Kolmogorov's heritage in mathematics

326 Pages·2007·2.64 MB·English

by  Eric Charpentier, Annick Lesne, Nikolaï K. Nikolski

Kolmogorov’sHeritagein Mathematics
PicturetakenfromtheKolmogorovArchivewithkindpermissionofA.N.Shiryaev
·
Éric Charpentier Annick Lesne
Nikolaï K. Nikolski (Eds.)
Kolmogorov’s Heritage
in Mathematics
With22 Figures
(cid:0)(cid:2)(cid:3)
ÉricCharpentier NikolaïK.Nikolski
InstitutdeMathématiques InstitutdeMathématiques
UniversitéBordeaux1 UniversitéBordeaux1
351coursdelaLibération 351coursdelaLibération
33405TalenceCedex 33405TalenceCedex
France France
[email protected] [email protected]
AnnickLesne
LaboratoiredePhysiqueThéorique
delaMatièreCondensée
UniversitéPierreetMarieCurie
Case121,Tour24-25-4èmeétage,pièce407
4PlaceJussieu
75252Paris
France
And
InstitutdesHautesÉtudesScientifiques
LeBois-Marie,35routedeChartres
91440Bures-sur-Yvette
France
[email protected]
Éditionoriginale:L’héritagedeKolmogorovenmathématiques(cid:2)c ÉditionsBelin–2004
OuvragepubliéavecleconcoursduMinistèrefrançaischargédelaculture–Centrenationaldulivre
ThiseditionwassupportedbytheFrenchMinistryofCulture–Centrenationaldulivre
CoverpicturetakenfromtheKolmogorovArchivewithkindpermissionofA.N.Shiryaev
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ISBN978-3-540-36349-1 SpringerBerlinHeidelbergNewYork
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Contents
Introduction
E´ric Charpentier, Annick Lesne, and Nikola¨ı Nikolski................. 1
1 The youth of Andrei Nikolaevich and Fourier series
Jean-Pierre Kahane .............................................. 7
2 Kolmogorov’s contribution to intuitionistic logic
Thierry Coquand................................................. 19
3 Some aspects of the probabilistic work
Lo¨ıc Chaumont, Laurent Mazliak, and Marc Yor ..................... 41
4 Infinite-dimensional Kolmogorov equations
Giuseppe Da Prato ............................................... 67
5 From Kolmogorov’s theorem on empirical distribution to
number theory
Kevin Ford...................................................... 97
6 Kolmogorov’s ε-entropy and the problem of statistical
estimation
Mikhail Nikouline and Valentin Solev ...............................109
7 Kolmogorov and topology
Victor M. Buchstaber.............................................139
8 Geometry and approximation theory
in A. N. Kolmogorov’s works
Vladimir M. Tikhomirov ..........................................151
9 Kolmogorov and population dynamics
Karl Sigmund ...................................................177
VI Contents
10 Resonances and small divisors
E´tienne Ghys....................................................187
11 The KAM Theorem
John H. Hubbard.................................................215
12 From Kolmogorov’s work on entropy of dynamical systems
to non-uniformly hyperbolic dynamics
Denis V. Kosygin and Yakov G. Sinai ..............................239
13 From Hilbert’s 13th Problem to the theory of neural
networks: constructive aspects of Kolmogorov’s Superposition
Theorem
Vasco Brattka ...................................................253
14 Kolmogorov complexity
Bruno Durand and Alexander Zvonkin..............................281
15 Algorithmic chaos and the incompressibility method
Paul Vitanyi ....................................................301
List of Contributors
Vasco Brattka Chalmers University of Technology
Laboratory of Foundational Aspects and Go¨teborg University
of Computer Science Sweden
Department of Mathematics http://www.cs.chalmers.se/∼coquand
and Applied Mathematics [email protected]
University of Cape Town
South Africa Giuseppe Da Prato
http://cca-net.de/vasco Scuola Normale Superiore
[email protected] Pisa, Italy
http://www.sns.it/scienze/menunews/
Victor M. Buchstaber docentiscienze/daprato
Steklov Mathematical Institute of [email protected]
Russian Academy of Sciences, and
Moscow State University, Russia Bruno Durand
http://www.mi.ras.ru/∼buchstab Laboratoire d’informatique fonda-
[email protected] mentale de Marseille
and School of Mathematics, The University of Provence (Aix-
University of Manchester, UK, Marseille I), France
[email protected] http://www.lif.univ-mrs.fr/∼bdurand
[email protected]
Lo¨ıc Chaumont
Laboratoire Angevin de Recherche Kevin Ford
en Math´ematiques (LAREMA) Department of Mathematics
University of Angers, France University of Illinois at Urbana-
