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20 Years of Soviet Mathematics
The following paper was written to celebrate the twentieth anniversary of the October Socialist Revolution of 1917.
Soviet Mathematics for 20 Years, UMN 1938 (4) (1938), 3-13. https://www.mathnet.ru/rm7112
Soviet Mathematics for 20 Years, UMN 1938 (4) (1938), 3-13. https://www.mathnet.ru/rm7112
20 Years of Soviet Mathematics.
The peoples of the Soviet Union celebrated the twentieth anniversary of the October Socialist Revolution, and together with them, the workers of the whole world and the best progressive figures of human culture marked this date with hope and joy.
The results of the grandiose work on the creation of a socialist society on the territory of one sixth of the globe are majestic, the results of those enormous victories in all areas of economic and cultural construction that were won by the peoples of the Soviet Union under the leadership of the party by Lenin and Stalin.
Among other indicators of the growth of the Soviet Union, the growth of Soviet culture, the growth of Soviet science are extremely characteristic. And this growth is even more indicative against the background of the cultural decline of the capitalist world. Fascism, with its hatred of culture, of human reason, with its return to the medieval worldview, is the most striking manifestation of this decline. In a country like Germany, the largest scientific centres have been destroyed (for example, the international centre of mathematical life in Göttingen has been virtually destroyed), and the greatest scientists (including the greatest mathematicians) have been expelled from Germany. Young scientists in capitalist countries, as a rule, do not find a use for their knowledge.
This cultural decline is countered by the incessant cultural growth of the Soviet country. Even in the first years of its existence, during the most difficult times of the civil war, the young Soviet state opened new higher education institutions, new universities, new scientific institutions. The first of the mathematical scientific institutes (at Moscow University) was founded in 1922. Over time, the network of scientific institutions, in particular mathematical ones, has grown steadily, especially during the reconstruction period. At present, there are more than ten mathematical and physical-mathematical institutes.
An extremely important circumstance for the development of science is the right of a citizen of the Soviet Union to education, realised in life and recorded in the Stalin Constitution. A young citizen of the Soviet Union who wants to devote himself to science has every opportunity to realise his desires - the state provides him with material support both during his studies at a higher school and during the period of preparation for scientific work (graduate school). In the Soviet Union there are no talents who perish because of need. What significance this has for the development of Soviet culture and Soviet science follows at least from the fact that, for example, the majority of our most prominent mathematicians came from the environment of Soviet graduate school (L S Pontryagin, A O Gelfond, L G Shnirelman, A N Kolmogorov, I G Petrovsky, M A Lavrent'ev, etc.),
The enormous scope of socialist construction, the creation of new technology served as a powerful stimulus for the development of the exact sciences, in particular mathematics. A significant portion of the young people who graduated from the mathematical departments of Soviet universities work in technical institutes. A number of young mathematicians who have come forward in recent years (S L Sobolev, S A Khristianovich, M V Keldysh, and others) began their scientific work in such scientific institutions as the Seismological Institute, the Hydrological Institute, the Central Aerohydrodynamic Institute, and others. The importance of mathematical sciences for Soviet technology has been reflected in the scope of mathematical education in the country. Hundreds of new mathematical departments have opened in new universities and technical colleges, and the circulation of university mathematical textbooks is measured in hundreds of thousands of copies, figures unheard of in any other country. Soviet youth shows a great interest in mathematical sciences, which is reflected at least in the success of mathematical Olympiads, in the requirements for mathematical literature, in the huge influx of candidates for the mathematical departments of universities (at Moscow University, for example, 8 applications were submitted for one vacant place), etc.
The growth of mathematical personnel and the increased interest in mathematical sciences led to the creation of a rich book of mathematical scientific literature in Russian (which was almost absent in pre-revolutionary Russia).
All these factors favoured the development of mathematics in the Soviet Union, its continuous growth and the continuous increase in its share in world science.
