Sobolev of the Euler School
On the centenary of Sergei L’vovich Sobolev, representative of the Russian mathematical school; his name is on the list of scientists whose creativity has contributed to the main intellectual treasures of the world
Mathematics studies the forms of reasoning. Generally speaking, differentiation discovers the trends of a process, and integration forecasts the future from trends.
The mankind of the present day cannot be imagined without integration and differentiation. The differential and integral calculus was invented by Newton and Leibniz. Euler used the concepts by Newton and Leibniz to bring up and cultivate the new mathematics of variable quantities, while making quite a few phenomenal discoveries and creating his own inexhaustible collection of miraculous formulas and theorems. Mathematical analysis remained the calculus of Newton, Leibniz, and Euler for about two hundred years.
The classical calculus turned into the theory of distributions in the twentieth century. As the key objects of the modern analysis are ranked the integral in the sense of Lebesgue and the derivative in the sense of Sobolev which apply to the most general instances of interdependence that lie beyond the domains under the jurisdiction of the classical differentiation and integration. Lebesgue and Sobolev entered into history, suggesting the new approaches to the integral and derivative which expanded the sphere of influence and the scope of application of mathematics.
Historic figures and discoveries deserve historical parallels and analysis. The gift of mathematics translates from a teacher to a student. The endless chain of alternating generations incarnates a mathematical tradition. Sobolev belongs to the school that originated with Leonhard Euler.“The еlder the school, the more valuable it is. The school is a set of creative techniques and traditions collected for centuries, verbal legends about scientists, passed away or alive, their manner of working, and their viewpoints on the subject of research. These verbal legends, which have been accumulated for centuries, are not intended for publication or communication to those who seem inappropriate—these narrations are treasures whose power can hardly be imagined or evaluated… If we try to find some parallels or comparisons, we can say that the age of a school together with the stock of its traditions and verbal legends is nothing but the energy of the school in implicit form” (Academician N. N. Luzin)
From Euler to Sobolev
Man is a physical object and as such can be partly represented by his world-line in the 4-dimensional Minkowski space-time. The longest part of the Euler’s worldline belongs to Russia. Born in Switzerland, Euler found his second homeland in Russia and is buried in St. Petersburg. Da Vinci of mathematics, he had become part and parcel of the Russian spirit. Our compatriots are proud to acknowledge Euler as the founder of the Russian mathematical school.
Euler’s efforts made Petersburg the mathematical capital of the eighteenth century. Daniel Bernoulli wrote to Euler: “I fail to convey to you quite properly how greedily they ask everywhere for the Petersburg memoirs.” He meant the celebrated Commentarii Academiae Scientiarum Imperialis Petropolitanae which became a leading scientific periodical of that epoch. The title of the journal changed many times and reads now as Proceedings of the Russian Academy of Sciences (Mathematical Series). The journal of the Petersburg Academy of Sciences published 473 Euler’s articles which were printed successively during many years after his death up to 1830.
At the turn of the nineteenth century, the center of mathematical thought shifted to France, the residence of Laplace, Poisson, Fourier, and Cauchy. The ideas of the new creators of mathematics were perceived by Ostrogradskii who studied in Paris after he was deprived of his legitimate Graduation Diploma of Kharkov Imperial University. The reputation of Ostrogradskii in France, as well as a few memoirs submitted to the Academy of Sciences, led to the recognition of his merits in Russia. Soon he became the undisputed leader of the Russian mathematical school.
Ostrogradskii, fully aware of Euler’s importance to the Russian science, vigorously raised the question of publishing Euler’s legacy. In the relevant memo, Ostrogradskii wrote: “Euler has created the modern analysis, enriching it more than all his predecessors and making it the most powerful tool of human mind.” A collection of 28 volumes was to be completed in 10 years, but the Academy found no finance either then or ever after…
Many other Russian mathematicians and mechanists were influenced by the research, teaching, and personality of Ostrogradskii. The Petersburg branch included P. L. Chebyshev, A. M. Lyapunov, V. A. Steklov, and A. N. Krylov. Among the students of Chebyshev we list A. N. Korkin and A. A. Markov, who taught N. M. Günter, the future supervisor of Sobolev’s graduation thesis. As his second teacher, Sobolev acclaimed V. I. Smirnov, a student of V. A. Steklov, who was supervised by A. M. Lyapunov. Such is the brilliant chain of Sobolev’s scientific genealogy.
