Andrei N. Kolmogorov (1903-1987) was one of the most prolific and influential mathematicians of the twentieth century. His contributions span several branches of pure and applied mathematics. His publications number about 500. The book under review, which is a translation of the original French edition (*L’héritage de Kolmogorov en mathématiques.* Éditions Belin, 2004.) attempts to illustrate how he dramatically changed the landscapes of the subjects he investigated. The editors express a hope that 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 reviewer would caution that at least a basic knowledge of the theory of functions and functional analysis, probability theory and mathematical statistics is required for many chapters.

In [1991], V. M. Tikhomirov had partitioned Kolmogorov’s contributions into three areas: order (mathematics and mechanics), chaos (probability theory and statistics), and information and algorithmic theories, where the latter two realms had no natural border. The book under review amplifies these areas with each chapter treating one of Kolmogorov’s research themes or a subject that was invented as a consequence of his discoveries. Twenty experts present his contributions, his methods, the perspectives he introduced, and the evolution of his research, in conjunction with examples of recent applications and its modern prospects. In most cases, they provide an overview of his major ideas (rather than detailed proofs). The editors refer the reader to Kolmogorov in perspective for a (more or less) complete bibliography of Kolmogorov’s publications.

The first chapter of the book under review by Jean-Pierre Kahane traces Kolmogorov’s early interest in Fourier series and his four publications that emanated from it. The author describes Kolmogorov’s early results (between 1922 and1924) which include a proof of the existence of a function whose Fourier series diverges everywhere, theorems on harmonic conjugation interpreted in terms of Fourier series, conditions for everywhere convergence of trigonometric, orthogonal and lacunary Fourier series, and discoveries about the order of magnitude of Fourier coefficients. Kahane not only places these youthful contributions of Kolmogorov in their historical context. He explains their significance in today’s world.

Kolmogorov’s contributions to intuitionistic logic are described by Thierry Coquand in chapter 2. He discusses Kolmogorov’s first paper [1925] in proof theory, published in Russian, describing the context in which it appeared. In this paper Kolmogorov defined and proved the soundness of an embedding of the classical logical system (the propositional calculus) into an intuitionistic system. Coquand analyzes this result noting its refinements, extensions and generalizations, and providing modern appraisals of its significance and limitations. Kolmogorov’s only other paper relating to mathematical logic was written in 1932. There he developed the calculus of problems as an interpretation of intuitionistic logic as formalized by Heyting. Coquand discusses how the late twentieth century redefinition of the calculus of problems in type theory supplies Kolmogorov’s calculus with an explicit notation for the solution of problems, bringing his pioneering work into a modern perspective.

Chapters 3 and 4 focus on probability. Kolmogorov’s classic *Grundbegriff* [1933] has been credited with inaugurating the modern era of probability theory. In it, Kolmogorov completed the solution of Hilbert’s sixth problem on the axiomatization of probability. He utilized measure theory, regarding a probability measure as a positive measure of mass one, to reformulate probability theory. This measure-theoretic approach to the subject has become standard today.

Loïc Chaumont, Laurent Mazliak, and Marc Yor believe, however, that this 1933 work does not constitute the most original creation of Kolmogorov in the field of probability. In chapter 3 they explore two remarkable contributions of Kolmogorov’s probabilistic work: his study of the various types of convergence for sums of independent random variables, and his revolutionary ideas about processes in continuous time. They discuss a joint paper [1925] of Kolmogorov and A. N. Khinchin which introduces techniques that form the basis for later developments of the theory of probability, notably in study of results of convergence for martingales.

The authors call Kolmogorov’s 1929 generalization of Khinchin’s 1924 law of the iterated logarithm one of Kolmogorov’s greatest achievements. Indeed, Khinchin’s own greatest achievement may be the law of the iterated logarithm, joining the law of large numbers and the central limit theorem as the classical limit theorems in probability theory. Kolmogorov would write about a dozen papers on probability between his first work with Khinchin and his *Grundbegriffe*. The last section of this chapter examines Kolmogorov’s study of Markov processes and his construction of the differential equations that the transition probability densities satisfy. The authors discuss one of the tools that Kolmogorov developed for the study of the pathwise properties of random processes, an effective criterion which guarantees pathwise continuity, and illustrate modern generalizations of this criterion.

