Andrei Nikolaevich Kolmogorov

April 25, 1903 - October 20, 1987

Curriculum Vitae & Biography

Kolmogorov in Perspective


Kolmogorov School Spiral

Ph.D. students and descendants of A. N. Kolmogorov

Scientist's Club of the Moscow State University and Moscow Mathematical Society
Kolmogorov Centennial Memorial Meeting
Great Hall of the Moscow State University

(Moscow, April 29, 2003)

Kolmogorov Centennial
Kolmogorov Signature

Russian Academy of Sciences (RAS) and Moscow State University (MSU)



(Moscow, June 16 - 21, 2003)

of Andrei Nikolaevich Kolmogorov
(25.IV.1903 - 20.X.1987)

Scopes and Themes

The Conference will cover the main areas of A.N. Kolmogorov's scientific interests with an emphasis on his vision of Mathematics in its fundamental unity as well as recent developments in the fields:
  • Dynamic Systems and Ergodic Theory
  • Theory of Functions and Functional Analysis
  • Theory of Probability and Mathematical Statistics
  • Turbulence and Hydrodynamics
  • Mathematical Logic and Theory of Complexity
  • Geometry and Topology

Invited Speakers

International conference devoted to the hundredth anniversary of A.N. Kolmogorov's birth
(Tambov, May 11-16, 2003 )


The conference is devoted to memory of
outstanding mathematician of modernity
Andrei Nikolaevich Kolmogorov

Third Annual Moscow Kolmogorov Readings - 2003

International Science Students Conference

Kolmogorov Specialized Physics and Mathematics School - Internat #18, Moscow
Lomonosov Moscow State University

May 5-7, 2003, Moscow   e-mail:

Second Annual Moscow Kolmogorov Readings - 2002

First Moscow Kolmogorov Readings - 2001

the abdus salam international centre for theoretical physics

ICTP-INFM Summer School

Transport, Reaction and Propagation in Fluids

8-12 September 2003, ICTP, Trieste, Italy

followed by conference on

Kolmogorov's Legacy in Physics:

One Century of Chaos, Turbulence and Complexity

15-17 September 2003, ICTP, Trieste, Italy

One-Day Workshop in Honor of the
One Hundredth Anniversary of the Birth of
Andrei Nikolaevich Kolmogorov

Complexity, Information, and Randomness:

The Legacy of Andrei Kolmogorov

Sunday, July 6, 2003, University of Aarhus, Denmark

in conjunction with

18th Annual IEEE Conference on Computational Complexity

Monday, July 7th, to Thursday, July 10th, 2003, University of Aarhus, Denmark

Centennial Seminar on Kolmogorov Complexity and Applications


2003, April 27 - May 5, Schloss Dagstuhl, D-66687 Wadern, Saarbrücken, Germany

dell' Istituto Nazionale per la Fisica della Materia (INFM)
e il Dipartimento di Fisica
Universita' degli Studi di Roma La Sapienza

organizzano una Giornata su


La Conferenza si terra' presso l'Edificio Fermi del Dip di Fisica
p.le Aldo Moro 2, 00185, Roma
AULA 1 ore 16 del 9 Maggio 2003

The University of London Inaugural Kolmogorov Lecture


Speaker: Professor Ray Solomonoff

27th February 2003 5:30pm, Main Lecture Theatre Royal Holloway, University of London


Discussion Leader: Israel M. Gelfand

January 21, 2003 4:30pm, Courant Institute of Mathematical Sciences, New York University

Fields Institute Kolmogorov Lecture Series 1998-1999

Chicago Kolmogorov Memorial Readings 2001

Dima Gordeyev. Teacher ( A. N. Kolmogorov ). 100x60 cm. Oil on canvas, 1980. Komarovka, Moscow Region.

A. N. Kolmogorov Bibliography

  • I. General List of the main publications by A. N. Kolmogorov
  • Report to the mathematical circle on covering by squares (1921)

    On operations on sets. II (1922)

    A. Kolmogoroff, Une série de Fourier-Lebesgue divergente presqne partout (1923)

    Sur l'ordre de grandeur des coéfficients de la série de Fourier-Lebesgue (1923)

    A. Kolmogoroff, Une contribution à l'étude de la convergence des séries de Fourier (1924)

    Sur la convergence des séries de Fourier (1924), jointly with G. A. Seliverstov

    La définition axiomatique de l'intégrale (1925)

    Sur le bornes de la généralisation de l'intégrale (1925)

    Sur la possibilité de la définition générale de la dérivée, de l'intégrale et de lasommation des séries divergentes (1925)

    A. Kolmogoroff, Sur les fonctions harmoniques conjuguées et les séries de Fourier (1925)

    On the tertium non datur principle (1925)

    Über Convergenz von Reihen, deren Glieder durch den Zufall bestimmt werden (1925), jointly with A. Ya. Khintchin

    Sur la convergence des séries de Fourier (1926), jointly with G. A. Seliverstov

    Une série de Fourier-Lebesgue divergente partout (1926)

    Sur la loi des grands nombres (1927)

    A. Kolmogoroff et D. Menchoff, Sur la convergence des series de fonctions ortogonales (1927)

    On operations on sets (1928) (in Russian)

    Sur une formule limite de M. A. Khintchine (1928)

    Sur un procédé d'intégration de M. Denjoy (1928)

