Prof. Arnold's seminar of 20 March

[Math]

M A T H E M A T I C S   S E M I N A R S   2006



Series of Lectures in "Experimental Discoveries of Mathematical Facts"


Thursday, 20 April, at 15.00 hrs.


Professor V.I. Arnold
(Steklov Mathematical Institute, Moscow, Russia)


Random permutations and
Young diagrams of Fibonacci automorphisms of tori.
(Lecture 2)


Abstract:

A permutation of the elements of a finite set defines the
partition  n = x_1 + x_2 +...+ x_y of the number  n  of the elements
of the set into  the lengths of the y cycles of the permutation, and
hence
it defines the Young  diagram (x_1 >= x_2 >= ... >= x_s).
The talk describes the averaged statistics of the parameters of these
diagrams for the n! permutations of the set and for the action of the
so called "Arnold's cat map" (that I call Fibonacci operator), defined
by the matrix

    2 1
    1 1  ,

on the finite torus  Z_m x Z_m , consisting of n = m^2 points.
The parameters of the Young diagram are: length x = x_1, width y,
fullness lambda = n/(xy).



VENUE:  MAIN LECTURE HALL
(ICTP Main Building, 1S Level)