The Russian people, in the difficult circumstances of Tsarist Russia, produced a number of first-class mathematicians, first of all Lobachevsky and Chebyshev, then Ostrogradski, Lyapunov, Markov, Voronoy, Zolotarev, and others. These scientists were effectively isolated. Even at Moscow University, the mathematics department had a provincial character for a long time. The connection between individual areas of mathematical work was weak: for example, the so-called St Petersburg school, which lived by the traditions of Chebyshev, with its high and unique algorithmic culture, was sharply opposed to the abstract set-theoretic Moscow mathematics. Scientific traditions, even those with a glorious past, sometimes turned into conservatism in conditions of isolation (for example, the negative attitude towards integral equations on the part of some prominent representatives of St Petersburg mathematics). Reactionary philosophical and political tendencies in Moscow mathematics (Bugaev, P Nekrasov, and others) had a negative impact on the development of science. Moving on to the Soviet period, we must first of all note that scientific work in mathematics did not cease even during the most intense years of the civil war. The entire Soviet period was accompanied by a continuous replenishment of the ranks of Soviet mathematicians with new personnel.
Three congresses are characteristic in this regard: the First All-Russian Congress in Moscow (1927), the First All-Union Congress in Kharkov (1930) and the Second All-Union Congress in Leningrad (1934). Each of these congresses demonstrated the achievements of new generations of Soviet mathematicians. At the same time, mathematicians who had formed as scientists in pre-revolutionary times received new incentives for scientific work and new talented students. Schools created on the eve of the revolution, such as the algebraic school of Academician D A Grave or the school of equations of mathematical physics of Academician V A Steklov, reached their heyday precisely in Soviet times.
In recent years we have no longer isolated scientists and no longer isolated mathematical schools, but a continuous front of mathematical research, covering all the main areas of mathematics, from its direct applications to the most general theoretical disciplines. In this article we will outline only the general line of development of the main directions of Soviet mathematics, illustrating it with some outstanding achievements. A more detailed analysis of the development of individual disciplines will be given in a number of articles in the following issues. Individual areas of work, mainly in the field of applications, are covered in previous issues of "Advances in Mathematical Sciences" (see, for example, the articles by S L Sobolev and S G Mikhlin in the first issue, S A Khristianovich and B B Devison in the second, N E Kochin in the third). A detailed analysis of the development of individual disciplines is given in a special publication of the USSR Academy of Sciences.
I. Number Theory and Algebra.
In number theory, pre-revolutionary Russian mathematics had a number of major achievements (the works of Chebyshev, Zolotarev, Voronoi, Markov). But the results obtained in Soviet times are especially striking. It is safe to say that in the field of number theory, Soviet mathematics has special achievements and plays a leading role in world mathematics.
First of all, it is necessary to note the cycle of works by Academician I M Vinogradov, which led him to results of increasing significance. One of his first works related to Waring's problem (to prove that every integer can be decomposed into a sum of a limited number of complete -th powers of integers). I M Vinogradov's new solution to this problem was included in the basic courses of number theory. In 1934-35 he gave an exceptionally precise estimate of the number . Namely, according to Vinogradov's estimate turned out to be of the order of ln , while the previous Hardy-Littlewood estimates for gave a value of the order of . Vinogradov's estimate is obviously already close to final. This work demonstrated the power of the new methods of estimating the so-called Weyl sums, which were exceptionally important for number theory, found by I M Vinogradov. They allowed I M Vinogradov to solve a number of extremely difficult problems, including an almost complete solution of the famous Goldbach problem on the representation of any number as a sum of three primes. This last work by I M Vinogradov is covered in a special article by N G Chudakov, published in this issue. I M Vinogradov applied his new methods to other extremely difficult problems in number theory. For example, I M Vinogradov gave a very precise estimate of the distribution of fractional parts of polynomials. Vinogradov's methods were used and developed by a number of Soviet mathematicians in solving other problems in number theory. Here, we should mention the works of N G Chudakov, V A Tartakovsky, B I Segal, R O Kuzmin, K K Marjanishvili, and others. For example, N G Chudakov fundamentally improved the value of the remainder term in the well-known asymptotic formula for the number of primes not exceeding a given limit.