    A. Kolmogoroff, Über die Summen durch den Zufall bestimmter unabhängiger Größen (1928)

    Bemerkungen zu meiner Arbeit "Über die Summen durch den Zufall bestimmter unabhängiger Größen" (1929)

    General measure theory and the calculus of probabilities (1929) (in Russian)

    Present-day controversies on the nature of mathematics (1929) (in Russian)

    A. Kolmogoroff, Über das Gesetz des iterierten Logarithmus (1929)

    Sur la loi des grands nombres (1929)

    Sur la loi forte des grands nombres (1930)

    A. Kolmogoroff, Zur topologisch- gruppentheoretischen Begründung der Geometrie" (1930)

    A. Kolmogoroff, Untersuchungen über den Integralbegriff (1930)

    Sur la notion de la moyenne (1930)

    A. Kolmogoroff, Bemerkungen zu meiner Arbeit "Über die Summen zufälliger Größen" (1930)

    A. Kolmogoroff, Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung (1931)

    Sur la probléme d'attente (1931)

    The method of medians in the theory of errors (1931) (in Russian)

    Eine Verallgemeinerung des Laplace-Liapunoffschen Satzes (1931)

    A. Kolmogoroff, Über Kompaktheit der Funktionenmengen bei der Konvergenz im Mittel" (1931)

    The theory of functions of a real variable (1932) (in Russian)

    Sulla forma generale di un processo stocastico omogeneo (Un problema di Bruno di Finetti) (1932)

    Ancora sulla forma generale di un processo stocastico omogeneo (1932)

    A. Kolmogoroff, Zur Deutung der intuitionistischen Logik (1932)

    Zur Begründung der projektiven Geometrie (1932)

    Introduction to the theory of functions of a real variable (1932), jointly with P. S. Aleksandrov (in Russian)

    Introduction to the theory of functions of a real variable, 2nd ed. (1933), jointly with P. S. Aleksandrov (in Russian)

    Grundbegriffe der Wahrscheinlichkeitrechnung (1933)

    A. Kolmogoroff, Beiträge zur Maßtheorie (1933)

    Zur Berechung der mittleren Brownschen Fläche (1933), jointly with M. A. Leontovich

    Sulla determinazione empirica di una legge di distribuzione (1933)

    Über die Grenzwertsätze der Wahrscheinlichkeitsrechnung (1933)

    A. Kolmogoroff, Zur Theorie der stetigen zufälligen Prozesse (1933)

    Sur la détermination empirique d'une loi de distribution (1933)

    On the question of suitability of forecast formulas found statistically (1933) (in Russian)

    10 papers in 1934

    4 papers in 1935

    17 papers in 1936

    A. Kolmogoroff, Zur Theorie der Markoffschen Ketten (1936)

    9 papers in 1937

    A. Kolmogoroff, Zur Umkehrkeit der statistischen Naturgesetze (1937)

    16 papers in 1938


    Complete list of Kolmogorov's works is published in the book Kolmogorov in Perspective.

    Most of published scientific papers and monographs of A.N. Kolmogorov are reproduced in a 3 Volume Set - Selected Works of A.N. Kolmogorov.

    See also more than a dozen of A. N. Kolmogorov's articles in Russian popular science journal for students Kvant (Quantum).

    Kolmogorov in Perspective
    Edited by A. N. Shiryaev / Published September 2000 by American Mathematical Society


    The editorial board for the History of Mathematics series has selected for this volume a series of translations from two Russian publications, Kolmogorov in Remembrance and Mathematics and its Historical Development. This book, Kolmogorov in Perspective, includes articles written by Kolmogorov's students and colleagues and his personal accounts of shared experiences and lifelong mathematical friendships. Specifically, the article, "Andrei Nikolaevich Kolmogorov. A Biographical Sketch of His Life and Creative Paths" by A. N. Shiryaev, gives an excellent personal and scientific biography of Kolmogorov. The volume also includes the following articles: "On A. N. Kolmogorov" by V. I. Arnol'd, "In Memory of A. N. Kolmogorov" by S. M. Nikol'skii, "Remembrances of A. N. Kolmogorov" by Ya. G. Sinai, "The Influence of Andrei Nikolaevich Kolmogorov on My Life" by P. L. Ul'yanov, "A Few Words on A. N. Kolmogorov" by P. S. Aleksandrov, "Memories of P. S. Aleksandrov" by A. N. Kolmogorov, "Newton and Contemporary Mathematical Thought" by A. N. Kolmogorov, and an extensive bibliography with the complete list of Kolmogorov's works--including the articles written for encyclopedias and newspapers. The book is illustrated with photographs and includes quotations from Kolmogorov's letters and conversations, uniquely reflecting his mathematical tastes and opinions.

    Copublished with the London Mathematical Society. Members of the LMS may order directly from the AMS at the AMS member price. The LMS is registered with the Charity Commissioners.


    Andrei Nikolaevich Kolmogorov. Biography.
    by V. M. Tikhomirov

    Kolmogorov Remembered
    by Leonid A. Bassalygo, Roland L. Dobrushin, Mark S. Pinsker

    Kolmogorov, Andrey Nikolayevich
    an article from the Encyclopædia Britannica

    Kolmogorov, Andrey Nikolayevich

    Automata and Life (1961)
    by A. N. Kolmogorov
    Prepared for publication and edited by N. G. Khimchenko (Rychkova) (Home Page).


