accessed from a phone, or at least from an Android one) displays a small ad-widget by default. I’ve noticed this on Vieux Girondin, Quomodocumque, What’s New, T.Gowers’s blog, etc… Here’s what it looks like:

If this is not the desired behavior (and some of the ads I’ve seen might not be what the owners like to be associated with; the one above is not the worst…) it is probably easy to disable the plugin in charge of the mobile theme (but I can’t check; although my own blog is Wordpress-powered, it is hosted at ETH and doesn’t have this plugin).
Also, unfortunately, I don’t know how to check the mobile version from a desktop or laptop computer (presumably one needs to tell the browser to pretend to be a mobile version, but I’ve never learnt how to do that…).

Kloostermania fans can look up what the new version 0.15 does on the Kloostermania page…

In particular, I’ve added the possibility to display Salié sums instead of Kloosterman sums. Precisely, it shows

instead of T(1,1;p) because the values of the latter are not as interesting, due to the fact that the root s of

modulo a prime are rather simple to compute….

This leads to an intriguing game to play: can you tell the difference between the two types of sums?

First, one must take a prime congruent to 1 modulo 4 (otherwise the Salié sum is zero in that case, which Kloosterman sums never are) for the question to be interesting. Then, there is a kind of theoretical/conjectural answer if you are allowed to look at many instances of the two sums (i.e., you can start the slideshow and observe it for a long time — skipping primes which are 3 mod 4 — without changing the type of sums): their distribution is not the same (conjecturally)! Precisely, the angles θp of the Salié sums, for primes which are 1 modulo 4, defined by

are equidistributed on [0,π] (for the Lebesgue measure; this is the wonderful theorem of Duke, Friedlander and Iwaniec), whereas one expects those of Kloosterman sums to be distributed according to the Sato-Tate measure

In particular, the Kloosterman sums should be more often “small”, in some sense, than the Salié sums since the density of the Sato-Tate measure vanishes at θ=0 (which corresponds to a maximal sum).

But what if you’re not allowed to start a long slideshow? For a fixed p, I don’t think one can expect to be able to guess more precisely, just from the values of the sums, than by throwing a coin and choosing Heads/Kloosterman, Tails/Salié. But I wonder if the shapes of the graph of partial sums (as drawn by the program…) could be used to extract more information to lead to a guess with better than even odds of being correct? Or if, at least, it could be used to shorten the length of time one would need to look at the slideshow before being sure of the answer from the distribution perspective?

For some reason, filling the MSC codes for my papers always feels a bit of a chore; I often have the impression that none of the headings really correspond to what I’ve done, and I’ve found in the past that looking for the right one in a PDF version of the classification is not very efficient. But the tiddlywiki version of the new 2010 classification promises to make things easier: it’s a single HTML file which, through some javascript magic, can be used to dig inside in outliner fashion (expand/unexpand), and which — once downloaded and stored locally — can be freely annotated. Moreover, I can even do that on my android phone and explore the minutiae of the classification during my tramway rides…

But I already noticed that I’ll probably continue feeling perplexed when selecting MSC numbers: for instance, there does not seem to be any item that fits the topic of expander graphs… It seems one has to use an unsatisfactory mixture of 05C40 (”Combinatorics : Graph theory : Connectivity”) and 05C50 (”Combinatorics : Graph Theory : Graphs and linear algebra (matrices, eigenvalues, etc.)”). I’m sure that if I (or anyone else) had pointed this out during the process that ended with the current version of MSC 2010, this would have been corrected, but it is now probably too late until the next revision…

It will probably not come as a surprise to most readers that I like exponential sums, especially over finite fields. I’m therefore very happy to announce a conference on the topic of

Exponential sums over finite fields and applications

that will be held at the Forschungsinstitut für Mathematik of ETH, during the week of November 1 to November 5, 2010. This conference is organized by N. Katz, P. Michel, R. Pink and myself.

The web site

http://www.math.ethz.ch/~kowalski/exponential-sums.html

is now up, with a preliminary list of speakers and an unofficial poster

(the official poster will come soon, its design will link the conference thematically with the other two number theory events that have been organized in Zürich this year, the Number Theory Days and the Rational Points conference.)

As indicated on the web site, there will be some support available for other participants, in particular PhD students and young researchers. Anyone interested in coming is invited to write to the contact email address: expsums@math.ethz.ch

At last! Yesterday, visiting the well-known Zürich rainforest, we finally saw one of their famed leaf-tailed geckos (these had otherwise seemed to be rather mythical creatures):

Leaf-tailed gecko

It is in fact not entirely surprising that we did not see one before; indeed, can you spot the one hiding in this other picture?

Where is the gecko?

(Click for a larger view)

There has been some recent lively discussion of ethical issues in hiring here.
Now what would you say of someone who, while Rektor of his provincial university in the town of G., visits a more prestigious foreign institution (no less than an Imperial one, in B.), accepts an offer to move there, does not mention it to anyone, twiddles and twaddles, and then reneges on the offer after he was supposed to have started teaching? Shocking, what?
I couldn’t help smiling when reading this synopsis of the employment history of L. Boltzmann, though the actual facts must have been quite distressing to the people involved (Boltzmann apparently once said that he knew much better “how to integrate than how to intrigue” — I’d like to see the original German, of course, but there was no reference in the book I’m reading). He was in Graz for 14 years, got an offer from Berlin in 1888, accepted, reneged (and had to be relieved of his duties by the Kaiser — though I don’t know which one; there were three in 1888, apparently)… and instead of staying in Graz as one might have expected after turning down one of the juiciest positions available at the time, he went on a true whirlwind of moves and changes during the following years (first to Münich, then to Vienna, then to Leipzig, then back to Vienna…)

I wish to thank the 50 individuals (or institutions) who bought a copy of my book on the large sieve last year! I’ll be happy to offer a beer (or any other drink) to any one of you who happens to spend some time in Zürich — even if this was an impulse buy due to its brush with literary celebrity, or a wish to see how the contents stacked up against such competition as Baboon Metaphysics.

Note: no need to bring a receipt to get the drink; I’m a trusting person. (Though there will be an interesting pigeon-hole problem once I’ve paid for 60 or 70 drinks…)

Irrelevant note: the happy few is a reference en anglais dans le texte to the quotation at the end of Stendhal’s La Chartreuse de Parme (one of my favorite French novels); I’ve just done my second Wikipedia edit by removing the claim that he used it for Le rouge et le noir (in the English entry).