The possibility of representing any number as a limited number of primes was first proved by L G Shnirelman in 1930 (see N G Chudakov's article in this issue). Shnirelman's works, which are among the best achievements of Soviet mathematics, demonstrated the importance of such general metric concepts as the density of a sequence of integers. They served as the beginning of research by a number of both Soviet (A Ya Khinchin, N I Romanov) and foreign scientists.
One of the most beautiful achievements of Soviet mathematics is A O Gelfond's solution of Hilbert's seventh problem, posed by the latter at the International Congress of Mathematicians in 1900: prove that every number of the form where and are algebraic, and is irrational, is a transcendental number. In 1928, A O Gelfond solved this problem for the case when is an imaginary quadratic irrationality. R O Kuzmin extended this proof to the case when is a real quadratic irrationality. Finally, in 1930, A O Gelfond completely solved Hilbert's problem.
B N Delone gave a deep study of indefinite equations of the third degree. In his research, B N Delone used geometric methods. In this geometric-arithmetical direction, V A Tartakovsky, D K Faddeev, O K Zhitomirsky, A D Aleksandrov worked. The works of B N Delone and his school are also connected with the theory of paralleloids used in crystallography.
The purely arithmetical direction is represented in our country primarily by B A Venkov. He conducted research on the theory of quaternions and was the first to obtain Dirichlet's formulas on the number of classes of quadratic forms with a given discriminant by purely arithmetical means.
The metric direction of number theory, i.e., research into the properties of irrational numbers for "almost all" numbers, is represented by a number of works by A Ya Khinchin.
The achievements of Soviet mathematicians in the field of algebra are also very significant. We have already noted the merits of the school of Academician D A Grave, from which came the main body of Soviet algebraists (N G Chebotaryov, O Yu Schmidt, B N Delone, and others: for more details on the creation of this school, see the article by N G Chebotaryov in the third issue of "Advances in Mathematical Sciences"). In Soviet times, the most prominent representatives of this algebraic school created a number of independent schools.
O Yu Schmidt and his students - A A Kulakov, V K Turkin, A P Ditsman, S A Chunikhin, and others - developed the theory of finite groups. Here, a number of results were obtained on the classification and enumeration of simple groups, and the study of special groups. O K Schmidt and A G Kurosh studied the theory of infinite discrete groups (decompositions into direct and free products). A G Kurosh and A I Mal'tsev gave a classification of infinite torsion-free Abelian groups.
The most important results in algebra were obtained by N G Chebotaryov. We will point out here his proof of Frobenius's conjecture on the existence of an infinite set of prime numbers belonging to a given class of permutations, and his profound research in the direction of the so-called thirteenth Hilbert problem. Namely, N G Chebotaryov gave the following principle for solving the generalised Klein problem on an equation (resolvent) with the smallest number of parameters to the solution of which the solution of a given algebraic equation can be reduced: if the Galois group of the equation is a subgroup of an -dimensional continuous group, then the solution of the original equation is reduced to the solution of an -parametric equation.
Further research in this direction led N G Chebotarev's students to a number of important results in the theory of Lie groups. We especially note the fundamental research of I D Ado, who managed to completely solve the long-standing problem of the local representation of Lie groups by linear permutations.
A number of results on the theory of generalised groups and semigroups were obtained by A K Suschkevich.
II. Theory of functions.
In the theory of functions of a real variable, two main directions can be noted. The first of them, constructive, founded by academician S N Bernstein, goes back to the famous works of Chebyshev on polynomials that deviate least from zero. Each continuous function can be assigned a unique polynomial of degree n that gives the best approximation of this function. S N Bernstein discovered the connections that exist between the structure of a function and the nature of polynomial approximations. For example, analytic functions are characterised by the fact that the degrees of their approximation by polynomials grow faster than a geometric progression. Thus, the opportunity opened up to study the theory of analytic functions of the real domain. These studies, begun by academician S N Bernstein in pre-revolutionary times, led him already in the Soviet era to the construction of a theory of quasi-analytic functions and to a number of excellent results in the theory of interpolation, on the extension of the theory to approximations by other classes of functions.
Among other works in this direction, we note the results of N I Akhiezer on polynomials of best approximation.