Mathematics in Russia in the 1930s
Great discoveries are signposts of the inevitable which are not erected without efforts. Necessity paves way through the impenetrable timberland of random events. Sobolev’s contributions belong to the epoch of tremendous breakthroughs in the world science.
The twentieth century is rightfully called the age of freedom. Freedom is a historical concept reflecting the manner of resolving the clashes between the individuals loose in diversity and the tight bonds of their collective coexistence. The historical entourage is an indispensable ingredient of any triumph and any tragedy.
Science has traveled from an individual solution to studying the function spaces, operators between the spaces, and the elements that are solutions.
The issue of conditions necessary for these generalized solutions to be classical becomes a problem in its own right.
We see that Sobolev distinguished the close connection of his theory with the Hilbert idea of socializing mathematical problems. Hilbert’s methodology rested on the Cantorian set theory.“In 1951, Sergei L’vovich delivered a course of lectures on equations of mathematical physics at Moscow State University. His speech was so lively and fast that we could not follow it in writing. Students passed him messages: “Sergei L’vovich, speak slower, please.” He tried to slow down, but in ten minutes was carried away again. At the same time, he would never distract from the lecture. He was extremely disciplined, and clearly read the course based on his own textbook. We were never afraid to take his exam. It did not use to take him long to make sure than the student knew the subject—he would give grades immediately.
At that time, we already knew that Sobolev was a famous mathematician, but we had no idea of his commitment at the other job with the Institute of Atomic Energy.” (V. I. Lebedev, doctor of physics and mathematics, professor, chief researcher of the Kurchatov’s Institute Russian Research Center and Institute of Computational Mathematics, RAS)
The idea of revising the concept of solution of a differential equation was in the mathematical air of the early twentieth century. Sobolev’s interest in this topic is undoubtedly due to Günter. In the obituary by Sobolev and Smirnov, they emphasized the role of Günter in propounding the Lebesgue idea of the necessity of a new approach to the equations of mathematical physics on the basis of the theory of set functions.
Sobolev learned about the ideas of functional analysis at the seminar headed by Smirnov. The program of the seminar included the study of the classical book by J. von Neumann on the mathematical foundations of quantum mechanics.
The ideas of von Neumann attracted another participant of the Smirnov seminar, Leonid Kantorovich, who was a university friend of Sobolev’s. In 1935 Kantorovich published two articles in Doklady AN SSSR. His articles were written in the spirit of Friedrichs and contained the distributional derivatives of periodic tempered distributions.
It seems absolutely improbable that Sobolev and Kantorovich, old cronies and members of the same seminar, could have been unaware of each other’s articles which addressed the same topic. However, neither of the two ever mentioned the episode in the future. It becomes clear that the 1930s were the years of a temporary detachment between Sobolev and Kantorovich, who cultivated a warm and cordial friendship to their last days. The clue to this predicament is the political events of the 1930s in the mathematical circles of Leningrad and Moscow.
The “Leningrad mathematical front” was deployed against the old mathematical professorate of the Russian northern capital. Günter, leading the Petrograd Mathematical Society from its reestablishment in 1920, was chosen as the main target of the offensive. Günter was not only accused of all kinds of misconduct, idealism, and neglect of praxis but also branded as a “reactionary in social life” and “conservative in science.” The “Declaration of the Initiative Group for Reorganization of the Leningrad Physical and Mathematical Society” as of March 10, 1931, containing dreadful accusations against Günter, was endorsed by 13 persons, among them I. M. Vinogradov, B. N. Delaunay, L. V. Kantorovich, and G. M. Fikhtengolts. Günter was forced to resign as the department chair and had no choice but to write a letter of repentance, which was nonetheless condemned by the “mathematician-materialists.” To Sobolev and Smirnov’s credit, they abstained from the public persecution of their teachers.The antidote was the close affinity between the scientific views of the teachers and students.
“War time and the subsequent ten years were extremely difficult for our father. Stalin liked to work at night, and everybody had to adapt to his schedule. Meetings were usually held at 3 o’clock in the morning. Therefore, when Sergei L’vovich came home and had a short time to sleep, we walked on tiptoes because we understood that we should not disturb him. His sleep and work were sacred for our family.
If we were too noisy, Sergei L’vovich would never scold us. It was our mother’s responsibility, and she managed to calm us down quickly.
