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Nikolay Luzin, his students, adversaries,and defenders

(notes on the history of Moscow mathematics, 1914-1936)

Yury Neretin

This is historical-mathematical and historical notes on Moscow mathematics 1914-1936.

Nikolay Luzin was a central figure of that time. Pavel Alexandroff, Nina Bari, Alexandr

Khinchin, Andrey Kolmogorov, Mikhail Lavrentiev, Lazar Lyusternik, Dmitry Menshov,

Petr Novikov, Lev Shnirelman, Mikhail Suslin, and Pavel Urysohn were his students. We

discuss the time of the great intellectual influence of Luzin (1915-1924), the time of decay

of his school (1922-1930), a moment of his administrative power (1934-1936), and his fall

in July 1936.

But the thing which served as a source of Luzins inner drama turnedout to be a source of his subsequent fame...

Lazar Lyusternik [351]

Il est temps que je marr ete: voici que je dis, ce que jai declare, etavec raison, etre inutile a dire.

Henri Lebesgue, Preface to Luzins book, Lecons sur les ensemblesanalytiques et leurs applications, [288]

, , ?

Alexandr Pushkin Poltava

There is a common idea that a life of Nikolay Luzin can be a topic of a Shakespearedrama. I am agree with this sentence but I am extremely far from an intentionto realize this idea. The present text is an impassive historical-mathematical andhistorical investigation of Moscow mathematics of that time. On the other hand, thisis more a story of its initiation and turning moments than a history of achievements.The notes contain English overview and the bibliography. The rest is written inRussian.

1. Overview (English)2. Was Luzin a victim of political persecutions?3. Moscow mathematics and Luzins thesis4. Luzins school: the creation and the chain reaction5. Decay6. The fate of set theory7. Mathematics and philosophy8. Students attack the teacher9. Over the top10. Young workers, left mathematicians, and red professors11. The year, which disappeared from annals

1

http://arxiv.org/abs/1710.10688v1

2

12. From a dialectic republic to the MechMathAddendum. Few remarks on historical context

13. In the next inning14. Historians and mathematicians15. Playing with fire16. After the shock-wave17. Autumn of Luzin18. Mathematics for non-mathematicians.19. AfterwordBibliography

1. Overview

1.1. Overview. In 1914 Nikolay Luzin1 (9.12.188328.2.1950), a student ofDmitry Egorov, returned to Moscow after a 3-year trip to Gottingen and Paris.In 1915 he published the thesis Integral and trigonometric series and in a shorttime initiated a mathematical chain reaction (a sentence of Lavrentiev [279]) inRussia. Since the fall 1914 he gathered a group of young mathematicians for awork in theory of functions of a real variable and in descriptive set theory. Thegroup included (in different periods of time) Luzins friends Vladimir Golubev, IvanPrivalov, Wac law Sierpinski, Vyacheslav Stepanov and Luzins students Pavel Ale-xandroff, Nina Bari, Alexandr Khinchin, Andrey Kolmogorov, Mikhail Lavrentiev,Lazar Lyusternik, Dmitry Menshov, Petr Novikov, Lev Shnirelman, Mikhail Suslin,Pavel Urysohn (also, Valery Glivenko, Lyudmila Keldysh, Alexey Lyapunov, andsome others). During the fall 1920-1921 the group was called Lusitania, in thehistorical-mathematical literature this term usually is extended to the whole Luzinsschool 1915-1935.

It is commonly recongnized that Luzin was the founder of the famous Moscowmathematical school of XX century and that Luzin and Lusitanians reached in 1920-30s a great mathematical breakthrough. Quite soon the domain of investigationswas extended to probability (Khinchin, Kolmogorov, Glivenko), topology (Alexand-roff, Urysohn, Pontryagin), number theory (Khinchin, Gelfond, Schnirelman), fluiddynamics (Lavrentiev, Golubev, M. Keldysh), variational and functional analysis(Lyusternik, Shnirelman, Kolmogorov, Raikov, Gelfand), differential equations(Stepanov, Petrovsky, Lyusternik), logics (Kolmogorov, Glivenko, Maltsev, Novikov),theory of function of a complex variable (Golubev, Luzin, Privalov, Lavrentiev).Menshov and Bari (and ocassionaly other Lusitanians) continued investigations intheory of functions of a real variable. Luzin himself with Novikov, L. Keldysh,Lyapunov continued works in descriptive set theory. Below we discuss this process,points of its branching, and the roles of Khinchin, Alexandroff, and Kolmogorov inthis development.

Starting late 1920s - early 1930s we observe the first students of Luzins studentsas Lev Pontryagin, Alexandr Gelfond, Anatoly Maltsev, Andrey Tikhonov, AlexandrKurosh, Mstislav Keldysh (but a continuation of the story is beyond the scope ofthese notes).

1The complete name: Luzin Nikolay Nikolaevich, Russian: (inthe old Russian orthography: ), the French transliteration usedin 1920-30s was Lusin Nicolas.

3

However, Luzins group decayed after a series of mathematical and personalconflicts in 1922-1930. Apparently, this process must be regarded as natural andpositive, but conflicts between Luzin and his students were unusually hot.

Let us say few words on another side of mathematical history of that time.Many Soviet scientists who became widely known as pure mathematicians parallellyworked in development of high technologies and in applications of mathematics fornatural sciences. This was one of reasons of favor of mathematics in the Sovietsociety. For mathematicians parallel activities gave a possibility for a wider view toscience. On a social level this gave to individuals certain degrees of independenceof a pure mathematical establishment. Also, this provided many positions for ma-thematicians.

Apparently, this process was initiated in the famous TsAGI (The central Aero-Hydrodynamical Institute), which became an initial point of a development of theaero-cosmic industry of the Soviet Union. The Institute was founded in December1918 by the mathematician, plane designer and professor of mechanics Nikolay Zhu-kovsky, he was enlisted by the aerodynamicist and mathematician Sergey Chaplygin.In the mid-20s, the complex analyst Golubev joined to aerodynamical and aero-mechanical investigations. In 1933, Golubev became the first dean of the famousMechanical and Mathematical department of the Moscow State University (belowMechMath)...

All biographies of Luzin published in 1950-1985 forgot his employment in TsAGI(1930-1936?). However, in this story, we will observe several well-known technologistsand applied mathematicians who supported Luzin in heavy moments of his biographyor mantained Luzins memory after his death2.

The time of our story was complicated and stormy. A picture of the Soviet politicsof 1920-30s contains a dense sequence of turning point, mistakes, crimes, correctionsof mistakes, apologizes to suffered, new mistakes, new crimes, big catastrophesand big successes. If to draw this politics as a graph on the paper, we get aconstructive proof of existence of a nowhere differentiable function. There were twopost-revolutionary tsunamis. The first was called The Year of the great break, itwas happened in 1929-1930. The second tsunami came suddenly in MayJuly 1937and finished in SeptemberNovember 1938 (in Soviet Union, this was called TheYear Thirty Seven).

Some Lusitanians were political radicals, in particular they participated in theGreat break and a local revolution in the Institute for Mechanics and Mathematicsof the Moscow University (1929-1930)...