Another direction in the theory of functions, more abstract, has now come close to the constructive one. It arose in Moscow in the school of Academician N N Luzin, shortly before the revolution. A whole generation of Moscow mathematicians currently working in various fields began with the theory of functions. G A Fichtenholz founded a school of function theory in Leningrad.
One of the major directions in the theory of functions is the qualitative study of the structure of functions and sets of the most general nature, the so-called descriptive theory of functions. M Ya Suslin, who died early, discovered and investigated, using the construction of P S Aleksandrov, a very important class of functions and sets - the so-called -sets. This theory of -sets was also developed in a number of works by Academician N N Luzin. Further generalisations of A-operations were given by A N Kolmogorov and L V Kantorovich. Generalising the theory of -sets, N N Luzin constructed a theory of projective sets.
Of the works of a descriptive nature, we also note the work of L V Keldysh (the structure of -sets), N K Bari (representation of arbitrary continuous functions by superpositions of absolutely continuous ones), and M A Lavrentev (topological invariance of Baire classes). We especially note the works of P S Novikov on implicit functions, which led him to theorems on separation, on the class of sets intermediate between projective and -sets, etc.
An even more extensive series of works concerned the metric theory of functions, which studied the basic operations of analysis - differentiation, integration, expansion into series, etc. Work in this direction began on the eve of the revolution by N N Luzin and his students. Here we note the research of A Ya Khinchin on asymptotic derivatives in the theory of differentiation. A large series of works was produced on the theory of integration (A Ya Khinchin, P S Alexandrov, and others); Let us especially note the general concept of an integral introduced by A N Kolmogorov.
A traditional topic of Moscow mathematicians was the theory of representability by trigonometric series. Here we note the following of the numerous results obtained: the construction by A N Kolmogorov of a summable function with an everywhere divergent Fourier series; his and A I Plessner's result that a trigonometric series with coefficients in the case of convergence converges almost everywhere, and a similar result by D E Menshov for any orthogonal sequences (with replaced by and with the impossibility of lowering the order of this factor, as proven by D E Menshov); A I Plessner's result on the relationship between convergence and summability of a given and conjugate series in the case of convergence of a trigonometric series; N K Bari's research on the uniqueness of trigonometric series; I I Privalov's result on the representability of conjugate functions by singular integrals; results by B M Gagaev and others.
Significant results in the theory of moments were obtained by M G Krein and N I Akhiezer. In the theory of almost periodic functions, the first generalisation of this theory was given by V V Stepanov.
The methods of the theory of functions of a real variable were widely used in various areas of analysis, and above all in the theory of functions of a complex variable. The works standing on the border of the theory of functions of a real and complex variable include the studies of D E Menshov on the conditions for the monogeneity of functions and a large cycle of works on the behaviour of a function on the boundary of a regularity domain (N N Luzin, I I Privalov, M A Lavrent'ev, V I Smirnov, G M Fichtenholz and others).
A large cycle of works was carried out by Soviet mathematicians on the geometric theory of functions of a complex variable. We note here the complete solution of the so-called rotation problem by G M Goluzin, the exact solution of the conformity radius by M A Lavrent'ev, the studies of I I Privalov, A F Bermant, and others.
Another direction is the study of entire functions, partly related to interpolation problems. Among the results in this direction, we note the works of V L Goncharov on the definition of a function from given values of it and its derivatives, the complete solution by A O Gelfond of the problem of uniqueness and construction of an entire function from a given countable set of characteristics, the research of M A Lavrent'ev on the structure of functions representable by polynomials, and some exact results of B Ya Levin on the relationship between the growth of an entire function and the distribution of its zeros. On the border with number theory are the studies of D D Mordukhai-Boltovsky on the hypertranscendence of the zeta function and N S Kotlyakov (summation formulas). Applications of the theory of functions of a complex variable to various problems of elasticity theory, aerodynamics and hydrodynamics and related technical problems were developed very intensively (Academicians S A Chaplygin, N I Muskhelishvili, V I Smirnov, N E Kochin, M V Keldysh, L I Sedov, and others).
