GANT

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…

A tale of two dichotomies

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

G/\Gamma

and left-wingers who prefer

\Gamma\backslash G

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

SL(2,\mathbf{Z})

and the redoubtable subscripters who will only write

SL_2(\mathbf{Z}).

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?

Kloosterman vs. Salié

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

S=-17.0711787748\ldots

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

\frac{|S|}{2\sqrt{p}}=\cos(\theta),

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…)

Roman dodecahedron

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).

Personal words

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.