At the first glance, at the end of 1920s fortune favoured Luzin. In 1927 he waselected as the corresponding member of the Soviet Academy of sciences, in 1929 as areal member. In 1928 Luzin was a vice-president of the International MathematicalCongress in Bolognia. In 1930 he published the monograph on descriptive set theory,Lecons sur les ensembles analytiques et leurs applications, which exposed researchesof Luzin and his students Suslin, Alexandroff, Novikov, Lavrentiev, Keldysh, Seli-vanovsky. In the same 1930 there appeared the first version of Luzins textbook onDifferential and Integral calculus for technical universities, later it it was revised

2In particular, Alexey Krylov, Sergey Chaplygin, Viktor Kulebakin, Alexandr Nekrasov, LeonidSretensky.

4

many times (13 editions 1930-1942, the total size of editions was 270 000, and 7editions of a completely new version 1946-1961, the total size 550 000). Apparentlyduring 25 years Luzins books were the most popular mathematical text-book inSoviet engineering education...

However, in mid 20s Luzin lost his significance as an intellectual leader of Moscowmathematics. It came the time of Khinchin, Kolmogorov, Alexandroff, Pontryagin(Alexandroffs student), Gelfond (a student of Khinchin and Stepanov)... Luzin didnot understand new mathematics, which was created in Moscow in 1920-30s.

After a long trip abroad, Luzin returned to Moscow in 1930 and met variousproblems in 1930-1933: a crisis of his own scientific program, conflicts with formerstudents, semi-fantastical situation in PhysMath Department of Moscow University3

in 1930-1931, heavy illnesses... However, after a reform of the Academy of Sciences1933-34, his formal social weight started to grow...

In March 1935 Luzin was confirmed as the President of the Mathematical groupof the Soviet Academy Sciences. We must explain the sense of this assignment.

First. The Academy of Science of the Soviet Union was a high collegium, whichleaded a development of sciences and high technologies. The position of an academicianwas life-long. A social status (and salary) of an academician was very high duringthe whole Soviet time. At that time the number of members of the Academy ofSciences was small (in 01.01.1936 the number of real members was 98 persons,this covered all natural and humanitarian sciences and also high technologies, eachperson had a great weight).

Second. In 1930-40s a person on an important position had a wide freedom ofinitiative and a big influence (but he was seriously responsible for results of hisactivity and his solutions).

In fact, Luzin became the main mathematician of the Soviet Union. He wasdeemed unfit this position and his activity implied a negative reaction of a majorityof Moscow mathematical community.

Unfortunately now the name Luzin in the common opinion is associated mainlywith the story of his fall in 1936 (and this story is painted in too bright colors), forthis reason we have to analyse it in these notes (Sect. 89, 15) and in the Overview.

Alexandroff, the main opponent of Luzin4, attempted to expel Luzin from theAcademy. Recall that a position of an academician was life-long. However, theAcademy charter included an item claiming that an academician can be expeledif he does not carry his duties or if his activity harms the Soviet state5. Alexa-ndroff decided to incriminate to Luzin numerous scientific-ethical and scientific-organizational accusations. Serpinski, who was a friend of Luzin, knew about Ale-xandroffs intention in July 1935 (sic!).

3We discuss this story below, due to a next turn of the politics of the Government and effortsof Alexandroff, Khinchin, Golubev, and others, in 1932-33 a situation was quickly normalized and1930s were a time of florescence of mathematics in Moscow.

4They had a heavy private conflict, details were hidden.5This item was introduced to the charter under a pressure of the Goverment. Literally: -

, -, , .

5

In 3 July 1936 the main Soviet newspaper Pravda published an anonymouspaper againist Luzin, apparently (for a detailed discussion see below Subsections8.28.8), the attack was initiated by Alexandroff (but certainly he was not an authorof the paper, on the other hand the text reflected points of view of some othermathematicians). This was a signal for a general offensive. The Academy organized aspecial Committee on Luzin affair that worked 7-15 July 1936. In the stenographicrecord of the Committee, former Luzins students Alexandroff, Khinchin, Kol-mogorov, Lyusternik, Schnirelman and other mathematicians (including AlexandrGelfond, Otto Shmidt, and Sergey Sobolev) tried to prove that Luzin was hardenedin sin. At the same time, Luzin was attacked by mathematicians on meetings inthe Steklov Institute and the Moscow University (in particular, by Lev Pontryagin,Felix Gantmakher, mechanician Nikolay Buchholtz).

Accusations of Alexandroff and Kolmogorov were scientific and ethical6. It issad to see that some participants of the attack pronounced speachs in a spirit of apolitical mistrust and invented new political charges.

Our knowledge of the future in past shows that this was a playing with fire.The expelling of Luzin from the Academy in 1936 could produce extremely heavyconsequences in the next, Thirty Seven, year. However this future was unknownto mathematicians and politicians (as main heroes of this year Nikolay Ezhov,Joseph Stalin, Robert Eihe, Efim Evdokimov, Nikita Khrushchev, Stanislav Kosior,Levon Mirzoyan, Pavel Postyshev, Mikhail Frinovsky...). Certainly, all7 participantsof the attack had no intention to imprison Luzin and could not imagine such anopportunity (the Code Penal ot that time was rigid but the list of crimes of Luzinwas outside it). But a danger of a further development of Luzin affair in a politicaldirection was serious even in 1936 (see a discussion below). Now it is known thatthe editor-in-chief of Pravda Lev Mekhlis8 was displeased with the final resolutionof the Academic Committee (supported by higher authorities), since its statementswere essentially softer than Pravdas clams.

Sergey Bernstein and the naval engineer Alexey Krylov capably defended Luzinin the Academic Committee. Not all Lusitanians joined to the attack. We knowthat Menshov and Bari resisted openly. Apparently, their reactions were predictable.Initiators of the attack reckoned on a joining of Lusitanians Novikov and Lavrentiev,who were offend by Luzin. They avoided a joining, as a result accusers could notprove one of their key ethical charges.

Obviously, the mathematical squabbling was potentially dangerous for the wholeAcademy9. We know that several powerful scientists tried to stop the dangerousdevelopments.

6According the Stenographic Record, a participation of Kolmogorov was minimal, Alexandroffseveral times defended Luzin from political accusations.

7All but one...8Mekhlis (1889-1953) later became famous as a pitiless, fanatical (and honest a certain sense)

figure. In 1940-1950 he was a minister [] of the Soviet Control (this ministry was the mainanti-corruption structure in the Soviet Union until 1950), in 1938-9.1940 and 7.1941-5.1942 hewas the main political commissar in the Red Army. After mid 50s he became one of referencedemons in expositions of the Stalin period of the Soviet history (apparently, his negative fame isa bit stretched). In 1936 Mekhlis was not a person, who solved the problem under the discussion,however there was a potential danger of his influence.

9From the letter of Vladimir Vernadsky to Alexandr Fersman (the most famous Russiangeologists of that time): we will go down the sloping plane.