Chapter 4 continues an investigation of Kolmogorov’s work in probability theory, focusing on infinite-dimensional Kolmogorov equations. Giuseppe Da Prato begins by summarizing Kolmogorov’s derivation of some important partial differential equations for transition probabilities from the Chapman-Kolmogorov equation, which is fundamental in the analysis of Markov processes. The author discusses the infinite-dimensional generalizations of these equations, seeking to demonstrate the “state of the art” in the domain of infinite-dimensional Hilbert spaces. Different methods for solving these equations are presented, and Da Prato describes active areas of research resulting from their applications to some relevant partial differential equations.

Statistics is the topic treated in Chapters 5 and 6. In the first of these, Kevin Ford takes the reader from Kolmogorov’s theorem on empirical distribution to number theory. He describes some new estimates for the probability that an empirical distribution function of uniform-[0,1] random variables stays on one side of a given line. Noting that Hardy and Ramanujan initiated the study of the statistical distribution of the prime factors of integers in 1917, the author discusses the applications of empirical distribution functions to number theory.

In chapter 6, Mikhail Nikouline and Valentin Solev discuss Kolmogorov’s epsilon-entropy, seeking to show the influence of Kolmogorov’s ideas on the development of statistical estimation. In 1933 Kolmogorov [1933a] determined the limit distribution of the normed deviation of the empirical distribution function F_{N}(x) from a continuous theoretic one F(x). Reversing the problem, so that the theoretical distribution is unknown and the empirical one is determined from the data, leads to the *Kolmogorov test*, which is used today in nonparametric statistics. The authors retrace the discovery of this goodness-of-fit test and describe later improvements that have interesting applications to number theory. They discuss Kolmogorov’s 1950s investigations of information theory and its relation with complexity theory, the theory of functions and statistical estimation and explain his notion of epsilon-entropy which has become a fundamental tool in non-parametric statistics for measuring the quality of estimators. They show how entropy is used to estimate a density, and give the reader a deeper understanding of how Kolmogorov’s ideas have inspired the results of several statisticians over the last fifty years in the area of non-parametric statistics on this problem.

Chapter 7 focuses on topology. Victor M. Buchstaber explains the remarkable results that Kolmogorov found between 1934 and 1937, in just thirty pages of his few papers on the subject. The author discusses the source of some of the ideas leading to Kolmogorov’s results. A prelude to the chapter introduces the founders of the Soviet school of topology, P. S. Urysohn and P.S. Aleksandrov, and describes their influence on their student Kolmogorov. Buchstaber explains why, at the first international topology conference held at the Institute of Mathematics of Moscow University in September 1935, the lectures of Kolmogorov and J. W. Alexander on the construction of dual complexes for quite general spaces attracted widespread attention. The author places the results obtained by Kolmogorov and Alexander in historical context and describes their impact on subsequent developments in algebraic geometry. He also provides a list of fundamental notions due to Kolmogorov who had introduced them to solve some important problems of general topology. These notions continue to enjoy new applications today.

Chapters 8 and 13 both include discussion of Hilbert’s 13^{th} problem and Kolmogorov’s role in its solution. In the first of these chapters, Vladimir M. Tikhomirov paints a broad picture of that topic, book-ended by geometry and approximation theory. The author examines six of Kolmogorov’s papers in approximation theory, and provides a list of mathematicians whose research was strongly influenced by Kolmogorov, with references to where the reader can learn more about their results. This discussion is preceded by an exploration of Kolmogorov’s ideas on geometry.

Throughout his lifetime, Kolomogorov distinguished three components of mathematical talent: algorithmic, logical, and geometric. He believed that geometric intuition plays a great role in almost all areas of mathematics, and counted this intuition among his own mathematical talents. The last years of his life were dedicated to high school mathematics education, where he sought to reorganize the school course in geometry building on geometric intuition.

