Andrey Kolmogorov — one of the greatest mathematicians of the XXth century
2024 Update: Kolmogorov-Arnold Network better MLPs in terms of both accuracy and interpretability! Check it out.
Andrey Kolmogorov is one of the best mathematicians of the XXth century who has worked and made top contributions to so many fields that for many scientists, it is hard to fathom how one person contributed so much to math and, indeed, many other sciences.
As one of the direct math descendants of Kolmogorov (who is my academic ‘grandfather’) and with Kolmogorov number 2 (my PhD supervisor is the last PhD student of Kolmogorov), I thought I would write a short article explaining his contributions to mathematics and many other fields that stay as relevant in the XXIst century as they were in the XXth century.
By establishing fundamental results in probability theory, topology, geometry, functional analysis, measure theory, theory of integrals, theory of approximations, theory of random processes, mathematical statistics, theory of algorithms, mathematical linguistics (precursor to NLP), theory of differential equations, dynamical systems, theory of information, intuitionistic logic, theory of turbulence, classical mechanics, algorithmic information theory, computational complexity, mathematical applications to problems of biology, geology, material science — there are so many fundamental contributions that permeate so many fields to this day that it is almost impossible to appreciate his contributions fully. The list of Kolmogorov books, papers, and writings alone includes 900 references.
Here are a few examples of his contributions and history to get a feeling of appreciation for one of the greatest mathematicians of all time.
Kolmogorov’s first mathematical discovery was published in his primary school journal, where he was the “editor” of the mathematical section of the journal. He was five years old; he noticed the regularity in the sum of the series of odd numbers is equal to the square of the median number (hello, Gauss!).
In one of the best schools he was enrolled in at the time, the teachers could not teach mathematics to Kolmogorov fast enough — he learnt math by devouring mathematics knowledge from 86 pieces ‘Brockhaus and Efron Encyclopedic Dictionary.’
Life in post-1917 revolution Russia was hard, and in 1918–1920, Kolmogorov had to support himself by working on constructing the Kazan-Ekaterinburg railway line whilst continuing to study on his own to complete high school exams.
When he was 17, he enrolled to study math at Moscow State University and passed all first-year exams in just a few months, after which he started research, gradually solving more complex problems.
In 1922, at the age of 19, Kolmogorov became internationally known by first creating the Fourier series that diverged almost everywhere and then the Fourier series that diverged at every point.
In 1924, Kolmogorov started to become interested in research in Probability Theory and in 1928, he was able for the first time to formulate necessary and sufficient conditions of the Law of Large Numbers that escaped other best mathematicians of the time for many decades. This problem was previously worked on by Chebyshev and Markov (senior, there were two Markov brothers who were mathematicians) in the XIXst century.
In 1931, Kolmogorov published ‘Analytical Methods in Probability Theory, laying the foundations for the modern theory of Markov processes and establishing deep connections between probability theory and the theory of differential equations. Minimal results connecting Markov chain problems with parabolic equations were obtained first by Laplace and later by Fokker and Planck, but those were just limited examples. Kolmogorov formulated the theory of Markov processes in the process of deriving both direct and inverse equations.
At the age of 30, Kolmogorov published his book, ‘Foundations of the Theory of Probability ‘Grundbegriffe der Wahrscheinlichkeitsrechnung’ that after centuries of betting applications finally took Probability to its rightsome place — one of the critical scientific fields and established his reputation as the world’s leading expert in this field.
At the end of the 1930s, Kolmogorov became increasingly interested in research on the problem of turbulence; for the first time, it was Kolmogorov and his students who clearly defined the mathematics of the field. It is interesting that in addition to unparalleled math skills, Kolmogorov also possessed great physics intuition, enabling him to derive the ‘law of 2/3s’ that is the fundamental law of nature with grandiose simplicity of its formulation — the average square of velocity difference at two points in turbulent flow is proportional to the distance r between these two points taken to the power of 2/3. When Kolmogorov formulated the law of two-thirds, the experimental data for turbulence were unavailable; later, many empirical researchers confirmed this law with high accuracy in natural (ocean, atmosphere) experimental conditions.
Kolmogorov’s contributions to physics (theory of turbulence, Kolmogorov developing complete theory of Kolmogorov Fokker — Planck equations) put him into the pantheon of great physicists like Newton and Kolmogorov was not even a physicist — the predominant bulk of his work was in mathematics.
In 1953–1954, Kolmogorov published several works related to the general theory of Hamiltonian systems and presented an extensive overview of the field at the 1954 Mathematical Congress in Amsterdam. The astronomy problem of the evolution of the orbits of 3 or more celestial bodies (astronomical objects) goes back to the times of Newton and Laplace; in the case of minor planets, this problem becomes part of a more general problem about the quasi-periodical movement of Hamiltonian systems as a response to small changes in Hamiltonian functions. Kolmogorov's works solved this problem under general conditions. Application of his framework allowed us to solve problems waiting for solutions for decades, including problems in astronomy and plasma physics (from Kolmogorov’s theorem, one can obtain results about the stability of magnetic field surfaces that are very important in plasma physics fields).