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Finally, Luzin lost his position of the main mathematician, in 1939 Kolmogorovoccupied it. Certainly, the both events were positive for mathematics. Fortunately,Luzin was not expelled from the Academy. The anti-Luzin company had a characterof a shock-wave. Its aftershocks were minimal (but they were strongly overestimatedin the common opinion after 1991).

Reading documents of 1936, we observe lot of famous respectable persons onthe both sides of the front. Later the Soviet mathematical community tried toforget this story. Positions of the both sides seem clear. The party of winners(which was more powerful) had reasons to hide details (see below quotations of theStenographic record of 1936). On the other hand, Luzins side preferred to forget theunhappy Luzins mathematical kingship 1935-1936 and numerous preceding heavyconflicts with Luzin participation. This produced a white gap in the literaturearound Luzin10. However, several papers commented a prehistory of 1936 werepublished by Lyusternik (1967), Lavrentiev (1974), Alexandroff (1979-1980).

A hidden dissent around Luzins name was stopped by Kolmogorov only afterthe death of Alexandroff (1982). During the next three years (1983-1985) the Sovietmathematical community widely celebrated the centenary of Nikolay Luzin. Thiswas a tardy but normal and human end of the story.

In my opinion, an intellectual history of the Moscow mathematical school 1915-1940 is more interesting and more pleasant topic than a story of a heavy scientificand personal conflicts.

Unfortunately, the end in 1983 was not an end.

1.2. Historical sources of Luzin affair. Our story before 1930 is a usualhistory of mathematics (with its own problems and tragedies). However the conflictof 1936 is a problem of both history of mathematics and history in the usual sense.It requires an investigation from the both sides.

Historical sources about 1936 that are available now are not perfect. But thecollection of sources is huge and it gives a possibility of investigation by logicaltools of historicas sciences.

The list of historical sources contains the followng groups of texts.

a) The Stenographic Record of the meetings of the special Committee of theAcademy of Sciences 7, 9, 11, 13, 15 July 1936 on the Luzin affair, [1]. It is a 200-pages battle scene of 1936 containing lot of records about Moscow mathematicallife 1911-1936.

b) Papers of Lazar Lyusternik [348][352], which are a mixture of recollectionsand a historical investigation. The author carefully avoids the dangerous 1936

10Of course, an oral tradition existed. The author of these notes enrolled at MechMath in 1975.Quite soon, I observed that all speeches about Luzin are ultra-positive but some worthful facesbecame wry hearing this (by mathematical reasons I joined to doubters). I remember sentencesthat Luzin was very powerful and was dethroned by Alexandroff in 1936. I remember approximatequotations , (fromPravda) and from Alexandroff (in Subsect. 2.3.2 we observe the true originof this sentence). There was a general certainty that after 1936 foreign publications of Sovietmathematicians were stopped (it is not a true, they were interrupted later by other reasons). Ihad no interest to learn more.

7

year. However, he tell us a lot about its prehistory. Apparently, [351] presented anofficial version of the leading group of Moscow mathematics of 1960s.

c) The biography of Luzin [449] composed by Nina Bari11 in 1950. It was discoveredat the end of 90s and published by Anna Tyulina in 2007.

d) Recollections of participants of the conflict, Pavel Alexandroff [65], [66] andLev Pontryagin [395], [394]. Recollections of beholders, Vadim Efremovich [125] andBoris Gnedenko [167].

e) Reminiscences of Mikhail Lavrentiev-jr., the son of Mikhail Lavretiev, [282],and of Sergey Novikov, the son of Petr Novikov and Lyudmila Keldysh, [389]. Also,late reminiscences of Alexei Gladky [163], a student of Petr Novikov.

f) Letters of Wac law Sierpinski, Henri Lebesgue, Arnaud Denjoy of 1936 publishedby the French historian Pierre Dugac [123] in the Russian periodical Istoriko-Matematicheskie Issledovaniya. Also, a letter by Andre Weil to Dugac, 1978 (in[123]). It seems that these letters are known only in Russian translations12. Also, acomment [469] of Andre Weil to one of his papers.

g) Papers in the Soviet political and scientific periodicals of 1936-1937, namely,Pravda [19], Russian mathematical surveys, [22], [23], Bulletin of the Academy ofSciences [32], Front Nauki i techniki [3][4].

h) Two letters of Lev Mekhlis, the editor-in-chief of Pravda, to Stalin andMolotov of July 1936, published in [111].

i) Several letters of 1936: Nikolay Luzin Aleksey Krylov 12.02.1936 [130],Nikolay Luzin to high authorities July 1936 [111], Petr Kapitsa Vyacheslav Molotov06.07.1929, [194], Vladimir Vernadsky Alexandr Fersman 03.07.1936, 07.07.1936[460], Nikolay Nasonov Alexandr Fersman 07.07.1936, [111], and Sergey ChaplyginVladimir Vernadsky, 11.07.1936 (see [483], [461]). I do not think that all letters ofthis kind were discovered by historians.

j) Vladimir Vernadsky diary [462].

k) Numerous papers about Luzin, 1946-1985 (biographies, reminiscenses, surveysof works, etc.). This family of texts naturally splits into several subfamilies:

Two texts 1946-1948, a congratulation paper [27] and brochure [129].

Obituaries 1950, [5], [30], [257], [17].

Memorial papers, 1951-1953, [170], [85], [86], [168], [216], [138], [436], [338],also a preface [388] published in 1958.

Ocassional papers 1960 -1966, [383], [84], [374], [267], [404]

Papers of 1974 in occassion of Luzins 90th birthday, [279], [270], [214].

Two papers outside a classification: paper of Alexandroff [63]-[64], 1977, andrecollections of Menshov [370], 1983.

Papers of 19831985 after Kolmogorovs turn, [271], [91], [371], [416], [242],[33], [456], [271] (a collection of papers), [455], [193], [451], [452], [372], [455], [6],and also [453], [199], [450].

11The typewritten manuscript was not signed. However, the autorship is doubtless, it isexplained in the introductory paper of the editor.

12There is the Polish translation from Russian [121].

8

Electronic archives of the Siberian branch of the Russian Academie of Sciencescontain three typescripts of Lyapunov [342], [341], [343], which were not publishedin that time, and unpublished typescript of Petr Kuznetsov [268].

At a first sight, this collection of apologies of Luzin contains nothing about 1936.This is not the case!

l) In 1966 Ivan Petrovsky (who was a rector of the Moscow State University atthat time) initiated a collecting of audio interview of Soviet scientists on history ofthe first third of XX century. In 2004 several mathematical interview (including Ale-xandroff, Boris Delauney, Lyusternik, Menshov, Pavel Yushkevich) were publishedin [272]. They do not contain revelations, however they are interesting from ourpoint of view.

1.3. The historiography. An epic scientific history of Moscow mathematicsof 1910-1940 was firstly represented by Boris Gnedenko [165], and by Alexand-roff, Stepanov, Gnedenko in [69], 1948. Later there were many other expositionsdeveloping this paper and its details. The work [349][352] of Lyusternik was a kindof a monograph split into a series of papers (also his joint works with A. F. Lapko[274][276] are important). In general, we follow the traditional scheme (but interruptour expostion imeddiately after the start of the mathematical chain reaction), inaddition we discuss a heavy pressure of non-solvable set-theoretical problems toLuzin and his group after 1925 and the opinion of Luzin on this topic (in early 40sNovikov broke through to logic but this is far outside our notes).

