Professor V.I. Arnold's forthcoming lectures at ICTP

[Math]

ICTP-SISSA Joint Colloquium
AND
Mathematics Seminars 2006

Experimental Discoveries of Mathematical Facts
Professor V.I. Arnold

Professor Arnold, currently visiting the ICTP, will give an ICTP/SISSA
Joint Colloquium and three lectures with the above theme, according to
the following schedule.

No previous knowledge is needed to understand the Colloquium and the  
lectures. Several problems will be enounced during the talks.


ICTP/SISSA Joint Colloquium

Tuesday, 11 April at 14:00

Complexity of Finite Sequences

Venue: Seminar Room, ICTP main Building, first floor
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Lectures

1) Statistics of topological structures of smooth functions, of
polynomials, of trigonometric polynomials, of affine Coxeter groups and  
the Sixteenth Problem of Hilbert.
     Wednesday, 5 April at 14:30

2) Random permutations and Young diagrams of Fibonacci automorphisms of  
tori.
     Thursday, 20 April at 15:00

3) Frobenius numbers, geometry and statistics of additive semigroups of  
integers.
     Wednesday, 26 April at 14:30

All the lectures will be held in the ICTP Seminar Room (Main Building,  
first floor).
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These four talks form the July 2005 lecture "Experimental discoveries  
of mathematical facts", read by V.I. Arnold at the Nuclear Center  
"Dubna" (Russian "CERN") to the highschool children (who understand new  
things better, than the Nobel prizes or Fields medal winners), and  
later (28 February 2006) at Universite' Paris-Jussieu Institut de  
Mathematiques (where it had been about 3 hours long for the totality of  
4 subjects).
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A detailed summary is given below.
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Complexity of Finite Sequences

Abstract: Everyone understands that the sequence 001001001001 (of 12
binary  numbers) is less complicated than the sequence 010010111001. The
talk  provides an exact mathematical meaning to this complexity notion  
in
terms of the graphs of mappings of finite sets to themselves, leading  
to a
hierarchy of the elements of a finite ring of functions and of its  
subring
of polynomials.  Experiments suggest that the most complicated  function
is the logarithm.  The ring of polynomials forms a binary tree with  
2^2^k vertices  (k=2,4,256, ...).

-----
Statistics of topological structures of smooth functions, of
polynomials, of trigonometric polynomials, of affine Coxeter groups  and
the Sixteenth Problem of Hilbert.

Abstract: One associates to a smooth function a graph, defined as the
topological  space whose points are the connected components of the  
level
hypersurfaces of the function.  The talk describes the structures of
these graphs for the Morse functions with a fixed number of critical
points and for the polynomials and trigonometric polynomials of given
degree or a given spectrum (provided, for instance, by the Newton
polyhedron of that trigonometric polynomial for an affine system of  
roots).

------
Random permutations and Young diagrams of Fibonacci automorphisms of  
tori.

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).
---
Frobenius numbers, geometry and statistics of additive semigroups of
integers.

Abstract: The Frobenius number N(a_1, ..., a_n), where the a_i are  
natural
numbers (with no common divisor greater than 1) is the minimal integer,
such that itself and all greater integers are representable as linear
combinations x_1 a_1 + ... + x_n a_n with nonnegative integral
coefficients  x_i . For instance, N(a,b)=(a-1)(b-1). But for n > 2 there
is no  explicit formula for N, and even its growth rate for growing
a=(a_1, ..., a_n) is unknown. The talk proves that it grows
at least as sigma^(1+(1/n-1)) and
at most  like (sigma)^2, where  sigma = a_1 + ... + a_n.

Both boundary cases are attained for some directions of the vector a,  
but
the growth rate depends peculiarly on this direction.  The average  
growth
rate has been studied experimentally and the talk will present the
empirical mean values ( for sigma = 7, 19, 41, 97 and 199). The observed
rate is (sigma)^P with  p ~ 2  at the beginning, declining to  p ~ 1,6  
for sigma between 100 and 200.  This confirms the  author's conjecture  
of 1999 that p tends to  1+1/(n-1) = 3/2  for large sigma.