  • N. G. Khimchenko -- How it was...

  • Automata and Life (text)

  • A. N. Kolmogorov -- Automata and Life (theses for his talk)

  • V. M. Tikhomirov -- A few words on topic: "Kolmogorov and cybernetics"

  • V. A. Uspensky -- Kolmogorov, as I remember him

  • A. N. Kolmogorov -- Mathematics - A Science and Profession
    Collected and prepared for publication by G. A. Galperin
    Published by Nauka, Moscow, 1988

    A. N. Kolmogorov -- Mathematics and its Historical Development
    Edited by V. A. Uspensky, collected and prepared for publication by G. A. Galperin
    Published by Nauka, Moscow, 1991

    See also the following Russian publications about A. N. Kolmogorov:

    Curriculum Vitae

    A. N. Kolmogorov family photo Home in Komarovka A. N. Kolmogorov at School A. N. Kolmogorov with his wife Anna Dmitrievna A. N. Kolmogorov Komarovka home in winter
    More photographs of A. N. Kolmogorov at the Kolmogorov School web site and at the Andrey Kolmogorov page of the History of Mathematics Archive.

    Andrei Nikolaevich Kolmogorov at his desk Andrei Nikolaevich Kolmogorov Andrei Nikolaevich Kolmogorov Andrei Nikolaevich Kolmogorov Andrei Nikolaevich Kolmogorov giving a lesson Andrei Nikolaevich Kolmogorov giving a lecture Andrei Nikolaevich Kolmogorov giving a lecture Andrei Nikolaevich Kolmogorov giving a lecture Andrei Nikolaevich Kolmogorov giving a lecture Andrei Nikolaevich Kolmogorov with tutors Andrei Nikolaevich Kolmogorov Andrei Nikolaevich Kolmogorov with young teachers Andrei Nikolaevich Kolmogorov with students at his school Andrei Nikolaevich Kolmogorov Andrei Nikolaevich Kolmogorov Andrei Nikolaevich Kolmogorov Andrei Nikolaevich Kolmogorov Andrei Nikolaevich Kolmogorov Andrei Nikolaevich Kolmogorov Andrei Nikolaevich Kolmogorov Click on any picture to enlarge.

    A. N. Kolmogorov

    Mathematics : Its Content, Methods and Meaning
    Edited by A. D. Aleksandrov, A. N. Kolmogorov, and M. A. Lavrent'ev
    1120 pages, Three Volumes Bound as One
    Published September 1999 by Dover

    This edition reprints in one volume the second edition of this title, which was published in three volumes by The MIT Press in 1969. The original edition was published in 1964, translated from the Russian. Eighteen Russian mathematicians survey the scope of math, from elementary to the advanced levels, writing to educate a lay audience (those with "secondary school mathematics") who are motivated to know more. Discussion includes both the origins and the development of analytic geometry, algebra, ordinary differential equations, partial differential equations, curve and surface theories, prime numbers, probability, functions of a complex variable, linear algebra, non-Euclidean geometry, topology, functional analysis, and groups and other algebraic systems.

    Selected Works of A.N.Kolmogorov : Mathematics and Mechanics, Volume 1
    Edited by V. M. Tikhomirov
    Selected Works of A.N.Kolmogorov : Probability Theory and Mathematical Statistics, Volume 2
    Edited by A. N. Shiryayev
    Selected Works of A.N.Kolmogorov : Information Theory and the Theory of Algorithms, Volume 3
    Edited by A. N. Shiryayev

    These three volumes devoted to the work of one of the most prominent twentieth-century mathematicians. Throughout his mathematical work, A.N. Kolmogorov (1903-1987) showed great creativity and versatility and his wide-ranging studies in many different areas led to the solution of conceptual and fundamental problems and the posing of new, important questions. His lasting contributions embrace probability theory and statistics, the theory of dynamical systems, mathematical logic, geometry and topology, the theory of functions and functional analysis, classical mechanics, the theory of turbulence, and information theory.

    The material appearing in each volume was selected by A.N. Kolmogorov himself and is accompanied by short introductory notes and commentaries which reflect upon the influence of this work on the development of modern mathematics. All papers appear in English -- some for the first time -- and in chronological order. The volume contains a significant legacy which will find many grateful beneficiaries amongst researchers and students of mathematics and mechanics, as well as historians of mathematics.

    Volume I: Mathematics and Mechanics

    This first volume contains papers in mathematics (excluding probability theory and information theory, which are the subject of the following two volumes), turbulence and classical mechanics. They include his famous paper on everywhere-divergent Fourier series, the concluding work on Hilbert's 13th problem, the fundamentals of the Kolmogorov-Arnold-Moser theory in classical mechanics, the fundamentals of the theory of upper homologies, an original construction of the integral, papers on approximation theory and turbulence, and much more.

    Volume II: Probability Theory and Mathematical Statistics

    This second volume contains papers on probability theory and mathematical statistics, and embraces topics such as limit theorems, axiomatics and logical foundations of probability theory, Markov chains and processes, stationary processes and branching processes.

    Volume III: Information Theory and the Theory of Algorithms

    This third volume contains original papers dealing with information theory and the theory of algorithms. Comments on these papers are included.