The second part of our bibliography [496][682] contains biographical referencesabout Russian mathematicians of 19001936.

Numerous biographical and semi-biographical papers about Luzin were publishedin 1950-1985 (the list is Subsect. 2.72.8). However these works carefully avoiddetails of Luzins biography after 1929. Apparently, this idea belonged to Golubev,it was perfectly realized in the Luzin biography composed by Golubev and Bari[170] 1951 (a type-written outline of Bari for that paper was accidentally preserved[449], it is the only known non-self-censored big text about Luzin written by acontemporary). This produced a big white piece in Luzins biography (even nowthe formal CV of Luzin is not completely clarified).

After deaths of all participants of the squabbling of 1936, due the Restructing(the Perestroika, which also was a restructing of the unpredictable past of Russia)a tragical intellectual hero Luzin was transformed to a person of political history.

In 1989 the well-known historian of mathematics Adolf Yushkevich (19061993)published a paper The academician N. N. Luzin affair [482]. It was the firstpaper telling about events on the Physical-Mathematical department of the MoscowUniversity in 1929-193113. Yushkevich said that the story of 1936 was inspired bythe Government and this was a lesson of Stalin for the Soviet intelligence. Claims

13This story was officially hidden due to the fall, arrest, exiling, and death of Dmitry Egorov(the Soviet State of 1950-70s criminated polytical arrests that happend starting December 1934,Egorov was arrested in 1930). Recall that Egorov was the adviser of Luzin and Ivan Petrovsky, theauthor of the Egorov theorem, and the main executive in Moscow mathematics in 1923-1929. Thestory was also non-officially prohibited due to the roles of several distinguished mathematicians inthese years (and also due to the roles of founders of the Soviet school of history of mathematics).The fall of Egorov is discussed in Subsections 11.4, 11.9, 12.8. We relatively confidently can showthat the arrest of Egorov was not related to his professional activity and professional problems.

9

of this type were obligatory for texts published during the Perestroika. In general,Yushkevich presented a good exposition of some events around mathematics.

In 1991 he published under the same title The academician N. N. Luzin affaira significantly different paper [483] (see a comparison in Subsect. 14.9 and 14.10).In particular, he claimed that Luzins story was supervised by Arnost Kolman14

and draws a picture of semi-innocent mathematicians. For the latter purpose heassiduously selected quotations from the Stenographic report [1], which was notpublished in that time...

The stenographic report was published in in 1998, its publication required anew transformation of the non-predictable past. This was done by the historian ofmathematics Sergey Demidov and the archivist Vladimir Esakov in the paper Thecase of Luzin in a collective memory of scientific community. [110]. Now Alexand-roff, Gelfond, Khinchin, Kolmogorov, Lyusternik, Schnirelman were transformed toyoung friends and ideological team-mates of Kolman who participated in an intrigueof Kolman and Lev Mehlis and in the Stalin lessons for scientists. The collectivememory of scientific community before [110] never heard such things.

Numerous later historical texts on Luzin affairs usually are popularization of[483] and [110].

New works about Moscow mathematical conflict of 1936 were published by SemenKutateladze [262][264], he returns to the traditional idea of an attack to Luzin fromthe mathematical community.

The author of the present notes have to analyze logical constructions of [483] and[110]. We also use numerous historical sources published during the last 20 years (theDugac collection [123], Vernadsky diaries [462], the manuscript of Bari [449], andPontryagin [395], Efremovich [515], Lavrentiev-jr. [282] recollections), which confirmthe traditional point of view existing before 1989. We accept an essential part ofKutateladzes logical argumentation. However, in my opinion he completely losesthe scientific side of the conflict, also his historical view contains some aberrationsrelated to a linear interpolation of the nowhere differentiable function mentionedabove.

These notes contain lot of quotations. They have different nature. On one handwe are trying to tell history by voices of its participants and creators, on the otherhand we must discuss certain non-obvious cases and this requires quotations foranalyzing of information.

14An adventurer with a little-investigated biography (18921979), see below Subsect. 10.8,14.12. An Austro-Hungarian citizen, got into the Russian captivity, in 1917-18 joined to Bolsheviks.An aggressive Marxist philosophical publicist in 1930s. In 1931 - early 1932 he occupied the positionthe President of the Association of Institutes for Natural Sciences of the Communist Academy.There were few institutes in the Association, but this position in 1931 was quite serious. TheAssociation finished its existence in 1932 (and 1932 was a beginning of the end of the CommunistAcademy itself). He was the head of the Science Department of the Moscow City Committee ofthe Communist Party in Spring (?) 1936 Spring 1937. He leaved memoirs [232], I comparedhis stories of 1920-30s with some formal historical facts and immediately observed lot of obviousbragging (see Subset. 10.8).In the modern historical literature he is regarded as one of the main ideologists of the Stalintime. However in [1] we read two ireful speaches of Gleb Krzhizhanovsky (a companion-in-armsof Lenin, a power engineer, a member of the Central Committee of the Communist Party, anda vice-President of the Academy) addressed to Luzin with the following sense: Who can believethat you, a Soviet academician, afraid of Raikov, Kolman, and Kagan?? This is irresponsibly! SeeSubsect. 14.13. Apparently, such claims indicate the real level of Kolman in a social hierarchy.

10

Now many historical sources are easily available in the Internet. The most ofRussian mathematical journals are available via the cite math-net.ru created bythe Steklov Institute

http://www.mathnet.ru/index.phtml?&option_lang=eng

Papers from other mathematical journals 1910-1940 usually can be easily find in astandard way.

A collection of sources related to the history of Russian mathematics of 1920-30now is present on my cite in Vienna:

http://www.mat.univie.ac.at/~neretin/misc/luzin/notes.html

I met lot of unexpected data in new historical sources during all time of work onthese notes, these data broke some my conjectures, so I do not think that this cannot happened again.

2. ?

. , . . 1936. . , 19301931 - (. . , . . , . . -). . . 30 ( ). - -? ?

2.1. , 1931-1936. . . . . [110] - 1930 . - . . 1931 ( .10.8, 14.12, - .14.16). 1936 ... 1930-1936.

16.12.1930 . 19291934 . -

15; 6.02.1931 - [2]; 19341937 . (), [252]. 1934-1936 - - , [449] ( , - .). -, - 16.

20.11.1930 ( ), 1936, . [1] .

15 1934. - . () (), - .

16 [449]: . . 1934. , .

11

1930 17- , , -, -. 1930, 1930, 1930, 1931, 1933, 1933, 1934, 1934,1935, 1935-1937; 1937, 1938, 1942, 33 - 270 000 - [248], 1931 . 1936 ( 7 18 [273]).

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23 1932 . , 25 - . H. . . .

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19 . . [246]. .

12

1933, 1935 , [315], 60 ., [319], 88 .

, 1935-1936, . . 20.

