The name Andre nikolayevich in this article is Colmo Golov (sometimes translated as "Andre Andrey Kolmogorov").
Colmo Golov, a genius among us: research style.
Now it is time to return to Kolmogorov's scientific career again and discuss some characteristics of his creative mode, which has contributed to the establishment of an extremely important scientific school in many aspects. Things can be presented in contrast. I would like to compare Kolmogorov's creative features with those of his outstanding colleagues and contemporaries.
I remember Colmo Golov once talking to us about who is the greatest mathematician of our time. During the conversation, everyone found that it was futile to try to entangle in any name: no agreement could be reached. The collection of the greatest mathematicians is relatively small, but it is clear: if you ask who are the 100 greatest mathematicians or the two greatest mathematicians, they will give roughly the same names. Later, in my conversation with my friends, this topic was involved more than once (especially when we were young). Who is in the first place-Kolmo Golov, vinogradov (I am, Mr. Hua in our country respects him very much, and they have a deep friendship), Bernstein (Snbernstein), Petrovsky (I. G. Petrovsky), L. Pontryagin (S. Pontryagin), Gail Fonder (I am just an China in the Soviet era. Of course, I admit that we can add the names of other great mathematicians in our country. Let's add Hilbert's name to this list-he was the most important mathematician in the first third of the 20th century, and Colmo Golov admired him so much that he personally wrote it into the encyclopedia of the Soviet Union. Many other foreign mathematicians' names will also be on our list-including Hadamard, Brouwer and G? Del, Siegel, E. Katan, A. Katan, Leberg, Levi, etc. )
Kolmogorov's creative way is fundamentally different from all the mathematicians mentioned above. Gelfond once said in a conversation, "Mathematics is a marathon." This is a profound thought. There is no doubt that Gelfond himself and all other mathematicians listed above are "marathon runners". Colmo Golov belongs to another kind of scientist (but I don't know anyone like him except himself). Andre nikolayevich is also a "marathon runner" of course, but he is mainly a sprinter on the way.
What does this mean? Over the years, whether in articles or in personal conversations, Colmo Golov often quoted a sentence from the mathematician Delong. Delong once spoke in front of primary school students at the closing ceremony of the Olympic Games, saying that the work of mathematicians is different from that of participants in the Olympic Mathematics Competition. It takes about 1 hour to solve an Olympic math problem, but it takes 5000 hours to solve a real and profound math problem. This value-5000 hours-represents the working characteristics of marathon mathematicians.
However, whenever he talks about himself, Andre nikolayevich shows obvious embarrassment. He can't do the famous 5000 hours. In an interview, he said: "I can work continuously for about a week, maybe two weeks at most, but I can't do more." About forty years ago, I first heard something similar: in class, he mentioned a much smaller number-he thought about the example of constructing a Fourier series that diverged almost everywhere for three days, and finally suddenly realized. In the early days, he called this result the most technically difficult of his existing achievements. The day after tomorrow, André Andrey Kolmogorov chose his most technically difficult result as the theorem that later led to the solution of the Hilbert 13 problem, and mentioned two weeks of unremitting thinking.
We can see that these examples reflect the unique style of Andre nikolayevich. He knows how to concentrate huge energy in a relatively short time. This accumulation of energy has produced a powerful effect, opening a huge crack in the seemingly indestructible fortress wall, causing dozens, sometimes hundreds and thousands of researchers to rush there. But Colmo Golov himself has left all this, and his thoughts have turned to other goals. This has happened many times before my eyes. Perhaps it will be interesting to browse the whole creative process of Kolmogorov from this angle.
Under the influence of Aleksandrov's course, Kolmo Golov completed the first important book describing set theory. He realized that Aleksandrov's main idea (constructing A- set) could be combined with Suslin's main idea (proving that A- set is wider than B- set), which laid the foundation of set operation theory. His tutor Luzin didn't understand this article at that time, so the first part of it was published seven years later-1928 (the second part was published as an appendix in the third volume of Selected Works of Colmo Golov in 1987). Andre nikolayevich didn't continue to write this question. Then the theory became very active, and the works of Andre nikolayevich became one of the sources.