It was a pleasure for the father to spend his time with the children. There were seven of us, and he often dreamt that we would live in different cities when we grew up. Then he would be able to visit us, one after another, when he grew old. Unfortunately, this dream never came true. He felt bad when he returned to Moscow from Siberia and never went anywhere after that. Sergei L’vovich was a very kind man. If I did not like something he told me, I could say: ‘Oh, it’s nonsense,’ and he did not get offended (though many people in those days would find such manners arrogant), but proved that I was in the wrong. He was fairly democratic; the only thing that we were not allowed to do was to offend the mother. In this aspect, he remained adamant.
In addition, he hated squabbling. We, children, did not always manage to live without quarrels, but he would not seek for a guilty person. If two children were fighting, both were guilty! This was his clear position in the rare cases when he had to separate us.
He did not like it when people told lies. A lie is made of half-truth and looks nice. He hated that…
Sergei L’vovich used to compose verses. When we lived in different cities, he would always send us letters with his own poems. They were versatile, about the whole wide world. Sadly, they are all lost.”
(E.S. Soboleva, lecturer at Moscow State University, Moscow)
The situation in the mathematical community hardly differed from the routine of the epoch. The old professorate was pursued in Moscow too. Kantorovich refrained from any offensive against Luzin, whereas Sobolev, regretfully, became an active member of the Emergency Commission of the USSR Academy of Sciences on the “case of Academician Luzin.”
Omnipresent was the tragedy of Russian mathematics. So were its triumphs.
Sobolev and the bomb
The power of man is his capability of creating and transferring intangible valuables. Mathematics saves the ancient technologies of impeccable intellectual conjurations. The art and science of provable calculi, mathematics resides at the epicenter of culture. The freedom of reasoning is the sine qua non of the human being’s personal liberty. Mathematics put in the foundation of mentality becomes the guarantee of freedom. The creative contributions of Euler and his best descendants provide a great many excellent examples, and the fate of Sobolev is no exception.
In the twentieth century mankind came to the edge of the frontiers of its safe and serene existence, exhibiting the inability to halt the instigators of the First and Second World Wars. The weapon of deterrence arose as a warrant of freedom. Invention and production of the A-bomb in the USA and Russia demonstrated the tremendous power of science, the last resort of mankind’s survival. Mathematicians may be proud of the valor of their colleagues in these exploits. Von Neumann and Ulam participated in the Manhattan project. Sobolev and Kantorovich were involved in the Soviet project “Enormous.”
Most documents on nuclear weapons production have been declassified and published, so we can feel the tension of the heroic epoch.
The start of the atomic project in this country is traditionally marked with Directive No. 2352ss of the SDC which was entitled “Organization of the Works on Uranium” and dated September 24, 1942. A few months later, in February 1943, the SDC decided to organize Laboratory No. 2 of the USSR Academy of Sciences for studying nuclear energy. I. V. Kurchatov was entrusted with the supervision of the Laboratory as well as management of all works related to the atomic problem. Sobolev was soon appointed one of the deputies of Kurchatov and joined the group of I. K. Kikoin, who studied the problem of enriching uranium with cascades of diffusive membranes for isotope separation.“Many people who saw Sergei L’vovich would say that he was a handsome person. He was tall and energetic, and had a light step. His speech was always very clear, and he was known to be good in debates. It was difficult to argue with him, because most often he was right. At the same time, Sergei L’vovich was always amicable and respected the opinion of other people.
Sobolev was a brilliant populist and used to give talks to various kinds of audience. He happened to explain the fundamentals of functional analysis to secondary school students. Though he did not dwell on the most complicated aspects, he skillfully gave a very clear explanation of the role and significance of this field of mathematics!”
(Academician, doctor of physics and mathematics, RAS Adviser, Yu. G. Reshetnyak, Sobolev Institute of Mathematics, SB RAS)
The Special Folder saves the report by Kurchatov and Kikoin as of August 1945. The preamble of this document reads: “As regards the methods for acquiring the atomic explosives (uranium—235 and plutonium—239) known abroad, namely, the method of the “uranium-graphite boiler,” method of the “uranium—deuterium boiler,” diffusion method, and magnetic method, the top officials of Laboratory No. 2 (Academicians Kurchatov and Sobolev together with Corresponding Members of the Academy of Sciences Kikoin and Voznesenskii) opine that the Laboratory has already obtained the data on the first three of these methods, which is enough for designing and erecting the facilities.”“At one of his anniversaries, Sobolev was awarded with an order. In his response speech, he told with tears on his eyes: ‘I always doubt whether I deserve the honor that you pay to me’.” (S. I. Fadeev, doctor of physics and mathematics, professor, chief researcher of the Sobolev Institute of Mathematics, SB RAS)
As early as in 1946, the first gaseous compressors were made and put into the serial production. Tests on enriching uranium hexafluoride began. The work required solving a host of scientific, technological, and managerial problems, which became Sobolev’s main business for many years.