The general theory of harmonic and subharmonic functions is adjacent to the theory of analytic functions. Here we note the research of I I Privalov, who extended a number of basic principles of the theory of harmonic functions to the theory of subharmonic functions, M A Lavrent'ev and M V Keldysh on the stability of the Dirichlet problem, M A Lavrent'ev on quasiconformal mappings.
Work on matrix calculus was carried out in various directions (I A Lappo-Danilevskii, M G Krein, F R Gantmacher, and others).
III. Analysis.
The focus of work on analysis in recent decades has been on the so-called boundary value problems of partial differential equations (sometimes vaguely called equations of mathematical physics). These questions play a fundamental role in the applications of mathematics to theoretical and applied physics, continuous medium mechanics, and a number of technical disciplines based on them.
In the first years of the revolution, one of the most prominent representatives of the old "St Petersburg" school was still working - Academician V A Steklov, whose students were V I Smirnov and A A Friedman. In the last decade, we have seen the increasing activity of the Leningrad school of young mathematicians trained by V I Smirnov. At the same time, mathematicians from Moscow and other centres were included in the range of these questions.
Let us move on to individual studies. S L Sobolev gave a new solution to the Cauchy problem for hyperbolic equations, based on the classical method of characteristics and having a number of advantages both of an algorithmic nature and in terms of the theoretical-functional generalisations obtained from it. S A Khristianovich obtained a similar solution to nonlinear hyperbolic equations.
I G Petrovsky conducted profound studies on the Cauchy problem for systems of differential equations. He identified a class of these systems, which can be called hyperbolic, for which the Cauchy problem has a solution in the region of non-analytic functions continuously dependent on the initial data, and a class of "elliptic" systems, to which the theorem of Academician S N Bernstein on the analyticity of solutions is extended.
We will not dwell here on the large cycle of works by V I Smirnov, S L Sobolev, E A Naryshkina on the so-called functionally invariant solutions of elasticity equations, already reviewed in "Advances in Mathematical Sciences", as well as on the works of N I Muskhelishvili on complex methods in elasticity theory, the works of S G Mikhlin, and others.
The theory of elliptic equations was also developed quite intensively. For example, the finite difference method for solving the Dirichlet problem (L A Lyusternik, I G Petrovsky), direct methods for solving polyharmonic equations (S L Sobolev), and the application of the integral equation method (N M Gunter, D I Sherman, V D Kupradze, and others).
In the field of parabolic equations, we note the works of I G Petrovsky, A N Tikhonov (in connection with problems of geophysics), and N S Piskunov.
In conclusion of the review of works on partial differential equations, we list the main directions in which their applications were pursued: the plane problem of hydrodynamics, a continuation of the classical studies of N E Zhukovsky and S A Chaplygin; problems of wave theory (N E Kochin, L N Sretensky, and others); problems of gas dynamics (I A Kibel, F I Frankl, S L Sobolev, S A Khristianovich); dynamic meteorology (A A Fridman, N E Kochin, I A Kibel); general problems of elasticity theory (G V Kolosov, N I Muskhelishvili and his school, S G Mikhlin); theoretical seismology (V I Smirnov, S L Sobolev, E A Naryshkina); theoretical hydrology (S A Khristianovich, B B Davison).
In connection with analytical works of an applied nature, work was carried out on numerical methods for solving problems of analysis (Academician A N Krylov, Academician S A Chaplygin, Academician N M Krylov, N N Bogolyubov, L V Kantorovich, D Yu Panov, S A Gershgorin, and P V Melentyev).
Very significant research was also carried out in the field of ordinary differential equations. First of all, we note the significant progress in the analytical theory of differential equations associated with the name of I A Lappo-Danilevskii. Using the theory of functions of a matrix variable developed by him, I A Lappo-Danilevskii obtained a number of algorithmic results on the main problems of Poincaré and Riemann. The works of I A Lappo-Danilevskii were continued by N E Kochin and the school of V I Smirnov.
A significant series of works is devoted to the qualitative theory of differential equations: A A Markov (study of almost periodic motions, introduction of an invariant measure in dynamic systems, etc.): V V Stepanov, A N Tikhonov, L S Pontryagin, Academician N M Krylov, N N Bogolyubov, V V Nemytsky.