Next is Colmo Golov's greatest discovery in the early stage of scientific research-he constructed a measurable function whose Fourier series diverged almost everywhere (as we just mentioned). Colmo Golov studied the theory of trigonometry and orthogonal series for some time, but his interest turned to probability theory (he worked closely with Qin Xin for several years). In the early days of 1930, his efforts ended with the completion of two works of fundamental importance: the paper "On the Analytical Methods of Probability Theory" and the monograph "Basic Concepts of Probability Theory". In addition to these marathon achievements, there are also some sprint achievements, especially his research in mathematical logic and his outstanding work in mathematical statistics and topology (in which he and American algebraic topologist Alexander independently introduced the most important topological concept-"homology"). It all happened in the 1930s. Here are two of his short essays on approximation theory, which laid the foundation for a new basic direction and solved the problem of dimension increase under open mapping. He devoted himself to theory of turbulence in the late 1960s and early 1960s. These studies also have a "marathon" component.
In the 1940s, Colmo Golov established the shooting theory (there is a "marathon" component here), which laid the foundation of the so-called branching process theory (this may be the result of "sprint").
Back in the 1950s. A sudden great insight led to the birth of KAM theory. Andrei nikolayevich himself only published two short articles in the journal of the Soviet Academy of Sciences, organized a seminar on mechanics and mathematics, and made the closing report of the Amsterdam World Congress of Mathematicians. From 65438 to 0955, information theory began to interest him. But at the same time, he "unexpectedly" almost completely solved Hilbert's 13 problem (at the cost of hard pressure, of course): he proved that any continuous function of four or more variables can be expressed as the superposition of continuous functions of three variables. Once again, he didn't continue to study the final solution to the problem. And left this problem to his student Arnold (then a junior) to solve.
.....1one day in the summer of 957, I arrived in komarov Card (there is a country house of Colmo Golov and Alexanderov). Teacher Kolmo Golov told me that the day before, when he was thinking about the structure of solving the Hilbert 13 problem, he suddenly realized that he had found an unusually simple new method to solve this problem, which strengthened Arnold's results. When I arrived, a short article for the journal of the Soviet Academy of Sciences had been written! The same is true of the von Neumann problem of dynamic systems (this problem has existed for more than 20 years and all experts in dynamic systems want to solve it), that is, whether the spectrum is a complete representation of dynamic systems. Another time happened when I visited komarov Card. Andre nikolayevich suddenly said: "Entropy is invariant, and it is not enough to have a spectrum." This epiphany once again led to a breakthrough, and several researchers rushed into the gap, including many first-class mathematicians; As often happens, Colmo Golov confined himself to a paper in the Journal of the Soviet Academy of Sciences, made only the first breakthrough, and then left. This is another example. One day, Andrei nikolayevich and I will go to Leningrad to attend a meeting. In the evening, we talked about different things in the corridor of the carriage. Suddenly he told me that he had just thought of this idea (right there, in a conversation! Under the linear mapping of linear topological space, entropy can also be invariant. Soon I wrote a short article, and many mathematicians became interested in this topic again. As far as I can remember, after that, Mr. Kolmo Golov never even thought about what would happen in this field.
It is easy to find that Colmo Golov has nothing in common with any of the "greatest" mathematicians listed above. The most striking contrast with Andre Andrey Kolmogorov is Hilbert. Because of the "sprint" feature of Kolmogorov's creative genius, he successfully broke through and opened up a lot of difficult problems and fields. In an article I wrote about Andre nikolayevich before, I listed about 40 fields of mathematics, natural science and humanities, and he left a basic mark in these fields (although he still didn't use up everything he created). In almost any sub-discipline, Andre nikolayevich's research is the work of pioneers, including the establishment of basic theories, while the remaining perfection in new fields is left to disciples and followers. In contrast, Hilbert devoted himself to eight subjects of pure mathematics for many years, sometimes even decades, trying to "find the foundation, root and core". Bernstein, vinogradov, Petrovsky and Pontryagin have similar research styles to Hilbert. Gelfond is a special case: he always cooperates with his colleagues, while other scientists on our list work alone. Like Colmo Golov, Gelfond has studied many fields, and he is undoubtedly a "marathon runner". )
To sum up, Kolmo Golov always generates many ideas and nourishes the students who work with him. As a matter of fact, Andre nikolayevich usually doesn't work with his students: he doesn't teach them according to the generally accepted meaning of the word "guidance". He just disseminates questions, assumptions, and shares ideas and methods-during lectures, walks and tea in komarov Ka Xiaoshe ... These strategic positioning problems are often not just a mathematical problem, but also contain broader scientific (or philosophical) significance. If a disciple sets foot on one of these roads, he can continue to improve himself without saying "everything is settled" rashly.
This article was first published on Zhihu Platform, /p/422726695. Here are more historical and mathematical stories about Kolmo Golov.