Kolmogorov's Heritage in Mathematics

A.N. 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 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. Each chapter treats one of Kolmogorov's research themes, or a subject that was invented as a consequence of his discoveries. The authors present here his contributions, 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 modern prospects. 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 generally by anyone who likes mathematical ideas. Rather than presenting detailed proofs, the main ideas are described, and a bibliography for those who wish to understand the technical details.

Table of Contents:
The youth of Andrei Nikolaevich and Fourier series.- Kolmogorov’s contribution to intuitionistic logic.- Some aspects of the probabilistic work.- Infinite dimensional Kolmogorov equations.- From Kolmogorov’s theorem on empirical distribution to number theory.- Kolmogorov’s ?-entropy and the problem of statistical estimation.- Kolmogorov and topology.- Geometry and approximation theory in A. N. Kolmogorov’s works.- Kolmogorov and population dynamics.- Resonances and small divisors.- The KAM Theorem.- From Kolmogorov’s Work on entropy of dynamical systems to Non-uniformly hyperbolic dynamics.- From Hilbert’s 13th Problem to the theory of neural networks: constructive aspects of Kolmogorov’s Superposition Theorem.- Kolmogorov Complexity.- Algorithmic Chaos and the Incompressibility Method.

Review :
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)