In Kazan, a number of mathematicians continue Lyapunov's research on the stability of motion (P G Chetaev, I G Malkin, K P Persidsky).
Let us move on to integral equations. Many works in this area are related to problems of mathematical physics. N M Gunther, systematically using the concept of a domain function and generalized integral equations, obtained a number of results in classical potential theory, hydrodynamic equations (proof of existence).
Methods of integral equations were applied to problems of elasticity theory (S G Mikhlin, D I Sherman), diffraction theory (V D Kupradze).
M G Krein and F R Gantmakher investigated equations with the so-called Kellogg kernels, for which properties similar to those of the Sturm-Liouville equations were established.
V V Nemytsky, N S Smirnov, A N Tikhonov (general functional equations of the Volterra type) worked on nonlinear integral equations.
Academicians N M Krylov and N N Bogolyubov (effective direct methods, especially the Ritz method), A M Razmadze (discontinuous solutions), and L A Lyusternik (finite-difference and functional-analytical methods) worked on the calculus of variations. In the theoretical-functional direction, N N Bogolyubov and M A Lavrent'ev worked.
From the generalisation of the methods of the theory of differential and integral equations, the calculus of variations, the so-called functional analysis arose, the issues of which have been intensively developed in recent years in Moscow, Leningrad, Odessa and other cities of the USSR. We note the works of I M Gelfand on the theory of integration and differentiation of abstract functions, the cycle of works by L V Kantorovich and his students on the theory of semi-ordered spaces, the works of M G Krein on the theory of convex bodies and its application to integral equations, G M Fichtenholz on linear spaces with a given convergence, etc. Young people take a very active part in this work.
From the works on the theory of Hilbert spaces, we note the work of A I Plesmer on maximal operators.
IV. Topology and Geometry.
A strong school was created in the field of topology in the USSR, which put forward a number of first-class mathematicians and took a leading place.
The beginning of this school was laid by the works of P S Urysohn and P S Aleksandrov in 1921-24.
In the initial period, topology in our country developed exclusively as a set-theoretic topology, studying various properties of abstract spaces. Among the works of this period, we note the dimension theory, which became classical, constructed by P S Urysohn, the studies on the metrization of abstract spaces by P S Aleksandrov and P S Urysohn, and the works of P S Aleksandrov and P S Urysohn on the construction of a general theory of topological spaces. A significant number of then-novice mathematicians were involved in research work on topology - L A Tumarkin. A N Tikhonov (theory of compact spaces), V V Nemytskii, and others. Later, along with set-theoretic topology, combinatorial topology began to develop. Here, one of the strongest Soviet mathematicians, L S Pontryagin, came forward. But, what is very important, both branches of topology did not develop in isolation, but in mutual connection. P S Aleksandrov developed the theory of "projective spectra", which allows us to consider each compact space as, in some way, a limit of ordinary complexes. In particular, this theory made it possible to substantiate the theory of dimension from a new side and to transfer the theory of topological duality to closed sets. These ideas received their most recent development in the theory of discrete spaces of P S Aleksandrov.
A major achievement of Soviet mathematics is the final research of L S Pontryagin on the theory of topological duality. The significance of these works went beyond topology and they led L S Pontryagin to the discovery of a new algebraic apparatus - the theory of "character groups" of a given discrete or compact group. The latter found a number of applications, in particular in the work of A N Kolmogorov (the theory of intersection rings for any complexes and compact spaces). L S Pontryagin is the creator of the general theory of topological groups. Based on the theory of characters, he proved a general theorem on their structure, which, in particular, contained a solution (for the case of commutative groups) to Hilbert's fifth problem on the introduction of analytic parameters in a group.
Independently of L S Pontryagin, A A Markov obtained a characteristic of n-dimensional vector manifolds using a different method.
L S Pontryagin also solved E Cartan's problem on the Betti numbers of compact groups.
