Andrei Nikolaevich Kolmogorov

Andrei Nikolaevich Kolmogorov was one of the greatest scientists in the Russian history. His work in probability theory, turbulence, and dynamic systems was fundamental and is now considered classic. The range of his contributions was enormous – from poetics to stratigraphy, from genetics to celestial mechanics, from topology to mathematical logic and algorithmic complexity theory.

Kolmogorov was born on April 25, 1903 in the central Russian city of Tambov. At 17 he finished the secondary school there and enrolled in the University of Moscow in 1920. He showed a keen interest in Russian history. His first work was a research paper on the registration of real estate in the medieval Novgorod republic. But when he found out that history professors required at least five different proofs of every assertion, he switched to mathematics, where one proof suffices! At that time Kolmogorov was interested in the ancient Russian arts as well and he retained this interest for the rest of his long life.

At the age of 19 Kolmogorov constructed an integrable function with a Fourier series divergent almost everywhere. This unexpected result created a tremendous sensation and made Kolmogorov an internationally recognized mathematician overnight.

At that time mathematics graduate students at Moscow University had to pass 14 examinations in various mathematical subjects, but it was possible to substitute an original article on a relevant topic in place of the exam. Kolmogorov never took any of the examinations, choosing instead to write the kind of papers he would make his life’s work. Even at the outset of his career, his articles contained new results in function theory, set theory, topology, mathematical logic, probability theory and other topics.

In May 1934, a little before James Alexander came up with the same idea, Kolmogorov introduced the cohomology ring, one of the most important topological invariants of a space. The idea came up to him from physics. He generalized such notions as the distributions of charges and currents in space, on surfaces, and on lines, considering the similar “functions of sets” for a more abstract mathematical situation.

Kolmogorov’s work in 1941 on turbulent motions changed the face of the theory of turbulence. Here he introduced the ideas of self-similarity and scaling, leading to the famous Kolmogorov law of 2/3. These ideas, and the modern developments they spawned, are now crucial elements of statistical physics and field theory.

What did Kolmogorov consider his most difficult achievement? His work, from 1955 to 1957, on the 13-th Hilbert problem involved the representation of continuous functions of many variables as the superposition of continuous single-variable functions and on the summation operation.

Kolmogorov’s last work before retiring from active research was dedicated to applying the ideas on information theory to the theory of algorithmic complexity and to the foundations of probability theory. He proved, for instance, that any “computer” containing N element of fixed diameter related to no more than k other elements by “wires” of fixed thickness may be packed in a cube with an edge of approximately N. He had guessed this result by starting from the observation that the grey substance of the brain (the neurons) forms its surface, while the white substance (the junctions) is inside.

In addition to his many mathematical theories, Kolmogorov expounded a theory of a more human sort: that it is impossible to do good mathematical research after the age of 60. And so, after half a century of original and often path finding work, he became a high school teacher. This was his main occupation for the last 20 years of his life. He was also appointed chairman of the Commission for Mathematical Education in the Academy of Sciences of the USSR and in that position instituted new programmes to more fully develop the research interests of schoolchildren.

In 1970, together with I.K.Kikiyin, Kolmogorov started a new magazine for Soviet youth – Kvant. He wrote articles for it and remained active in managing it right up until his death in 1987.

A.N. Kolmogorov stood out among the great mathematicians of the 20^th century in that he revolutionized both mathematics and physics, much as Newton had done two centuries earlier. His mind roamed freely in many fields and tirelessly sought connections. A brilliant guesser and a hard worker, Kolmogorov was a mentor to students and younger colleagues.

(Abridged and adapted from V. Arnold, Physics Today)

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