Kolmogorov's Heritage in Mathematics

ISBN-13: 9783540363491 / Angielski / Twarda / 2007 / 328 str.

Eric Charpentier; Nikolai Nikolski; Annick Lesne

Opis

In this book, several world experts present (one part of) the mathematical heritage of Kolmogorov. Each chapter treats one of his research themes or a subject invented as a consequence of his discoveries. The authors present his contributions, his methods, the perspectives he opened to us, and the way in which this research has evolved up to now. Coverage also includes examples of recent applications and a presentation of the modern prospects.

Recenzje

From the reviews:

"Andrei N. Kolmogorov (1903-1987) was one of the most prolific and influential mathematicians of the twentieth century. ... this collection would be read by those who hold only a bachelor's degree in mathematics, computer science or physics, and more generally, by those interested in mathematical ideas. ... This impressive collection gives the reader an understanding of the depth and breadth of Kolmogorov's contributions to mathematics. Each chapter includes a substantial list of references that invite further reading." (Elena Anne Marchisotto, MathDL, December, 2007)

"The object of Kolmogorov's Heritage in Mathematics is to enable the general mathematical reader (educated to master's level) to grasp something of Kolmogorov's achievements. ... I can judge, it is successful in its aim. ... those mathematicians with a sense of history, who wish to place themselves within the general culture of mathematics, will find this book a worthy memorial to a great man." (T. W. Korner, SIAM reviews, Vol. 50 (4), December, 2008)

Spis treści

Introduction: Eric Charpentier, Annick Lesne, Nikolaï Nikolski .- The youth of Andrei Nikolaevich and Fourier series: Jean-Pierre Kahane .- Kolmogorov's contribution to intuitionistic logic: Thierry Coquand.- Some aspects of the probabilistic work: Loïc Chaumont, Laurent Mazliak, Marc Yor.- Infinite dimensional Kolmogorov equations: Giuseppe Da Prato.- From Kolmogorov's theorem on empirical distribution to number theory: Kevin Ford.- Kolmogorov's -entropy and the problem of statistical estimation: Mikhail Nikouline, Valentin Solev.- Kolmogorov and topology: Victor M. Buchstaber .- Geometry and approximation theory in A. N. Kolmogorov's works: Vladimir M. Tikhomirov.- Kolmogorov and population dynamics: Karl Sigmund.- Resonances and small divisors: Etienne Ghys.- The KAM Theorem: John H. Hubbard .-From Kolmogorov's Work on Entropy of Dynamical Systems to Non-uniformly Hyperbolic Dynamics: Denis V. Kosygin, Yakov G. Sinai.- From Hilbert's 13th Problem to the theory of neural networks: constructive aspects of Kolmogorov's Superposition Theorem: Vasco Brattka .- Kolmogorov Complexity: Bruno Durand, Alexander Zvonkin.- Algorithmic Chaos and the Incompressibility Method: Paul Vitanyi.

Blurb

A.N. Kolmogorov (b. Tambov 1903, d. 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 new point of view: in a few magnificent pages, which are models of shrewdness and imagination, and which astounded his contemporaries, he changed drastically the landscape of the subject.

Most mathematicians prove what they can, Kolmogorov was of those who prove what they want. For this book several world experts were asked to present one part of the mathematical heritage left to us by Kolmogorov.

Each chapter treats one of Kolmogorov's research themes, or a subject that was invented as a consequence of his discoveries. His contributions are presented, his methods, 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 current prospects.

This book can be read by anyone with a master's (even a bachelor's) degree in mathematics, computer science or physics, or more generally by anyone who likes mathematical ideas. Rather than present detailed proofs, the main ideas are described. A bibliography is provided for those who wish to understand the technical details.

One can see that sometimes very simple reasoning (with the right interpretation and tools) can lead in a few lines to very substantial results.

The Kolmogorov Legacy in Physics was published by Springer in 2004 (ISBN 978-3-540-20307-0).