1934-36, [110]( - ). , 1918.21 1934. 1934-1936 , - ( ). , .

22 1935 - 1936, , .., . - 5 , 23

(. .8.3).

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2.2. , 1936-1950. , . . [263]:

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13

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, 318pp, [323]. , 1948.

, . (, ) . [449], . . [451], . . [483] - 1941 . . . [386], 1943 , [252] 1946 . -, , 1941 [449]. , 1947 . . . -, 1950. , , 1947 ().

1943-1947 . 1944-1950 . . 1945 , [30]. 1946.

, . - 1949, 1952, 1953, 1955, 1958, 1961.

1948. . .

28 1950.

.

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. . . [] - (1911), - . . . , - - . 1915 , , , . . ,. . , , . . (18941919) . . (18981924).

1947 - 30 ( +). -

14

. 1946 . . - . . 1948 . . . . -- . , . .

2.3. . - 1936.

1) , . [131], 24 : , , -, 19 1937. ( ):

... , . , -, . , , , .

- - . , - , , - .

.

2) , [246]:

, 28 1937 . ( )

, , , . , , , , , , -, , , , , . ., , . ., . . ....

, . , -, . , ( - ) , . ( ,

24 , -, - .

15

, , , , ). .

, [53] - - - , 1937, 8-9. , - , 1936. 15:

, . - , - ( - , - , ..).

. a [ ] - - .

, - , , . - , , , , , . , ( ) - ( ). , , , , , , - a [ ] .

- . . , . . . , ( ) . , - . , , ( ) .

(, -, ). 39 (585-591) -, , , :

16

- : (.), , (.), , , .

, - : 30 , . , ( , , . .9.6). , , .

1936. , . -, , 25.

1938. . . [378]

- , .. .

, . , , , . .

, ( ) (1938-1939.) (. 102).

3) , - . 1937 [22] ( ). [24] ( 1938.) 20 . 4 , . . - - 4, -3, -5, - 9. .

1938. 20. . . . . , [140]. ( , ).

4) : .. . , .-. . . , . . . . ( ) () 20.07.1936.

5) -, 1940. [2] 2.09.1938 1.08.1939.

25 ( , - ), , 1937. [360]. , , ( [232] , - , .., 1937.). , , .

17

. . ([]), 20.10.1937. ( , ).

-, 1936. -, (, , - ). , , , . ...

2.4. . - .

, , -, . -, . 1936-1953 ( ), . , .

(1880-1968) (1942), 2 (1945, 1953), - (1944).

26 (1863-1945) , 1941. 3 (1939, 1943, 1945). (1943). 1951-1956. 12 ( 17).

, 1951. , 1941-1977.

(1901-1961) .

:

(1901-1975). 1945. - .

(1900-1980). 19391941. 19451948. . 1946, 1949. II 1944. 1945, 1948, 1953; 1953.

(1869-1942),

. , 1941. 1942 . . 1948 .

-, ( - ) . , 27. , .

26 -. . 1898 . 19081910 . . . - . . ..

27 , . , . . [163].

18

, .

2.5. . , . ( ).

(1896-1982), , 1943.

(1903-1987). - 1939. 1941. , 1940. 1944, 1945.

(1899-1981). , 1946.

28 . 1937 - , 1939. ( - ). , 1944. 1945-49 . 1946-49 - . 1951. - .

(18941959), , 1941. - 1944, 1945. 1953.

. . ( ) 1938 . , , . . .9.11

. , -.

29 (1892-?). , -, 1938. .

30 (1981-1943). - (19321938); (19381942). , 1943. (1936), (1940), (1940). -- (1943).

(1906-1968). , 1945, 1945. , 1953. - , 1953

(1908-1988). 1942. , 1945. , 1953.

28 - [449], ( , . .8.21)., - (-) . , 50 computer science.

29 . , 42:1, 44:4. .30, , ,

, 1930-1945. 1965-1972.

19

(1901-1971). , , 19361938(?)31. 19351971 . ( - ).

(1908-1989), 1941, 1951, 1953. 1951. 1945, 1949, 1951, 1953.

(1900-1938)32, () 1935-1937. 02.08.1937. .

(1891-1956). 19351937. , 1932-1938, 1937, 1932, 1937, 1953, 1937, 1945, 19391942- .

, , - .

, , . [4]: - . . , .-. . . , .-. . . , . . , . . , . . , . . -, . . , . . , . . , . . .

. , . , , -, . . . , (, , ). . . . .

- (1892-1938), - . 1938 . (. . , . . , . . ) .

, , - . , -, , , - .

31 - -. . . , . .10.10 1929-30. ( ), -. , , , . 1934. - , 5 , (, , , ). - () 1934-38(?) . 1936., , . , - .

32 . 1924-29. - 1929-30. . 10.2, 13.1. . . -., . .8.6

20

.

2.6. 1936 . , 30 . -

. , . , , -

. . , . , .

:

33. . . . .

, - ( ) , -- ( ) . , .

2.7. 1953 . a) . -, [5] , (- ). , , . 34. , , . [589].

[30] , , 35. [17] ( ) [257], . . -36.

b) 5- 1950 (. . , , . . ), . , - ( ;

33 . - (real analysis), , , - ().

34 http://odasib.ru/OpenArchive/Portrait.cshtml?id=Xupavl635188979823906250305635, 1939. ( -

), (Stephan Banach) 1946., ( ) 1982.

36 (1891 1970) ( c1939), - - . (1939-1941) - ( ).

21

, -37; , ). [338] .

c) 1951-1953 9 , . [327][329], [128], , , , , [85], , [216],38, 39 [168], 40 [436], 41 [138].

d) 1951 . -, [170], [86] .

1953. , [330] - Lecons sur les ensembles analytiques et leurs applications, .

e) 1953, 1958, 1959 - [331][331]. - . , (744.), - . , .

(-), 42, , , , .

, (, , ) .

, - , , , , , . , - , , -, 43. , , .

37Ljapunow, A. A.; Stschegolkow, E. A.; Arsenin, W. J. Arbeiten zur deskriptiven Mengenlehre.Berlin, 1955. iii+108 pp.

38 . ., , . ., 38:3-4 (1931). .

39 (1911?). -, . - . ./ [267], [270], [271], [269].

40 (1902-1973), , , .- 1939, . .

41 (18931983), . - , . .

42 (18831957), --, - , 1946 , 1952 . 1937 (, , ), 1938, ( - ) , 1943.

43 [340] .

22

2.8. , 1960-1985. . , , ,

1960. [383]1961. [84]1963. 44 [374]1965. . . [267].1974. [279], [214], . . [270].1977. [63], [64] ( )1983. [370], [371], [242], . . [416],

[91], . . , . . [453],1984. . . [271] ( ), . . , . .

[456], [33] (, ), [6] . . [450].

1985. . . [451], [452], [193], [455], ,. . , .. [372].

1950-1985 , , -, , . ( ). , , , .

2.9. 1950-85. 1. -, - , . , . - 1934 45. , , , 46,, , , - , - - .. ... -, , , . . . - 1989-91 [482], [483].

.