Sobolev joined both the plutonium—239 and uranium—235 groups. He organized and coordinated the work of calculators, dealt with the problem of control of industrial isotope separation, and was responsible for minimizing production losses. His role in the atomic project became more important.
The test of Joe—131 took place near Semipalatinsk at 8 a. m. local time on August 29, 1949. Exactly two months later, more than eight hundred staff members of the atomic project were decorated with various state orders. Sobolev was awarded with the Order of Lenin.
It was in the mid 1949 that Laboratory No. 2 was renamed to become the Laboratory of Measuring Tools of the Academy of Sciences, abbreviated as LIPAN in Russian. In the LIPAN Sobolev wrote the main book of his life, Some Applications of Functional Analysis in Mathematical Physics.
The atomic project enriched Sobolev’s scientific and personal potential. To his last days, computational mathematics occupied a prominent place in his creative activities. From 1952 to 1960, he held the chair of the department of computational mathematics at Lomonosov State University. Later in Siberia, Sobolev propounded the theory of cubature formulas, which is wondrous in the beauty of its universality. He synthesized the ideas of the classical approximative methods and distribution theory.
The work in the LIPAN added many bright colors to Sobolev’s views of mathematics. Those years made him understand that, in many cases, actual presentation of a reasonable solution on a deadline is more important than the abstract problem of existence of a solution.
New derivative—new calculus
Sobolev’s contributions are connected with reconsideration of the concept of solution to a differential equation. He suggested that the Cauchy problem be solved in the dual space, the space of functionals, which meant rejection of the classical view that any solution of any differential equation presented a function. Sobolev proposed to assume that a differential equation is solved provided that all integral characteristics of the behavior of the process under study are available. Moreover, solution as a function of time may fail to exist at all rather than stay unknown for us temporarily. In fact, science has acquired a new understanding of the key principles of prognosis.
As long ago as in 1755, Euler gave the universal definition of a function, which was perceived as the most general and perfect. In his celebrated course in differential calculus, Euler wrote: “If, however, some quantities depend on others in such a way that if the latter are changed, the former undergo changes themselves, then the former quantities are called functions of the latter quantities. This is a very comprehensive notion and comprises in itself all the modes through which one quantity can be determined by others. If, therefore, x denotes a variable quantity, then all the quantities which depend on x in any manner or are determined by it are called its functions.”
The generalized derivatives in the Sobolev sense do not obey the Euler’s definition of a function. Differentiation by Sobolev implies a new conception of interrelation between mathematical quantities. A generalized function is determined implicitly from the integral characteristics of its action on each representative of some class of test functions chosen in advance.“Certainly, Sobolev played an important role in the nuclear project. But why was he invited to the Institute of Nuclear Energy, being a specialist in an absolutely different field?
At that time, there were practically no electronic machines; instead of processors, calculations were performed by… young women, because they make less mistakes than men.
Imagine a room with many desks and girls sitting at these desks. Each girl had to do a certain number of calculation operations a day. It was hard work, and the girls did not even know the purpose of these calculations.
Sobolev was the ‘marshal’: it was he who divided a problem into fragments that were further partitioned into smaller pieces by his ‘generals’ and ‘colonels’. The tasks at the lowest level were rather simple: calculate this or that.”
“That was how he was used at that work. When he was asked: ‘Why were you chosen?’, he answered: ‘Others could have done that too, but I must have been better.’”
(M.D. Ramazanov, doctor of physics and mathematics, professor, chief researcher of the Institute of Mathematics and Computing Center, RAS Ural Branch)
The discoveries by Newton and Leibniz summarized the centuries-old prehistory of differential and integral calculus, opening new areas of research. The achievements of Lebesgue and Sobolev continued the contemplations of their glorious predecessors and paved the turnpike for the present-day mathematicians.
