ICTP/SISSA Joint Colloquium in Mathematics on 20 February 2008

[Mabilo Koutou]

ICTP/SISSA Joint Colloquium in Mathematics

Wednesday, 20 February, at 15.00 hrs.



Professor Vladimir I. Arnold

Steklov Mathematical Institute, Moscow, Russia

"Permutations"



Abstract:

Random permutations of N elements have peculiar statistics of the  
lengths of the cycles (discovered by V.L. Gontcharov, 1942).

The number  of the cycles of a random permutation of N points grows with

N (in the mean) as 1+ 1/2 +1/3 + … + 1/N  ~ 0,58 + ln N.

The Fibonacci cat map is the action of the matrix  [] permuting the  
n^2 points of the finite torus Z/nZ x Z/nZ. Such algebraically  
defined permutations have quite different statistics (from that of  
the random ones).

Example: Some permutations of 100 points have periods exceeding 230  
million.

The length of the period of the cat map, permuting 150x150 pixels, is  
only 300. The study of similar examples (begun in the 1990s  by I.  
Persival and F. Dyson) has led to unexpected mathematical theorems.

Example: The number of those permutations of 2N points, all whose  
cycles are of even length, is a square of an integer, namely, of  
(2N-1)!! =1.3.5…..(2N-1).



Venue:  ICTP Lecture Room C, Main Building