A) 1948 . - . 2-3 - . . (, ) - . - . . . , -, !

B) , - 1

44 (1911-1978), .45 , ,

.46 30 . -

-. , , , . .

23

1930 1931 . -, (. , ); , , [53] [240] 1937-38.), . , , , . , 1930-1931 - .

, [269],. . , 1971:

. . 10 1931 . ( 25.IX. 1931 .) .

, - 47.

[65],

. . . . . - , . , , , .

1931 . . . - , . . . -. ( ). , .

1930-31 ( 11), , 1942 , , -, . 1930 , 1932 . - , . [55], [57],, . . [67], . . [260].

, . . , ....

, 1989 .

.

2.10. . 2. , , . - , . . [271] 1984.. 1985 , , . . [451]

47, , , , .

24

Curicullum Vitae . CV (, -, ). , , 1985 , ( -) , !

1930 , . 1951 [170]:

30 40- , . . . , . . : .

(??)

. . . - , ,...

( , , ?)

[] 1934 ., -. . . ; .

(-, , , 40 ; , , - [449], ).

-....

. - (, , -) , 1935-1936 . 1936 ...

- . , [349][352] - (1967-1970), 113 . :

, , [] - : , - - , 30- , , . - . , , , ?

, . -, 35-36 , - 30. - , .

25

, 1936 , 30 . , , 1935-36 .... - , 1951 , ...

2.11. , 1975. 1966. XX . . . , . . . - , , , . . -, . . , , , . . , . . ,. . 2005. [272].

, , , - ...

. .[ ]: . , , ...

. .: . 14- 30- ....

[ ]. .: , 30- ?. .: , . , -

, . . ?

. .: , .. .: , ?. .: .. .: . ( .)

2.12. . -:

1973 1974. 1976. - 1978 - 1980 - 1981 -1982 - 1983 - 1987 - 1988 1988 03.01.1989 48

1936 [394] 1978 . :

1936 . . . . , .

48 .. .., 1985.

26

. . -. . . ,, . . -, . . . - ....

: 1936 . . . . . 1939 ., - , , , -, .

, 1989 . , , -. .

. . 1903 -1989. . 1906-1993. . 1912-1995. . 1905-2012 , (

), 1974 , 1951 ,, . . . . .

-, , - 1936 . 1974 [62] 49, 1977 - [63], [64] . [65], [66] ( ) (. .8.7).

1980-1983. , 1983 , .

3.

3.1. CV 1930 .9 (27 ) 1883 ;1893-1901 - ( );1901 ; 1905 1906

;1907 . . ;1909 ;1910 -; 1910-1912 ;

49 . , -, .

27

1911 ;1912 ;1913 (?) 1914 - ; 1914 -; 1915 - . -

1915. 1916 (242);

27.04.1916 . , ;

1917 ? 1930 ;1918, 1920. - -

;1919, -, - .. - 50;1920-1925 - [411]; 1925- 1926 - ; 1927 - - ; 1927 - ;1.07.1928. 1.09.1930. 51;1928 - -

;1929, . , -

, ; 1929 c - . . .

, ;1930

; (?) 1930 20.11.1930 - ;16.12.1930 - ( ).

3.2. . [390]:

. - . . ; : , -, .

( ) . - -, ... -, : ,

50, . . -. .., 15 1919. . . , [191].

51 :. : 19051906 . 19101912 , . 19121914 . 19251926 9 . 19261927 5 . 19281929 - . .

28

. -. . , . , . ! , . , , , , . - . - , , , ,

, [170]:

. . ; , . - . . , - : -, (, ). . . , ; , - , , . , - . , - . ; . . , - , , . , - , , .

[449]:

...... ,

- . - -

, , - , , , . . , . - -, ....

29

1901 . . . - - . , , ; , .

. ., ....

. . , - - . . , - . , - . . -; . . - , , . - ..... [] , - -, - .

, 1905 ( ). :

, ; 1905 , . ....

. . -, , . , , - . , , - (. . -52. - . .) . . , . -

52 (18831963), , - , , , , , ( , ). . , 1904. , 1905. . 1907-1908 , , . 1917 - - . , . , - . , , , 5 , : , , , , - ... , , ...

( , ). , 23 1928. ( - ), - , [137]. 1928-29.

30

, . , 1905 . - 17 . . , ...

. . . . -, .. . ., - 53, - . , , - . . . , , [. .], , , . . - .

[169], , 1942:

, 1905 -, . , . , . , , , , , . , , , , , , , , .

, . , - , . 10, , - . , . ., -, , . , , .

:

. . 1906 , . . , - ; -,

. (Vito Volterra). -, 1940 . . 1941-1942 . . -, . 1984 , . , , -. -, , [569].

53 , - ...

31

; , College de France , - . . - . . - . ; - ....

. . 1906 .

:

... - . . - . 1907 . . . - . . . , . , , , , . - - , , .

21 1907, [247]:

... , . , . - . , , : , . , , , -, , -, , .

, ; .

:

... , -, 1905 , - . - , : . . - , , - . . . - - . 1906-1907 . .

32

, , . , 600 . . . ....

. .. , , . - , , , . ., -, . . ..... . ....

1909 . . , - - , . . . 1909 . , , . . , . . . . . . . . . , . . .

, 1910 , 27 , - , , , -.

3.3. XX . , - XX ( 54) ,

(1848-1921) (18371903) (18681922) (1869-1931) (1869-1947) (1847-1921) (1863-1927) (1858-1923) (18531924) (1869-1942)

54 20-30 - , . , -. , , . , .

33

-, ( ): .. - , 55. , , , , - ( -). , -, - , , , - .

XX , , , , . - [70].

[XIX] - , . -.

, [67]:

, . -, , . (, . . . . ). , , . . (18691931) . . (18831950), - ( , ).

3.4. XX . , , . . -, . . , . . , . . , . . . . .

1890 . . [598], [599] - . - , , (Ferdinand Georg Frobenius)., , . , . . , . . , . . - ( ) . .

, - . , ... , ,

55 , . . . [96], . [95].

34

, -.. ; , - , -....

, - , . , , . - . ( )

, - .

,

- .

C -, (, - )56.

, , , . (TheodorMolien) . (Joseph Wedderburn) , c . , - .

1900. ( ).

:

[ , , , ] , : , .

; . . - . . . . . : -, , . .

. . . . 1926:

, - ( -), npoeccop, . . . ,

56, , -, .

35

, -, , .

, [457] 1921., - 57,

p(n) =exp

{

23 (n 1/24)

}

4

3(n 1/24)

[

1

3

2(n 1/24)

]

+O(

exp{

6

n})

e

2/3n

4

3n

- [Godfrey H. Hardy, Srinivasa Ramanujan], [186], 1918, ).

58. 1926 , , , -, . - ... , -.

. . . , XIX XX [478], ( -)

, , .

-.

3.5. . [58]:

- . - . . . . - . , . . (18691947) - , , - . , - , . . ( XX ). - - . . (1901). . . . .

57, n p(n) - n n = k1 + + km, k1 > . . . > km.

58 . . [406]: [] (18831947), , ,

, , , ., , .