    Mathematics of the 19th Century: Constructive Function Theory According to Chebyshev, Ordinary Differential Equations, Calculus of Variations, and Theory of Finite Differences
    Edited by A. N. Kolmogorov and A. P. Yushkevich
    Mathematics of the 19th Century: Geometry, Analytic Function Theory
    Edited by A. N. Kolmogorov and A. P. Yushkevich
    Mathematics of the 19th Century: Mathematical Logic, Algebra, Number Theory, Probability Theory
    Edited by A. N. Kolmogorov and A. P. Yushkevich


      Volume I

    • Function Theory According to Chebyshev
      1.2 Functions of Minimal Deviation from Zero
      1.3 Continued Fractions

    • Ordinary Differential Equations
      2.1 Summary of the Development of Ordinary Differetial Equations in the Eighteenth-Century
      2.2 The Problem of Existence and Uniqueness
      2.3 Integration of Equations in Quadratures
      2.4 Linear Differential Equations
      2.5 The Analytic Theory of Differential Equations
      2.6 The Qualitative Theory of Differential Equations

    • The Calculus of Variations
      3.2 Calculus of Variations in the First Half of the Nineteenth Century
      3.3 Calculus of Variations in the Second Half of the Nineteenth Century

    • The Calculus of Finite Differences
      4.1 Interpolation
      4.2 The Euler-Maclaurin Summation Formula
      4.3 Finite-Difference Equations

    • Bibliography
      Index of Names

      Volume II

    • Geometry
      1.1 Analytic and Differential Geometry
      1.2 Projective Geometry
      1.3 Algebraic Geometry and Geometric Algebra
      1.4 Non-Euclidean Geometry
      1.5 Multi-Dimensional Geometry
      1.6 Topology
      1.7 Geometric Transformations

    • Analytic Function

    • Literature
      Index of Names

      Volume III

      Introduction to the English Translation
    • Mathematical Logic

    • Algebra and Algebraic Number Theory

    • Problems of Number Theory

    • The Theory of Probability

    • Addendum by O. B. Sheinin
      Index of Names

    Introductory Real Analysis
    by A. N. Kolmogorov and S. V. Fomin
    Published June 1975 by Dover

    Elements of the Theory of Functions and Functional Analysis
    by A. N. Kolmogorov and S. V. Fomin
    Published March 1999 by Dover

    The Thirteen Books of Euclid's Elements, Books I-II
    The Thirteen Books of Euclid's Elements, Books III-IX
    The Thirteen Books of Euclid's Elements, Books X-XIII
    (Translated with introduction and commentary by Sir Thomas L. Heath)
    Published 1956 by Dover

    What Is Mathematics?: An Elementary Approach to Ideas and Methods
    by Richard Courant and Herbert Robbins, Revised by Ian Stewart
    Published 1996 by Oxford University Press (2nd Edition)
    COURANT, ROBBINS: What Is Mathematics?: An Elementary Approach to Ideas and Methods

    Translated into Russian by A. N. Kolmogorov and with introduction by A. N. Kolmogorov
    COURANT, ROBBINS: What Is Mathematics?: An Elementary Approach to Ideas and Methods

    The Principia : Mathematical Principles of Natural Philosophy
    by Sir Isaac Newton
    (New Translation by I. Bernard Cohen and Anne Whitman)
    Published 1999 by University of California Press

    Foundations of Geometry
    by David Hilbert
    (Translation by Leo Unger)
    Published 1988 by Open Court (2nd Edition)

    Foundations of the Theory of Probability
    by A. N. Kolmogorov
    The translation is edited by Nathan Morrison
    Originally published 1933 in German as "Grundbegriffe der Wahrscheinlichkeitrechnung"
    English translation published 1950 by Chelsea Publishing
    Most recent 3rd Russian Edition was published 1998 by Phasis, Moscow

    Full text of the 1st Russian Edition (1936) is available at web page maintained by Vladimir Vovk


    • Elementary Theory of Probability:
      1.1 Axioms
      1.2 The relation to experimental data
      1.3 Notes on terminology
      1.4 Immediate corollaries of the axioms; conditional probabilities; Theorem of Bayes
      1.5 Independence
      1.6 Conditional probabilities as random variables; Markov chains

    • Infinite Probability Fields:
      2.1 Axiom of continuity
      2.2 Borel fields of probability
      2.3 Examples of infinite fields of probability

    • Random Variables:
      3.1 Probability functions
      3.2 Definition of random variables and of distribution functions
      3.3 Multi-dimensional distribution functions
      3.4 Probabilities in infinite-dimensional spaces
      3.5 Equivalent random variables; various kinds of convergence

    • Mathematical Expectations:
      4.1 Abstract Lebesgue integrals
      4.2 Absolute and conditional mathematical expectations
      4.3 The Tchebycheff inequality
      4.4 Some criteria for convergence
      4.5 Differentiation and integration of mathematical expectations with respect to a parameter

    • Conditional Probabilities and Mathematical Expectations:
      5.1 Conditional probabilities
      5.2 Explanation of a Borel paradox
      5.3 Conditional probabilities with respect to a random variable
      5.4 Conditional mathematical expectations

    • Independence; The Law of Large Numbers:
      6.1 Independence
      6.2 Independent random variables
      6.3 The law of large numbers
      6.4 Notes on the concept of mathematical expectation
      6.5 The strong law of large numbers; Convergence of a series

