Author: Kowalski

I am a professor of mathematics at ETH Zürich since 2008.

Buffon’s needle

As a result of recent moves, the (almost) complete set of Buffon’s monumental Histoire naturelle belonging to my father’s family has recently arrived here in Zürich (it comes from my grand-father, who was director of the Muséum d’histoire naturelle de Nantes). I will keep these in my office for the moment, as it definitely lends it a very scholarly air…

(As far as I can see from the web page above, what is missing from our set is the Histoire naturelle des poissons, which was not written by Buffon anyway, but by the Comte de Lacépède, who also wrote the volumes about snakes, which we do have).

Many probabilists know Buffon for his annoying habit of dropping needles on the parquet, and finding the value of π after doing this sufficiently many times. This game was indeed included in his natural history, more precisely in the Essai d’arithmétique morale (or “Essay of moral arithmetic”) in Volume VII of the Suppléments — at least, it is there in my family’s edition, though it is missing from the web site containing Buffon’s works, where the Essai is in Supplement volume 4.

Here are pictures of the first pages of the description of the problem (click for readable larger picture):

Buffon’s needle

and

Buffon’s needle, 2

Notice the delightful typography and orthography: the “s” that looks like an integral sign (and is barely distinguishable from an “f”), the way the past tense is written demanderoit instead of the current demanderait, etc.

Some things I didn’t know about groups

Experimenting with finite groups recently, I’ve learnt a few things I was completely unaware of:

(1) there are so many distinct groups of order 512 (or 1024, 1152, 1536 and 1920) up to isomorphism, that Magma and GAP are not able to recognize them (they have databases of groups of order up to 2000, except for 1024, but given one abstract group of this order, they can not pinpoint which element of the database it is). Precisely, there are 10494213 distinct finite groups of order 512 up to isomorphism, 157877 of order 1152, 408641062 of order 1536 and 241004 of order 1920. These numbers are several order of magnitude larger than what I would have guessed if asked point-blank.

(2) the order of groups of a fixed Lie type, say X(q), where X could be PSL, is not always monotonic with respect to the variable q; for instance:

gap. Order(PSL(3,17));
6950204928
gap. Order(PSL(3,19));
5644682640
gap. Order(PSL(3,29));
499631102880
gap. Order(PSL(3,31));
283991644800

(Of course, this is then easy to check, and the point is that when q-1 is divisible by 3, the order drops as the center of SL(3,q) becomes a bit bigger).

Random matrices, L-functions and primes

This could be the title of a book, of a research paper, but it is really (for the current purpose) the name of the conference that Ashkan Nikeghbali (from Universität Zürich) and myself are organizing in late October — from October 27 to October 31, to be precise –, thanks to the support of the Forschungsintitut für Mathematik from ETH Zürich.

Despite the late announcement, this has been in preparation for a few months already, but we are both not always the most efficient among organizers. However, there is at last some kind of web page, although for the moment it is not really more informative than the poster that will soon be printed and distributed everywhere (click for full-size picture):

Poster for the conference

The chameleon in the picture, in case you are wondering, is one specimen from the Zürich rainforest. It has no special meaning apart from that.

(The following

Alternate poster for the conference

almost became the poster, but it was finally decided that the other one was more memorable).

We hope that this conference will be an occasion to further the links between probability theory and number theory, in particular through the three topics of the title.

2, 3, 13, 19, 17, 11, 23, 5, 10, 14, 20, 21, 16, 22, 18, 15, 4, 6, 12, 9, 7, 8

The rather bizarre sequence in the title is the ordering of the integers from 2 to 23 which is induced by the values of the Chebotarev invariant of the group An. In other words, if n comes before m in this list (for instance, n=20, m=9), it means that on average, you will need to pick fewer conjugacy classes in An at random to obtain a set that necessarily generates it, than you need to do for Am.

This illustrates some non-obvious property of this invariant; for instance, it is easier to “fill up” A21 than A8, despite the fact that the first has more maximal subgroups (14 against 6) and many more conjugacy classes (408 against 14) — in fact, there’s probably some money to be made by setting up betting games based on this type of observations.

The actual values of the invariants are in fact fairly close (except for the degenerate cases 2 and 3), ranging from 4.01632 or so to 4.939. It is not at all clear (to me) what the limiting behavior will be as n grows; I had first a vague guess that the sequence of invariants would grow to infinity (maybe slowly), but the data is more consistant with a convergence (or an oscillating behavior), which is what D. Zywina suggested to me for the symmetric group — where the behavior is roughly similar but the ordering is different.

I have been able to do these computations thanks to having again access to Magma; and I once more can’t help expressing my admiration for the incredible work that this team has done. (Admiration and gratitude which also apply equally to the members of the other teams working on algebraic-computation packages such as GAP, Pari/GP and Sage, even though for this particular type of computations, Magma is currently far ahead).