Tikhomirov gives evidence of Kolomogorov’s geometric abilities, starting with a discussion of his two geometric papers. The first (1930) studies spaces of constant curvature, and the second (1932a) explores projective space, both from an axiomatic point of view. Next, the author explore what he calls “geometric motives” in several “non-geometric” papers by Kolmogorov, including those devoted to the notion of measure and topological vector space. He begins the story with the 13^{th} Hilbert problem that focuses on one of the central questions of analysis regarding functions many variables. Kolmogorov would ultimately prove that any continuous function of several variables may be represented by means of superposition of continuous functions of one variable and addition. This is known as *Kolmogorov’s Superposition theorem*. Vasco Brattka analyzes this theorem from the point of view of computational analysis in Chapter 13. He traces the impact of the theorem, from its refutation of Hilbert’s conjecture about the impossibility of the solution of the general 7^{th} degree equation to research devoted to questions of smoothness that initially followed from it, to its application to neural networks.

Mathematical ecology is the subject of chapter 9. It addresses Kolmogorov’s contributions to the deterministic theory of population dynamics in brief notes of 1936 and 1972 emanating from the predator-pray equation. Karl Sigmund describes the famous model of Vito Volterra that explains a surprising discovery about the number of predators and prey in the Adriatic after World War I. Volterra used an ordinary differential equation to show why the number of predators had increased while the number of prey had diminished. His model implies that a prey population, in the absence of predators, would grow exponentially toward infinity. But Volterra’s method of expressing the rates of growth of populations by explicit functions depending on a small number of parameters, and then solving the equations explicitly, can only apply to highly simplified models. Sigmund demonstrates how Kolmogorov approached the problem qualitatively (using methods of H. Poincaré) in a way that enables the specification of an ecological system not by giving precise analytic expressions for the rates of growth, but by setting conditions for the signs of their partial derivatives. The author describes the impact of Kolmogorov’s work in the context of other contributions to the problem (notably by G. F. Gause). He explains why Kolmogorov’s general approach, which produces a model of biological communities made up of three or more species, today gives some of the results which are most useful for ecological applications and most interesting from a mathematical point of view.

Chapters 10, 11, and 12 are all focused on dynamical systems. In the first of these, Étienne Ghys provides a historical survey of the problem of stability of movements in celestial mechanics. He gives an elementary introduction to the Kolmogorov-Arnold Moser (KAM) theorem according to which the solar system is probably almost periodic. Kolmogorov had announced this theorem during the International Congress of Mathematicians held in Amsterdam in 1954, presented his proof in talks, but never published it. V.I. Arnold published the first proof of it in 1963, subsequent to the publication of an article by J. Moser which established a different but related result in 1962. These proofs were extremely complicated. Ghys attempts to give the reader a sense of the KAM problem by treating a simplified example inspired by it, which he solves using Fourier series. In doing so he reveals the role of resonance and small divisors.

In Chapter 11, John H. Hubbard continues the discussion of the KAM theorem, first in the context of two examples: the solar system and the forced pendulum. In the second part of the chapter the author sketches the main ideas of a 2002 proof of the theorem (which was an improvement of a relatively simple one published in 1984). The section contains nice illustrations, but to understand the proof, the reader should have an understanding of differentiable manifolds, flows of vector fields, etc.

Kolmogorov believed that general measure preserving dynamical systems are mixtures of quasi-periodic motions and motions with positive entropy. Chapter 12 takes the reader from Kolmogorov’s work on entropy of dynamical systems to non-uniformly hyperbolic dynamics. Denis V. Kosygin and Yakov G. Sinai illustrate why Kolmogorov’s 1958 paper, where he introduced the concept of entropy of a dynamical system, can be considered the starting point of the modern theory of deterministic chaos. Until that time, the main method of studying dynamical systems was the spectral one, influenced by J. von Neumann’s metric classification of dynamical systems with pure point spectrum. But Kolmogorov (and Sinai) noted that there exist deterministic systems with nonzero entropy. The modern notion of *hyperbolicity* of a dynamical system was developed as a result of the investigation of such systems. Kosygin and Sinai describe this discovery and discuss the latest results on the subject.