Sobolev was among the pioneers of application of functional analysis in mathematical physics, propounding his theory in 1935. In the articles by Laurent Schwartz, who came to the similar ideas a decade later, the new calculus became comprehensible and accessible for everyone in the elegant, powerful and rather transparent form of the theory of distributions, which utilized many progressive ideas of algebra, geometry, and topology.“At one of the parties, Sergei L’vovich was presented with a tie. It seemed to be a trifle, but he was so happy to get this present. He had many awards, regalia, and honors, but they were so formal that he did not take them to heart. Knick-knacks seemed to make him much happier. Once I accidentally heard his talk on the phone with a highly respected person. They were discussing some urgent problem. The tensions were growing, the discussion reached its extreme point, and Sergei L’vovich hung up the receiver. I asked him how much time he would needed to recover from this stress, and he answered: ‘Never mind. I’ll get back to work and I forget everything in five minutes.’”
(N. G. Zagoruiko, doctor of technical sciences, professor,
chief researcher of the Sobolev Institute of Mathematics, SB RAS)
Lavish was Sobolev’s appraisal of Schwartz’s contribution to the elaboration of the Fourier transform technique for distributions: “The generalized functions, in much the same way as the ordinary functions, can be subjected to the Fourier transform. We may say even more: In the classical calculus, the Fourier transform was confronted with many considerable difficulties such as divergence of integrals, impossibility of interpreting the resultant infinite expressions in a definite sense, and so on. The theory of generalized functions eliminated most of these difficulties and made the Fourier transform a powerful tool of analysis.”
Differential calculus of the seventeenth century is inseparable from the general views of classical mechanics. Distribution theory is tied with the mechanics of quanta.
We must emphasize that quantum mechanics is not a mere generalization of classical mechanics. Quantum mechanics presents a scientific outlook based on new laws of thought. Classical determinism and continuity swapped places with quantization and uncertainty. It was in the twentieth century that the mankind raised to a completely new comprehension of the processes of nature.
Similar is the situation with the modern mathematical theories. The logic of these days is not a generalization of Aristotle’s logic. Banach space geometry is not an abstraction of the Euclidean plane geometry. Distribution theory, reigning as the calculus of today, has drastically changed the whole technology of the mathematical description of physical processes by means of differential equations.
Sobolev heard the call of the future and bequeathed his spaces to the mankind. His discoveries have triggered many revolutionary changes in mathematics whose progress we are happy to observe and follow.
The terminal series of Sobolev’s mathematical articles was devoted to the subtle properties of the roots of the Euler polynomials…
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Kutateladze S. S., Sergei Sobolev and Laurent Schwartz, Vestmik RAN, 2005, Vol. 75, No. 4, pp. 354—359.
Johann von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955.
Dubinina L. G. and Ovchinnikova I. N. (eds.), Nikolay Petrovich Dubinin and the 20th Century, Nauka, Moscow, 2006.
Ramazanov M. D. (ed.), Sergei L’vovich Sobolev. Pages of Life in Contemporaries’ Reminiscences, Inst. Math. Comp. Center, Ural Branch RAS, Ufa, 2003.
Smirnov V. I.and Sobolev S. L., Biographical essay [Nikolay Maksimovich Günter (1871—1941)], in: N.Günter. Theory of the Potential and its Application to the Basic Problems of Mathematical Physics, GITTL, Moscow, 1953, pp. 393—405.
Sobolev S. L., Introduction into the Theory of Cubature Formulas, Nauka, Moscow, 1974.
Sobolev S. L., Selected Papers, Vol. 2, Inst. Math. SB RAS, Novosibirsk, 2006.
Fedoveev N. P. et al. (eds.), Philosophical Problems of Modern Natural Science, Izd. AN SSSR, Moscow, 1959.
Euler L., Foundations of Differential Calculus, Springer, 2000.
Lutzen J., The Prehistory of the Theory of Distributions, Springer, New York etc., 1982.
Schwartz L., A Mathematician Grappling with His Century, Birkhauser, Basel etc., 2001.
The editorial board and the author are grateful to E. S. Soboleva; Press Secretary of SB RAS Presidium O. V. Podoinitsyna; librarians of the Institute of Mathematics, SB RAS; and director of the publishing house T. N. Rozhkovskaya for their assistance in preparing this publication.The paper involves materials from the archives of S. L. Sobolev’s family, Sobolev Institute of Mathematics SB RAS, and SB RAS Presidium