36

- . : , , . , , - . . (1914), - . . . -, . . . -, !

, . 1914-1915 ( ) ...

[449]:

... [] - , - , . . , . . , . . . - . 1905 , , , , , , ... ... . . , , , - ...

3.6. . . . 59 ( - 1921 ) , [91], 1983:

. . . (, . . , - ). . .

, , , - . .

( ) - . - ( ), , . - . , :

Jules Tannery (1848-1910) , . - . [320]( . 18.3), .

59 , , , . , 1965, 1.

37

(1869-1947) 1902 - . 1907 (340) - . , - ( , , ) . , - - . , - . 20 , 1930 . . 1946 , (. [259], . . ), .

3.7. . , - , . , - . - , - .

1905-1921 , , - .

. - (-, - , -, [449], [276], [67]). , , -, . , - . , - .

( . . ) 1921 . 1923 .

, [240] 1938, , - , , . 1946 [57], 01.12.1948 - 25- . [29]. .

1923 , - , -, - . 1930 . -. 1924 - .

38

1922 .. -

1922 . . 60.

(, -, ), - ( ). , ( , . [108]).

20 - , - , , , .

( - ) - . 20 , - ( .10.1),, , 1929 .

, , :

(1884-1954) (1891-1941) (1889-1950) (1901-1973) (1902-1973) ( -

)) (1983-1964) (1882-1963) (

) (1895-1932) 18831963 (?61)

, 62 . , , - (1916) ( ) - ( [539]), -,

60, - - .

61 , , . , , .

62 . 20 . . . . . -. - , . .

39

63. , 1918-30. .

, , , .

, . -.

, , , - . - 1940-44 1951-1973. - (1918-19), (1921-23), (1933-1934) (1944-1952), - (19361941, 19421950), , , - - (1944). - .

3.8. . . . - [269] 1975. 25 , - ( [96]) , . [519], , . .

-, 240- ( ) [516] 1901.

ds2 =

xdx2 +

ydy2

. -. , x = y = . f(x)dx2+g(y)dy2, - : x y = const ( ) . - ( ), , , - . ( ).

R3. - , , -, ( ). , -, .

. -, , - R3

63 [56], 1945: , - . . (. 1884), .

40

. , , , . ...

( ). - ( , ) .

Lecons sur lessystemes orthogonaux et les coordonnees curvilignes (Gaston Darboux) [106], - (E)- ( - Compt. Rendus).

-, . -

, 90 ,

jxj

dx2j WDVV-, . [119].

3.9. . , 1910 . :

... . ., .... , , [Edmund Landau]. . . , . . , . . . . -; , . . - , , . - , . ., , ....

, .. 1912 , - - . ....

, 06.04.1912 [112]:

- Fourier, . , [ [301] , ( )]... , Continuum problem - ( )64. , , - ( ), , ( ) (). , - . . -. (Hilbert, Landau).

. . . [644]:

64 -...

41

1912 . . . . - , . ., -, . - . , - . , . ...

. . . . - . . , , - ....

. . - 1915 . - , . . . . ....

, , , .

:

, - ....

. . , - . . , - ; - . . , (2000. ), 22 ....

. . . , - , (, , , , [Charles Picard,Jacques Hadamard, Emile Borel, Henri Lebesgue] ) - , - , , , .

, - , ( ). Comptes Rendus - ( Comptes Rendus ) . . , . 1936 , - .

42

3.10. . , ., , [85]-[86], [452]). .

[300], 1911, F (z) =

kzk, ,

|z| = 1. , , , - .

.

Tn(z) = 1 + z + z2 + + zn, z = ei

z = 1 |Tn(z)| n + 1, (, ), |Tn(z)| > n.

Sn :=

l>0: 2l61

[

zl(n+1)T (e2lz)]

Sn n. :

F (z) =

n=0

anzrnSn(z),

an , , an > 1/n, - rn , , p > q zj Sp - Sq. ,

ckzk . ,

, M

k=N ckzk,

... -.

, 1912, [301], : f > 0 g(x), , f(x) 6= g(x) C > . C -. .

1912 , .. f(z) , F (x) , F (x) = f(x) ..65

, , , . ,, x2 [1, 1]. x1 , . , - - k(x), , . (F (x) +k(x)) = F (x) , .., - . , x1+k(x) .

65 (Moonens,Pfejfer, 2008): U Rn F : U Rn . - G : U Rn, G U F U .

43

x1 + k(x) , x2 () .

, - , . g(x) > 0 - G(x) , |G(x)| < . C, f . g, f C . G. C (. [303], 14-15).

3.11. . 1911. , 1912 - 7 ... - - .

, 7 1914 [247]:

. , . -, , , , , . -. .

, , . , -, , ; ....

[369]:

, , , , . - , 14- . - . . , , -

, - . -, ( , ), .

3.12. . 1914 - 1915 ,, . 1916 242- . [303].

. - . , ( .) ( ) -. , , ( ) ( , , ,

44

, ). , ..,

:= limt10

ckt|k|eik.

, -.

(, , [Oskar Perron]), - , -. , ( ) ( ). , , .

, , - , . .

, u(x, y) - u(x, y) + iv(x, y), u v . u(0, 0) = 0. ( ) . - ei = x+ iz,

J(

n

(an cosn+ bn sinn) =

n

(bn cosn+ an sinn).

,

Jf() =1

v.p

2

0

cot(

2

)

f() d :=

:= lim+0

1

(

f(+ ) f( ))

cot

2d.

, , f - L2.

, , , ( [188] , - ). , L2 , .

(, -):

45

... , , - , 66.

- , , , - . , -, . , ( ), .

1916 , - . , 100 , 100 - .

. . , , , , , -, , , , , , ....

[128] :

. . , - . , . . . , , , , - .

, . . , , , - - . . .

13 1916 . . . .

. . .

4. :

4.1. . 1915-1935 :

(1884-1954)

66 : - A lim0

12

(A [x , x + ]) 0 1 . , , , , .

46

(1891-1941) [Wac law Franciszek Sierpinski] (1882-1969) (1889-1950)

:

(1896-1982) (1901-1961) (1897-1940) (1904-1976) (1903-1987). (1900-1980) (1899-1981). (1911-1973) (1892-1988) (1901-1975) (1894-1919) (1898-1924) (18941959) (1905-1938)

, 1914-1916 , , , , , (1923-1924) (1933).

, , ( -), 1914 . () . - 67. 1917 , . 1920 . (Mazurkiewicz, Stefan) 3. (Janiszewski, Zygmunt) Fundamenta Mathematica, .

, -. .

67: , , 1916 ., [636].

47

, , - 68,69 ( )

(18931983) [-] 70

(1900-1967) (1895-1932) (18961962) (19051944) (19031951), .. . ( 1901 - 1967) (1900?-1968?) (1905-1997). . .

. , .

4.2. 1916-1917. 19161917 Comptes Rendus - , , , .

a) . , - . , - -, .. - , , . , -. , -

68 (18891976). , , [259], , -, ( ). , , , - . 1928 , , . 1930 . - 1935 ( 46 !). , Compt. Rendus, Compositio, Bull. AMS,Annals, Annali di Pisa, Tohoku. . - , , -, . MR0186765, MR0176935, MR0186560, MR0000656, MR0001814, MR0001813 MathReview ( , , J. W. Tukey E. Mendelson).