    • Appendix--Zero-or-one law in the theory of probability

    • Bibliography

    • Notes to supplementary bibliography

    • Supplementary bibliography

    Limit Distributions for Sums of Independent Variables
    by B. V. Gnedenko and A. N. Kolmogorov
    Translated from the Russian, annotated, and revised by K. L. Chung
    With Appendices by J. L. Doob and P. L. Hsu
    Published by Addison-Wesley, 1954, 1968 (Second Edition)

    The Kolmogorov Legacy in Physics: A Century of Turbulence and Complexity (Lecture Notes in Physics, 642)
    Edited by Roberto Livi and Angelo Vulpiani
    Hardcover: 246 pages; Publisher: Springer Verlag; (February 2004) ISBN: 3540203079
    English translation from the French edition L'héritage de Kolmogorov en physique published September 2003 by Belin, Paris
    The Kolmogorov Legacy in Physics: A Century of Turbulence and Complexity (Lecture Notes in Physics, 642)
    Table of Contents

    Kolmogorov Pathways from Integrability to Chaos and Beyond 3
    From Regular to Chaotic Motions through the Work of Kolmogorov 33
    Dynamics at the Border of Chaos and Order 61
    Kolmogorov's Legacy about Entropy, Chaos, and Complexity 85
    Complexity and Intelligence 109
    Information Complexity and Biology 123
    Fully Developed Turbulence 149
    Turbulence and Stochastic Processes 173
    Reaction-Diffusion Systems: Front Propagation and Spatial Structures 187
    Self-Similar Random Fields: From Kolmogorov to Renormalization Group 213
    Financial Time Series: From Batchelier's Random Walks to Multifractal 'Cascades' 229

    L'héritage de Kolmogorov en physique
    Edited by Roberto Livi and Angelo Vulpiani
    Published September 2003 by Belin, Paris
    L'héritage de Kolmogorov en physique
    Table des matières (Contents)

    Préface Yakov G. Sinai
    Introduction Roberto Livi et Angelo Vulpiani

      Première partie : CHAOS ET SYSTÈMES DYNAMIQUES

      Roberto Livi, Stefano Ruffo, Dima Shepelyansky
      1 Une perspective générale
      2 Deux degrés de liberté : l'application standard de Chirikov
      3 De nombreux degrés de liberté : l'expérience numérique de Fermi, Pasta et Ulam
      4 Seuils énergétiques
      5 Spectres de Lyapounov et caractérisation de la dynamique chaotique
      6 Ordinateurs quantiques et chaos quantique
      Alessandra Celletti, Claude Froeschlé, Elena Lega
      1 Introduction
      2 Mouvements stables
      2.1 Systèmes intégrables et non intégrables
      2.2 Théorie des perturbations
      2.3 Le théorème de Kolmogorov-Arnold-Moser
      2.4 La stabilité d'un modèle associé au problème des trois corps
      3 Mouvements instables
      3.1 Théorème de Nekhoroshev
      3.2 Outils pour différencier le chaos de l'ordre
      3.3 Représentation du réseau d'Arnold dans un modèle hamiltonien simple
      Arkady Pikovsky et Michael Zaks
      1 Introduction
      2 Suite de Thue-Morse : un exemple non trivial de séquence symbolique complexe
      3 Attracteurs à spectre fractal : du codage symbolique aux singularités des temps de retour
      4 Les spectres fractals en hydrodynamique laminaire
      5 Conclusion


      Massimo Falcioni, Vittorio Loreto, Angelo Vulpiani
      1 L'entropie en thermodynamique et en physique statistique
      2 L'entropie dans la théorie de l'information
      3 L'entropie dans les systèmes dynamiques
      4 Complexité algorithmique
      5 Complexité et information en linguistique, génomique et finances
      5.1 Du jeu en bourse à l'estimation de l'entropie
      5.2 Recherche d'informations pertinentes
      5.3 Entropie relative et écart entre séquences
      5.4 Compression de données et mesures de complexité
      Giorgio Parisi
      1 Complexité algorithmique
      2 Quelques propriétés et paradoxes apparents de la complexité
      3 La profondeur logique
      4 Apprentissage par l'exemple
      5 Apprentissage, généralisation et propensions
      6 Une approche statistique des propensions
      7 Une définition possible de l'intelligence
      Franco Bagnoli, Franco Bignone, Fabio Cecconi, Antonio Politi
      1 Notes historiques
      2 Les contributions directes de Kolmogorov
      3 Information et biologie
      4 Les protéines : un exemple paradigmatique de complexité

      Troisième partie : TURBULENCE

      Luca Biferale, Guido Beffetta, Bernard Castaing
      1 Introduction
      2 Théorie de Kolmogorov 1941
      2.1 Symétries de Navier-Stokes
      2.2 Anomalie dissipative
      2.3 Loi des 4/5 et auto-similarité
      3 Théorie de Kolmogorov 1962
      3.1 Intermittence et loi d'échelle anomale
      3.2 Cascade multiplicative
      3.3 Approche multifractale
      3.4 Tests sur les hypothèses de Kolmogorov
      4 L'héritage de Kolmogorov sur la turbulence moderne
      4.1 Universalité des fluctuations aux petites échelles
      4.2 Turbulence anisotrope
      Antonio Celani, Andrea Mazzino, Alain Pumir
      1 Introduction
      2 Turbulence d'un scalaire passif
      3 Le modèle de Kraichnan et ses prolongements
      4 Du côté de la turbulence de Navier-Stokes
      5 Conclusion
      Massimo Cencini, Cristobal Lopez, Davide Vergni
      1 Introduction
      2 Propagation de front dans l'équation de diffusion non linéaire
      3 Systèmes de réaction-diffusion en physique, en chimie et en biologie
      3.1 Systèmes de réaction-diffusion à multi composants
      3.2 Systèmes d'advection-réaction-diffusion