Let us now turn to the branch of geometry called "geometry in the large" and which studies the properties of a geometric figure as a whole, using topological methods along with differential-geometric methods. An example of such studies is the proof by L A Lyusternik and L G Shnirelman of the existence of three geodesics on convex surfaces of genus zero (a question first posed by H Poincaré).
The works of S E Cohn-Vossen are very interesting; he gave a complete study of the behaviour of geodesic lines on unbounded surfaces with positive Gaussian curvature (carried out by the author in the Soviet Union, where he emigrated from Nazi Germany). A D Aleksandrov obtained a number of results on the rigidity of surfaces and the generalisation of Minkowski's metric and differential-geometric theorems.
A significant number of works were produced on tensor analysis. In Moscow, this is being done by a group of mathematicians headed by V F Kagan [V F Kagan, P K Rashevsky, A P Norden, V V Wagner, G B Gurevich, G M Shapiro, Ya S Dubnov, and others]; in Kazan, by P A Shirokov. Here, they are working on both the apparatus of tensor analysis itself and on its applications to classical and multidimensional differential geometry.
Along with this, work continued on classical differential geometry (S P Finikov, D M Sintsov), on projective geometry (A K Vlasov, N A Glagolev).
Separately, we should mention the works of N G Chebotaryov (transfer surfaces) and I G Petrovsky (on ovals of algebraic curves).
V. Probability Theory.
In probability theory, we had the glorious traditions of Chebyshev, Lyapunov and Markov. Their research was worthily continued by Soviet mathematicians, who retained the leading place for Soviet mathematics in this area. The immediate successor of the research of the aforementioned classics of Russian science was Academician S N Bernstein. In his famous memoir (1926), the conditions for applying the Gaussian law to sums of dependent random variables were significantly expanded and the justification for applying this law to sums of vectors of any number of dimensions was given for the first time. In the same memoir, interesting applications to natural science were obtained.
In 1923-26, on the basis of the theory of functions of a real variable, the Moscow school of probability theory was formed. Here we note the asymptotic formula of A Ya Khinchin for the deviation from the mean position (the law of the iterated logarithm), the development of A N Kolmogorov's axiomatics of probability theory, etc.
The greatest achievement of the Moscow school was the fundamental research of A N Kolmogorov (1931) on the theory of random processes (this work, like a number of other works in this area, was published in the next issue of "Advances in Mathematical Sciences"). In this work, integral and differential equations were established for processes "without after effect", which for some important cases turn into ordinary parabolic equations. This was the beginning of the systematic use of analytical theory in the corresponding probabilistic studies. This theory was intensively developed by both Soviet and foreign scientists. In connection with it, new methods for proving limit theorems were obtained (S N Bernstein, I G Petrovsky).
Later, stationary random processes were studied in detail in the works of A Ya Khinchin (the most general formulation of Birkhoff's theorem) and E E Slutsky.
The study of continuous random processes led to the study of "random functions", i.e. probability distributions in functional spaces (E E Slutsky, A N Kolmogorov).
V I Romanovsky (Tashkent) gave a number of deep works on Markov chains. Of the general problems of mathematical statistics, the most successful were, firstly, the issues of statistical determination of hidden periodicities and, secondly, the issues of statistical determination of distribution functions. E E Slutsky and V I Romanovsky worked in the first direction, and A N Kolmogorov, V I Glivenko and, especially, I V Smirnov worked in the second.
Recently, classical questions of addition of independent random variables have also been developed (A Ya Khinchin, G M Bavli).
A number of works by Soviet mathematicians on probability theory arose in connection with important applications to natural science and national economic issues (meteorology, crop forecasting, a number of issues related to the problem of crowding, for example, automatic telephony, crystallisation of metals, etc.).
The above list of the main works of Soviet mathematicians, despite its incompleteness, nevertheless gives an idea of the scope of mathematical work in the USSR. At the same time, the significant majority of the works cited relate to the second decade.
The continuous growth of Soviet culture, the ever new problems that mathematics faces in connection with socialist construction, the incessant work of the Soviet government to develop science, the scientific enthusiasm of Soviet youth - all this is the guarantee of the further flourishing of Soviet science, and in particular Soviet mathematics.
Last Updated June 2025