Selected Works I: Mathematics and Mechanics

Note that this is a reprint of an edition originally published in 1991.

 

Andrei Kolmogorov (1903-1987) was one of the great mathematicians of the 20th century. He was the founder of the Soviet school of probability theory (effectively axiomatizing the subject), information, ergodic theory, mechanics and the theory of turbulence. His interests ranged over very large areas of modern science and mathematics and his name is enshrined in KAM theory and in the solution (with V.I. Arnol’d) of Hilbert’s thirteenth problem. The papers in this collection were chosen by Kolmogorov himself and the editor V. M. Tikhomirov. The complete project includes three volumes. 

 

This collection of his early papers (up to early 1960’s with a last paper from 1972) showcases Kolmogorov’s development of interests through a broad range of classical mathematics. His first paper featuring a construction of a Fourier series of a summable function which is divergent almost everywhere is here (1922). The discovery made quite an impression in the larger analysis community and it was followed by a large output of papers by him on convergence conditions for such series. Many of these relate to showing convergence of a Fourier series follows from various summability conditions on the coefficients involving enlargements of square summability using logarithmic multipliers. Through the seven or so papers collected here we see his mastery of detailed “hard” analysis as well as a burgeoning interest in integration, means and set theory. Most of these are doubtless outgrowths of his advisor’s (N. N. Luzin) interests. However, Kolmogorov is already pursuing much larger agendas here and investigating generalizations of integration theories that push the boundaries of that subject. The work of Denjoy, Frechet and Moore function as starting points for this work and several papers on the subject feature the use of homogeneity and integral means which presage his work in probability theory. His work there leads to some very interesting sidelights on intuitionist logic and axiom schemas as well as set operations (all of which today would likely be grouped under the rubric “descriptive set theory”).  There are also some biographical essays here that stress Kolmogorov’s wide range of interests and the lucky “accident” which led to his adoption as student by Luzin.

 

In 1929 another lucky event occurred which led to an early interest in topology as a field of endeavor.  Kolmogorov took a boat trip on the Volga with Paul Aleksandrov and some of the results of that trip are visible here in his work in algebraic topology. Some early work on combinatorial topology and duality is followed by work on homology rings of complexes and compact spaces. One witnesses the early development of algebraic topology through reading these – there is no mention yet of manifolds or exact sequences and much of the techniques used involve Betti numbers and groups and Kuratowski-style set topology. The last of these papers address relative cycles and the Alexander Duality Theorem, coverings and topological vector spaces (though not yet named as such). The exciting drumbeats of the accompaniment to his life increase until a thunderous change in dynamic in 1937.

 

The paper “A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem” (Bull. Moscow Univ., Math. Mech. 1:6 (1937)) announces the next change of direction in Kolmogorov’s interests. The paper seems to introduce the existence of traveling waves in diffusion problems and marks Kolmogorov’s entry into the world of partial differential equations. The work here is all in the form of the classical analysis of functions. It is the 1941 paper“Local structure of turbulence in an incompressible viscous fluid at very large Reynold numbers” (Dokl. Akad. Nauk SSSR 30:4 (1941)) which introduces work following G.I. Taylor and von Karman. Papers on isotropic turbulence, dissipation of energy and more showcase work based on symmetry conditions, scaling and dimensional analysis applied to the hard problems of non-linear turbulence. Throughout these and the subsequent papers collected here we see the development of modern dynamical systems. Invariant tori, integrals of motion, ergodic theory and operator approaches appear here (1953) and it is instructive to read the address of Kolmogorov at the 1954 International Congress of Mathematics to see how the general outlook that has dominated in mathematical mechanics until the present time came to be. I must stress how readable these papers are – they address the subject forthrightly and can be read profitably by any modern mathematician. Kolmogorov’s work on transitive measures and spectra can be read as the culmination of so much of his early interests. The selection of papers enables a very readable view of the history at the individual level for this purpose. To conjecture beyond this will require a professional historian of mathematics (which I am not) whose specialty is 20 th century work. At this level of the individual, at least, the story seems clear and believable. I find it inspiring, as well, to see how ideas develop through looking at the work of this brilliant mathematician. The fantasy of “the idea which appears out of nowhere” is not invoked in any way and one sees the consistent hard work for which Kolmogorov was famous on every page. His prolixity is to be admired and celebrated.

 

The last big feature of this volume is disclosed in a sequence of papers that are devoted to the solution of Hilbert’s 13 th problem. The first such paper announces Kolmogorov’s proof that “any continuous function of an arbitrarily large number of variables is representable as a finite superposition of continuous functions of at most 3 variables” (1956). This is followed by several other papers which mention V.I. Arnol’d’s contribution to the reduction to 2 variables and the last of the set features the construction of the functions associated with their solution to the problem. This incredible achievement also features some early concepts of complexity theory in the form of “amounts of information” needed to specify a function belonging to a special class of functions. These ideas have gone to degrees of development way beyond what is represented here but it is enlightening to see the seeds of such developments. Truly, Kolmogorov was a giant in the mathematical world but also a pivotal figure who spread his ideas far and wide and created the language of the world of modern applied mathematics.

 

This selection of Kolmogorov’s work is followed by a short (110 pages) commentary which helps the reader to locate the importance of that work in the larger world of mathematical research. The writing in this commentary is done by some very famous figures in their own right and is worth reading for more details of the historical record.

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Jeff Ibbotson is the Smith Teaching Chair in Mathematics at Phillips Exeter Academy.