      Giovanni Jona-Lasinio
      1 Introduction
      2 Bref historique
      3 La spirale de Wiener, et les processus apparentés
      4 Le groupe de renormalisation : idées générales
      5 Le groupe de renormalisation : un point de vue probabiliste
      6 Une propriété des champs aléatoires autosimilaires critiques
      7 Structure multiplicative
      8 Théorèmes limites et universalité des phénomènes critiques
      9 Conclusion
      Jean-Philippe Bouchaud et Jean-François Muzy
      1 Introduction
      2 Caractéristiques universelles des séries temporelles des rendements
      3 Des lois d'échelle multifractales aux processus en cascade
      3.1 Comportement multi-échelle des rendements d'actifs
      3.2 Le paradigme de la cascade
      3.3 L'héritage de Kolmogorov, turbulence et finance
      4 Marche aléatoire multifractale
      5 Conclusion

    TURBULENCE: The Legacy of A. N. Kolmogorov
    by Uriel Frisch
    Published 1994 by Cambridge University Press

    TURBULENCE: The Legacy of A. N. Kolmogorov

    This textbook presents a modern account of turbulence, one of the greatest challenges in physics. The state-of-the-art is put into historical perspective five centuries after the first studies of Leonardo and half a century after the first attempt by A.N. Kolmogorov to predict the properties of flow at very high Reynolds numbers. Such "fully developed turbulence" is ubiquitous in both cosmical and natural environments, in engineering applications and in everyday life. First, a qualitative introduction is given to bring out the need for a probabilistic description of what is in essence a deterministic system. Kolmogorov's 1941 theory is presented in a novel fashion with emphasis on symmetries (including scaling transformations) which are broken by the mechanisms producing the turbulence and restored by the chaotic character of the cascade to small scales. Considerable material is devoted to intermittency, the clumpiness of small-scale activity, which has led to the development of fractal and multifractal models. Such models, pioneered by B. Mandelbrot, have applications in numerous fields besides turbulence (diffusion limited aggregation, solid-earth geophysics, attractors of dynamical systems, etc). The final chapter contains an introduction to analytic theories of the sort pioneered by R. Kraichnan, to the modern theory of eddy transport and renormalization and to recent developments in the statistical theory of two-dimensional turbulence. The book concludes with a guide to further reading. The intended readership for the book ranges from first-year graduate students in mathematics, physics, astrophysics, geosciences and engineering, to professional scientists and engineers.

      Table of Contents

    • Preface
    • Introduction
    • Why a probabilistic description of turbulence?
    • Probabilistic tools: a survey
    • Two experimental laws of fully developed turbulence
    • The Kolmogorov 1941 theory
    • Phenomenology of turbulence in the sense of Kolmogorov 1941
    • Intermittency
    • Further reading: a guided tour
    • References
    • Author index
    • Subject index
    (See also Kolmogorov's turbulence definitions from his famous K41 paper)

    K41 A. N. Kolmogorov. Dokl. Akad. Nauk SSSR, 30;4:3201, 1941.

    An English translation of this paper was recently republished (Translation by V. Levin):

    A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Proc. R. Soc. Lond. A, 434:9-13, 1991

    in the book Turbulence and Stochastic Processes: Kolmogorov's Ideas 50 Years On (1991)

    and in the book Selected Papers on Adaptive Optics and Speckle Imaging (1994)

    An Introduction to Kolmogorov Complexity and Its Applications
    by Ming Li, Paul Vitanyi (Home page)
    Published January 1997 by Springer-Verlag New York (2nd Edition)
    An Introduction to Kolmogorov Complexity and Its Applications
    See also Kolmogorov Complexity and Solomonoff Induction Mailing List
    and Special Issue on Kolmogorov Complexity,
    The Computer Journal, Volume 42, Issue 4, 1999.
    Edited by Alexander Gammerman (Home page), and Vladimir Vovk (Home page).
  • Kolmogorov Complexity: Sources, Theory and Applications
  • Generalized Kolmogorov Complexity Generalized Kolmogorov Complexity
    by Jürgen Schmidhuber (Home page)

    Hilbert's 10th Problem (Foundations of Computing)
    by Yuri V. Matiyasevich (Home page)
    Published October 1993 by MIT Press
    Hilbert's 10th Problem (Foundations of Computing)
    See also Hilbert's Tenth Problem page

    The Honor's Class: Hilbert's Problems and Their Solvers
    by Benjamin Yandell
    Published December 2001 by A K Peters, Ltd.
    The Honor's Class: Hilbert's Problems and Their Solvers
    See also an original Hilbert's address in German
    Mathematische Probleme
    (Vortrag, gehalten auf dem internationalen Mathematiker-Kongreß zu Paris 1900 ) Von David Hilbert

    Mathematical Problems (in English)
    (Lecture delivered before the International Congress of Mathematicians at Paris in 1900)
    by Professor David Hilbert

    Hilbert's Problems (in Russian) edited and with introduction by P. S. Aleksandrov
    ALEKSANDROV: Hilbert's Problems

    See also Millenium Prize Problems by Clay Mathematics Institute

    The Millenium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time
    by Keith J. Devlin
    Published October 2002 by Basic Books
    DEVLIN: The Millenium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time

    Russian Mathematicians in the 20th Century
    Edited by Yakov Sinai
    Published October 2003 by World Scientific


    In the 20th century, many mathematicians in Russia made great contributions to the field of mathematics. This invaluable book, which presents the main achievements of Russian mathematicians in that century, is the first most comprehensive book on Russian mathematicians. It has been produced as a gesture of respect and appreciation for those mathematicians and it will serve as a good reference and an inspiration for future mathematicians. It presents differences in mathematical styles and focuses on Soviet mathematicians who often discussed "what to do" rather than "how to do it". Thus, the book will be valued beyond historical documentation.