The last two chapters deal with the complexity of description, which Kolomogorov defined in 1965. Given that any information may be encoded as a bit string (a finite sequence of bits), Kolomogorov complexity centers on the idea that a long string of information bits needed to define a given object can sometimes be reconstructed from a short description. In such a case, the Kolomogorov complexity of the string is defined as the length of its shortest description. But most strings have no description significantly shorter than the string itself, and are called incompressible or random. Kolmogorov’s definition of complexity merged probability theory and the theory of algorithms, offering a new perspective on both subjects.

In chapter 14, Bruno Durand and Alexander Zvonkin give nice applications of Kolmogorov’s version of algorithmic complexity, with their descriptions of complexity proofs of Gödel’s incompleteness theorem and the Berry paradox, both suggested by Gregory Chaitin. They show how Kolmogorov complexity provides a procedure for producing true but unprovable propositions. The authors explore the connections between complexity and randomness, tracing developments in the efforts of Kolmogorov and others (e.g., Martin-Löf, Levin, and Schnorr) to characterize randomness using several types of descriptional complexity.

Paul Vitanyi, in chapter 15, extends the discussion of randomness, explaining how Kolmogorov complexity can express randomness in determinism and gives an approach to formulate chaotic behavior. He introduces the Incompressibility Method, giving examples of the use of Kolmogorov complexity arguments as a proof technique in number theory and graph theory. The author shows how this method, which has produced simple proofs of known results and solutions to open problems, can be used as a technical tool to quantify the unpredictability of chaotic systems.

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. However, the book would have benefited from a comprehensive index which guides the reader to topics and to works of Kolmogorov that are revisited in different chapters. There are minor errors (spelling mistakes, imprecise translations) that could have been caught by better editing.

**References**

Hardy, G.H., Ramanujan, S. The normal number of prime factors of a number *n*. *Quart. J. Math., 158*, 76-92 (1917).

Khinchin, A.Y., Kolmogorov, A.N. Über das Gestz des iterierten Logarithmus. *Math. Ann., 101,* 126-135 (1929).

Kolmogorov, A. N. On the principle of the excluded middle (Russian). *Matematicheski Sbornik, 32*, 646-647 (1925). English translation in *From Frege to Gödel. A source book in mathematical logic, 1879-1931* (J. van Heijenoort, Editor, 1967).

___________ Zur topologisch-gruppenteoretischen Begründung der Geometrie. *Nachr. Ges. Wiss. Göttingen, Fachgr. I (Mathematik), H. 2,* 208-210 (1930)

____________Zur Deutung der intuitionistischen Logik. *Mathematische Zeitschrift 35*, 58-65 (1932).

____________Zur topologisch-gruppenteoretischen Begründung der projektiven Geometrie. *Ann. Math. 33*, 175-176 (1932a)

____________*Grundbegriffe der Wahrscheinlichkeitsrechnung*. Springer, Berlin (1933). English translation *Foundations of the Theory of Probability* (Chelsea, N.Y., 1950).

____________Sulla determinazione empirica di una legge di distribuzione. *Giornale Istiuto. Ital. Attuari 4,* 32-91 (1933a)

____________Sulla teoria di volterra della lotta per l’esistenza. *Giornale Istituto Ital. Attuari, 7*, 74-80 (1936)

____________ A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces. *Dokl. Akad. Nauk SSSR (N.S.) 119*, 861-864 (1958)

____________The quantitative measure of mathematical models in the dynamics of populations (in Russian). *Problems of Cybernetics, 25*, 100-106 (1972).

Tikhomirov, V. M. Editor *Selected works of A. N. Kolmogorov*, Kluwer Academic, Dordrecht (1991).

Elena Anne Corie Marchisotto is professor of mathematics at California State University. She is co-author with James T. Smith of The Legacy of Mario Pieri in Geometry and Arithmetic. She is planning subsequent books on Pieri's research in projective and algebraic geometry, and the philosophy of science . In addition to geometry and the history of mathematics, her research interests includes mathematics education. She is the co-author with Zalman Usiskin, Anthony Peressini, and Dick Stanley of *Mathematics for High School Teachers, an Advanced Perspective* .