In 1955, Kolmogorov turned his attention to the information theory, where he made significant, unexpected and bold contributions, developing Shannon’s ideas, synthesising them first with Kolmogorov’s theory of approximations and later with the ideas from the theory of algorithms. During 1955–1956, Kolmogorov introduced the notion of ‘ε-entropy’, the quantity of information required to approximate the function from this class with accuracy ε. As Kolmogorov has shown, for r differentiable function of n variables, ε-entropy grows as ε^(n/r) — i.e., the required quantity of information to approximate a function with accuracy ε increases proportionally to the number of independent variables divided by the degree of smoothness as measures by r.
This research turned Kolmogorov's attention to the so-called Hilbert’s 13th problem (set out in a celebrated list of 23 problems compiled in 1900 by David Hilbert). The problem asked to prove that a continuous function of 3 variables can’t be represented as a composition of continuous functions of two variables. In 1956, Kolmogorov arrived at the unexpected result. A continuous function of any number of variables can be represented as a composition of functions of three variables. The problem was reduced to representing function defined on universal trees in 3-dimensional space. This problem was successfully solved under the guidance of Kolmogorov by his PhD student Arnold in 1957., The solution contained the result opposite to that postulated in Hilbert’s 13th problem — every continuous function of 3 variables CAN be represented as a composition of functions of two variables. In the same year, 1957, Kolmogorov made the final step, proving that the continuous function of any number of variables can be represented as a composition of continuous functions of one variable and addition.
Hilbert’s thirteenth problem is one of the 23 problems set out in a celebrated list compiled in 1900 by David Hilbert. It entails proving whether a solution exists for all 7th-degree equations using algebraic (variant: continuous) functions of two arguments. It was first presented in the context of nomography, and in particular, “nomographic construction” — a process whereby a function of several variables is constructed using functions of two variables. The variant for continuous functions was resolved affirmatively in 1957 by Vladimir Arnold when he proved the Kolmogorov–Arnold representation theorem, but the variant for algebraic functions remains unresolved.
He has also actively taught generations of students at Moscow State University for decades and was a role and aspiration model for millions of children interested in math and science.
Around 1936 Kolmogorov contributed to the field of ecology and generalised the Lotka–Volterra model of predator–prey systems.
In a 1938 paper, Kolmogorov “established the basic theorems for smoothing and predicting stationary stochastic processes” — a paper that had significant military applications during the Cold War.
In 1939, he was elected a full member (academician) of the USSR Academy of Sciences — a pantheon of top scientific minds.
During World War II, Kolmogorov contributed to the Russian war effort by applying statistical theory to artillery fire, developing a scheme of stochastic distribution of barrage balloons intended to help protect Moscow from German bombers.
In his study of stochastic processes, especially Markov processes, Kolmogorov and the British mathematician Sydney Chapman independently developed the pivotal set of equations in the field, which were named the Chapman–Kolmogorov equations.
In 1971, Kolmogorov joined an oceanographic expedition aboard the research vessel Dmitri Mendeleev, where he solved many high-impact problems.
Another famous mathematician, Vladimir Arnold, once said:
“Kolmogorov — Poincaré — Gauss — Euler — Newton, are only five lives separating us from the source of our science”.
Kolmogorov’s mathematical legacy is eternal for Probability Theory and other fundamental fields, including complexity theory.
In 2021, scientists used his work to move forward frontiers in quantum computing.
References:
- Kolmogorov Arnold Network! Paper. Code.
- https://en.wikipedia.org/wiki/Turbulence#Kolmogorov%27s_theory_of_1941
- https://encyclopediaofmath.org/wiki/Epsilon-entropy
- https://en.wikipedia.org/wiki/Hilbert%27s_thirteenth_problem
- https://ru.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BB%D0%BC%D0%BE%D0%B3%D0%BE%D1%80%D0%BE%D0%B2%2C_%D0%90%D0%BD%D0%B4%D1%80%D0%B5%D0%B9_%D0%9D%D0%B8%D0%BA%D0%BE%D0%BB%D0%B0%D0%B5%D0%B2%D0%B8%D1%87
- To the 80th anniversary of Kolmogorov http://www.mathnet.ru/links/c9fc3bce559dd8d013b55bc815e37bb9/rm2938.pdf
- Talk with Andrey Kolmogorov (interview to Russian STEM magazine for kids ‘Quant’, 1983, reprinted in 2020)
PhD Thesis (2021): ‘Algorithmic Randomness and Kolmogorov Complexity for Qubits’