    The editor, Professor Yakov Sinai, a distinguished Russian mathematician, has taken pains to select leading Russian mathematicians — such as Lyapunov, Luzin, Egorov, Kolmogorov, Pontryagin, Vinogradov, Sobolev, Petrovski and Krein — and their most important works. One can, for example, find works of Lyapunov, which parallel those of Poincaré; and works of Luzin, whose analysis plays a very important role in the history of Russian mathematics; Kolmogorov has established the foundations of probability based on analysis. The editor has tried to provide some parity and, at the same time, included papers that are of interest even today.

    The original works of the great mathematicians will prove to be enjoyable to readers and useful to the many researchers who are preserving the interest in how mathematics was done in the former Soviet Union.--

    Golden Years of Moscow Mathematics
    Edited by Smilka Zdravkovska and Peter L. Duren
    Published January 1994 by American Mathematical Society


    This volume contains articles on Soviet mathematical history, many of which are personal accounts by mathematicians who witnessed and contributed to the turbulent years of Moscow mathematics. In today's climate of glasnost, the stories can be told with a candor uncharacteristic of the "historical" accounts published under the Soviet regime. An important case in point is the article on Luzin and his school, based in part on documents only recently released. The articles focus on mathematical developments in that era, the personal lives of Russian mathematicians, and political events that shaped the course of scientific work in the Soviet Union. Another important feature is the inclusion of two articles on Kolmogorov, perhaps the greatest Russian mathematician of the twentieth century. The volume concludes with an annotated English bibliography and a Russian bibliography for further reading. This book appeals to mathematicians, historians, and anyone else interested in Soviet mathematical history.

    • A. P. Yushkevich -- Encounters with mathematicians
    • S. S. Demidov -- The Moscow school of the theory of functions in the 1930s
    • E. M. Landis -- About mathematics at Moscow State University in the late 1940s and early 1950s
    • B. A. Rosenfeld -- Reminiscences of Soviet mathematicians
    • V. M. Tikhomirov -- A. N. Kolmogorov (see also his Biography)
    • V. I. Arnol'd -- On A. N. Kolmogorov (see also An Interview with Vladimir Arnol'd)
    • M. M. Postnikov -- Pages of a mathematical autobiography (1942-1953)
    • B. A. Kushner -- Markov and Bishop: An essay in memory of A. A. Markov (1903-1979) and E. Bishop (1928-1983)
    • I. Piatetski-Shapiro -- Étude on life and automorphic forms in the Soviet Union
    • D. B. Fuchs -- On Soviet mathematics of the 1950s and 1960s
    • A. B. Sossinsky -- In the other direction
    • S. S. Demidov -- A brief survey of the literature on the development of mathematics in the USSR
    • S. S. Demidov -- Bibliography (Russian)

    Quantum : The Magazine of Math and Science
    A. N. Kolmogorov co-founded in 1970 and was First Deputy Editor-in-Chief of the Russian popular scientific journal for students 'Kvant'.
    Current Editor-in-Chief is Yu. A. Osipian

    English translation Quantum was published from 1990 to 2001 by the National Science Teachers Association (NSTA). It was a lively, handsomely illustrated bimonthly magazine of math and science (primarily physics).

    Full texts of many issues of 'Kvant' is currently avalable from the Russian web site of MCCME, Moscow Center for Continuous Mathematical Education. Moscow Center for Continuous Mathematical Education

    See also "Math in Moscow" - a program in English for undergraduates and graduate students at the Independent University of Moscow.

    Full texts of A. N. Kolmogorov's articles in Russian journal Kvant (in Russian):

    Algebra and Elements of Analysis (A High School Textbook for 10-11 grades)
    Edited by A. N. Kolmogorov, A. M. Abramov, Yu. P. Dudnitsyn, B. M. Ivlev, S. I. Shvatsburg
    Published by Prosveshchenie, 2001 (11th Edition)
    Algebra See also Algebra I. M. Gel'fand and A. Shen': Algebra by I. M. Gelfand and A. Shen

    Full text (in Russian) of the article by G. V. Pukhova A. N. Kolmogorov and Summer School at Lake Rubskoye at web site of the Math Department, Ivanovo State University.
    See also full text (in Russian) An Inroduction to the book 'A Summer School at Lake Rubskoye', Published 1971 by A. N. Kolmogorov, I. G. Zhurbenko, G. V. Pukhova, O. S. Smirnova, S. V. Smirnov.

    Theory of Probability and Its Applications
    A. N. Kolmogorov founded in 1956 and was Editor-in-Chief of the Russian journal 'Teoriya Veroyatnostei i ee Primeneniya'.
    Current Editor-in-Chief is Yu. V. Prokhorov who is Kolmogorov's student.