http://math.univ-angers.fr/∼ Champaign, USA
chaumont http://www.math.uiuc.edu/∼ford
loic.chaumont@univ- angers.fr [email protected]
Thierry Coquand E´tienne Ghys
Department of Computer Science Unit´e de math´ematiques pures et
and Engineering appliqu´ees (UMPA)
VIII List of Contributors
CNRS and E´cole normale sup´erieure and International Institute for
de Lyon, France Applied Systems Analysis, Austria
http://www.umpa.ens-lyon.fr/∼ghys http://homepage.univie.ac.at/
[email protected] Karl.Sigmund
[email protected]
John H. Hubbard
Department of Mathematics Yakov G. Sinai
Cornell University, Ithaca, NY, USA Department of Mathematics
http://www.math.cornell.edu/People/ Princeton University, USA
Faculty/hubbard.html [email protected]
[email protected]
J.-P. Kahane
Valentin Solev
D´epartement de Math´ematiques
Laboratory of Statistical Methods,
d’Orsay
Steklov Institute of Mathematics,
University Paris-sud, Orsay, France
RAS, Saint Petersburg,Russia
http://www.academie-sciences.fr/
http://www.pdmi.ras.ru/staff/solev.html
Membres/K/Kahane JP.htm
[email protected]
[email protected]
Vladimir M. Tikhomirov
Denis V. Kosygin
Moscow State University and
Courant Institute of Mathematical
Independent University of Moscow,
Sciences
Russia
New York University, USA
[email protected]
Paul Vitanyi
Centrum voor Wiskunde en Infor-
Laurent Mazliak
matica (CWI), Amsterdam
Laboratoire de Probabilit´es et
The Netherlands
Mod`eles Al´eatoires
Institutdemath´ematiquesdeJussieu http://www.cwi.nl/∼paulv
University Paris VI, France [email protected]
http://www.proba.jussieu.fr/users/
Marc Yor
lma/mazliak.html
Laboratoire de Probabilit´es et
[email protected]
Mod`eles Al´eatoires
Mikha¨ıl Nikouline (M. Nikulin) Institutdemath´ematiquesdeJussieu
Statistique Math´ematique et ses University Paris VI, France
Applications http://www.academie-sciences.fr/
University Victor Segalen (Bor- Membres/Y/Yor Marc.htm
deaux II), France
http://www.sm.u-bordeaux2.fr/stat Alexander Zvonkin
[email protected] Laboratoire Bordelais de Recherche
en Informatique (LaBRI)
Karl Sigmund University Bordeaux I, France
Faculty for Mathematics http://www.labri.fr/perso/zvonkin
University of Vienna [email protected]
Introduction
E´ric Charpentier1, Annick Lesne2, and Nikola¨ı Nikolski3
1 Institut deMath´ematiques de Bordeaux, University Bordeaux I, France
http://www.math.u-bordeaux1.fr/imb/
[email protected]
2 Laboratoire de physiqueth´eoriquede la mati`ere condens´ee, UniversityParis VI,
and Institut des HautesE´tudes Scientifiques,Bures-sur-Yvette,France
http://www.lptmc.jussieu.fr/users/lesne/
[email protected]
3 Institut deMath´ematiques de Bordeaux, University Bordeaux I, France
http://www.math.u-bordeaux1.fr/imb/
[email protected]
Translated from the French by Elizabeth Strouse
Andrei Nikolaevich Kolmogorov (Tambov 1903, Moscow 1987) was one of
the most brilliant mathematicians that the world has ever known. Incredibly
deep and creative, he was able to approach each subject with a completely
newpointofview:inafewmagnificentpages,whicharemodelsofshrewdness
and imagination, and which astounded his contemporaries, he changed dras-
tically the landscape of the subject. Most mathematicians prove what they
can, Kolmogorovwas of those who prove what they want.
In this book we haveaskedseveralworldexperts to present(one partof4)
the mathematical heritage left to us by Kolmogorov5. Each chapter treats
one of Kolmogorov’s research themes, or a subject that was invented as a
consequenceofhisdiscoveries.Wepresentherehiscontributions,hismethods,
the perspectives he opened to us, the way in which this research has evolved
up to now, along with examples of recent applications and a presentation of
the modern prospects.
We hope that this book can be read by anyone with a master’s (or even
a bachelor’s) degree in mathematics, computer science or physics, or more
4 Inthebook Kolmogorov in perspective (History of Mathematics, Vol.20, Ameri-
can Mathematical Society, 2000) one can find a (more or less) complete bibliog-
raphy of Kolmogorov’s work, which consists of about 500 publications. A lot of
theseare at the heart of very active research going on today
5 A book entitled The Kolmogorov Legacy in Physics has already appeared
(Springer,2004).Itcontainscontributionstodynamicalsystems,complexity,tur-
bulenceand probability. Westrongly recommend it to mathematicians