    Theory of Probability and Its Applications is a translation of the Russian journal Teoriya Veroyatnostei i ee Primeneniya, which contains papers on the theory and application of probability, statistics, and stochastic processes.

    Russian Mathematical Surveys
    A. N. Kolmogorov was a founding member of the editorial board of the Russian journal 'Uspekhi Matematicheskikh Nauk' from 1934 till his death in 1987.
    He was Editor-in-Chief from 1946 till 1954 and from 1982 to 1987.
    Current Editor-in-Chief is S. P. Novikov

    Russian Mathematical Surveys is the English translation of the Russian bimonthly journal Uspekhi Matematicheskikh Nauk, founded in 1936. The English language version is a cover-to-cover translation of all the material: that is, the survey articles, the Communications of the Moscow Mathematical Society, and the biographical material.

    Portraits of A. N. Kolmogorov by his former student Dima Gordeev (click to enlarge)

    Kolmogorov 1 2 3 . . . . . . . . . . . . . . .
    State of the Art

    Internat #18 alumni web site, alumni club, and questionnaire.

    Kolmogorov Specialized Physics & Mathematics School - Internat #18 at Moscow State University

    Internat18 - Kolmogorov School
    School was founded by A. N. Kolmogorov in 1964. A. N. Kolmogorov was life-long Chairman of the Board of Trustees. The School was named after A. N. Kolmogorov in 1988.

    Lomonosov Moscow State University
    A. N. Kolmogorov was student (1920-1925), graduate student (1925-1929), researcher (1929-1931), and professor (1931-1987) at the University.

    MechMath - Faculty of Mechanics and Mathematics
    A. N. Kolmogorov was Dean of the MechMath Faculty (1954-1958) and Head of the Mathematics Division (1954-1956 and 1978-till his death in 1987).
    From 1931 A. N. Kolmogorov was Head of graduate school of MechMath Faculty - Director of the Institute of Mathematics and Mechanics.
    In 1951 he was once again appointed Director of the Institute of Mathematics.

    Department of Probability Theory
    A. N. Kolmogorov was Founder (1935) and first Head of the Department (1935-1966).
    From 1966 to 1995 head of the chair was his student B. V. Gnedenko.
    From 1996 another student of A. N. Kolmogorov professor A. N. Shiryaev became head of the chair.

    Department of Mathematical Logic and Theory of Algorithms
    A. N. Kolmogorov was second Head of the Department of Mathematical Logic (1980-1987) after first Head A. A. Markov (1959-1979).
    From 1988 to 1993 head of the chair was V. A. Mel'nikov.
    Head of the Department since January 1995 is professor V.A.Uspensky who is Kolmogorov's student.

    Russian Academy of Sciences
    A. N. Kolmogorov was elected Full Member (Academician) of the Division of Mathematical and Natural Sciences since 29.01.1939.
    He was elected Member of the Presidium of Academy and Head (Academician-Secretary) of the Division of Physical and Mathematical Sciences.

    Steklov Mathematical Institute of the Russian Academy of Sciences

    Department of Probability and Mathematical Statistics of the Steklov Mathematical Institute
    Founded by A. N. Kolmogorov in 1938 who was Head of Department until 1958 (excluding 1946-1948 when A. Ya. Khinchin occupied this position). After 1960 it is headed by Yu. V. Prokhorov who is Kolmogorov's student.

    A. N. Kolmogorov was Head of the Turbulence Laboratory (1946-1949) at the O. Yu. Shmidt Institute of Theoretical Geophysics of the Russian Academy of Sciences. His student A. M Obukhov became Head of the Turbulence Laboratory in 1949 and later Founding and life-long Director of the Institute of Atmospheric Physics of the Russian Academy of Sciences. His other student A. S. Monin became Director of the Institute of Oceanology of the Russian Academy of Sciences.

    A. N. Kolmogorov was President of the Moscow Mathematical Society from 1964 to 1966 and from 1973 to 1985. P. S. Aleksandrov was President of the Moscow Mathematical Society from 1932 to 1964. Current President of the Moscow Mathematical Society is V. I. Arnol'd who is Kolmogorov's student.

    Small Hall of the Conservatory of Moscow Academic Philarmonia
    P. S. Aleksandrov and A. N. Kolmogorov were the life-long Season Tickets holders...
    Johann Sebastian Bach: Concerto for 2 violins & strings in D minor ("Double"), BWV 1043 (P. S. Aleksandrov's and A. N. Kolmogorov's favorite)
    Wolfgang Amadeus Mozart: Symphony No.40 in G minor, K.550 (P. S. Aleksandrov's and A. N. Kolmogorov's favorite)
    Johann Sebastian Bach: St. Matthew Passion
    Christoph Willibald Gluck: Orfeo ed Euridice

    1. Gift of 59 books from the Class of 1976, 1 September 2002.
    2. Gift of 9 books from the Class of 1976, 7 March 2003.
    3. Gift of ~80 books from the Class of 1976, 11 June 2003.
    4. Gift of ~35 books from the Class of 1976, 6 December 2003.
    Some recent additions - Gifts to the Kolmogorov Library. 5, 6, 7, ... Thanks for your support.

    Send your comments and suggestions to
    Kolmogorov Library Project

    Kolmogorov Library Project

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