69 , [65]: . - : . , . . - . , .

70 C. R. 184, 1311-1313 (1927); . ., 35:3-4 (1928), 379413; Fundam.Math. 21, 20-28 (1933). , [276] , .

48

-, . - -, , . 71.

. , . , - . G. . , . -, .. [286], -, , , -, 1. , , .

, , . - , - , - ! - . , . , - ( - , ).

1916 . ( ) . , . : - . - . A-. , Xk1,...,kn , .

k1,k2,k3,...

n

Xk1,...,kn

, - A- (, , - ). , , A- - ( ).

-, -, , - 1916 , (Felix Hausdorff).

2) . 1916 - , . -, . ,

71 -, - , , - . , .

49

. , - , . , ,

cn =

2

0

ein d().

cnein, ,

. , . , ( , - ).

(Antoni Zygmund) 60 :

... [ ] - ...

-, . , (Frederic Riesz) 1918.,[403].

3. . 1916 - mainstream, - . , , (- , , , , , -). , , (OskarPerron) . , , - , . . [200], . 1916 - . , , .

[200][202], - . ( , JaroslavKurzweil), - .

, , , -, .

4. . XX , , , -. - , , . , ,

50

. 1916 72... , .

, 1917 , A-, , A-. , , , , G-. , , 73.

, , , , . , , .

, 191516. . .

4.3. . - 1911 , - 1912 , 1913. 1914 . 1917 Comptes Rendus (, , ).

, , . , , , . , .

, - . 74. - .

. [62]:

[] 2- . , , -, . , -, , , ,

72 , - . , . .

73 Comptes Rendus - [304] .

74 : ( 58) . . . , . . . . 6 1914.

, . . (1914 . ), , Dirichlet , , (B). 1914,

51

18- . , - , ; - , , . , .

, . , : 75 ( , ). - , !.

, . , , , , , , , -.

- [432] , 1964:

1916 . , , . , - 1905 . Journal desMathematigues. , , - , , . - , , - , -. ........ , , , , - 10 . , , - , , ....

, . , -, -. , .

75 . . .

52

; -.

4.4. . 1911-1914. 1915. . 1917 - ( 76), ( ), [337] (- [53] , 1919).

, .. , - ( , , ). 1917 , . 20 77,

-, , - . (. ), , .

,, R\Q. , - ( ) - . - , .

, , , [311], 1930 . -, , 1950, [415],[76], [216], [214], [455], [193]. (1924-25).

, , R Rn, - -78. , ( )

76 Sur un mode de convergence de lintegrale de Dirichlet -.- , 1934,.6, . 3, . 1-4 20, [326].

77, , , (Motokiti Kondo).

78 - . , - , -. , , G- ( - ) - (, [49]). , R \ Q. - NN, .. (l1, l2, . . . ), . (l1, l2, . . . ), (m1,m2, . . . ).

53

. . [258].

-. , - (. [192], [193]). - , . [197], . [152], .

, - . Ams(M) M , . Ams(M) , gj Ams(M) g, A, B M -

(

gj(A) B)

(

g(A) B)

(, - ). ( 30 ) Ams(M). , - Ams(M) . , Ams(M) Ams(M) , (g, h), , g h. , , ; , , g, h, . [189], [143]. , - (, ).

4.5. -. , - , , - , , , , ( ) .

, 1917-1918 - , - , , (1921-1923).

, -, - - . , ,

k , . - 1/k. -. [0, 1] NN (, R \Q (R \Q) [0, 1]). , . , - . , , 20 , [311] R R \ Q ( , , ) , , - . [311] ( - R , ). -, . .7.7 [307] ..., , - () .

54

, , , -. .

1915 - - . 1918 . 1918 , , , , , 79. ( , , , , ).

, -.

1919 .

. 20

, , . , , 1922.

4.6. . , 1920-40 , , (Paul Levy). ? .4.64.10 , .

, 1922 . :

, - - -, 1922, 5, 2741.

, - -, 1922, 5, 4248.

. . -

-, 1925, 8, 3237.

, , , , , 1922 - (, , - ). [204]. - [207][208]. , a1, a2, . . . , q1, q2,. . . . (..)

limn

na1 . . . an = C, lim

nnq1 . . . qn = D,

79 [276] , .

55

C =

j=1

[ (j + 1)2

j(j + 2)

]log2j

,

,

D = e2

12 ln 2 .

[206], 1926, -. a . (t) t > 0, , t2(t) . -

1t(t) dt . a

p/q, ,

a p

q

< (q).

, a .

1922 - . . [351],

1922 . . . ( 4 -). . . . .

[58]:

- , . . 19251926 . , - , . . . ( - . . ), . . , . . ,. . . , , - , . . ,, .

- 30, .. 4.9. , , -

20 - , . . , . . , . . , . . . 1934 . , , .

, -. 1924 [205] . . s(n) , n . - ? . -,

lim supn

s(n)2n ln lnn

= 1.

56

[203]. 1925 -

. .

j , . , .

j 80. , [351]:

, - - , , . . . . - -. . . - . -, , . . , .

, - , .

4.7. . 1920 (, -). [418]:

- - . . . - (, , -). : , - -, - , .

, - .. . 1920 - 81 - XVXVI , ... -

1921 - . [242], . . 1983.:

. .?

, . . . . , , . . .

80 L2-, .81 , [245].

57

, . -, . . - . . . - . ( - 82). . . , (, ) .

1922 L1, , [234]. - , [238]. [244]:

- - , - , B- A-83. , B- ( ) -...

... 19211922 . . . , , 1926 .

, - . - .

[242]:

1925 . . - . . . (, , : 1934 .) 1924 . . 1924 . . . ( . . ).

1925 tertium non datur, [236]. , XX . , (Luitzen EgbertusJan Brouwer, 1881-1966). , , . , A A . , Z A, B, . . .

82. [235].83 .

58

, Z A, B, . . . , .

, , , . -, .

1925. - 1927 , [237], 1932 60- Grundbegriffe derWahrscheinlichkeitsrechnung, [239] .

.

4.8. . - [165], 1948:

1921. - . . . . - . .

. . - , 15- . . . . 19181919 . .

:

. . , - 1921 . 1922 .84, . . 85, - . 1922 . . . . . . . . 3- - -. . . . . ,. . , . . , . . . -, .

: , , 86.

84 :. . .6. . . . . .

85. . X .86 [369]:

[] 1916- 17- , , , - . . . - , , , , , , - . , .

59

1923 [71]. , [72], 1922, 1929 .

, 1924 , . , 1927 [454].

U, - : A -, B , A . A U B U ( , - ).

, , . , - M U . . - K U U (. . -, 2000). , U ; , - U (. . , 1990) .

, , - ( , ). , , - . , 1951 , . -, 1986 (Miroslav Katetov) . , , -, (.., 2004)87.

, 1922-1924 - 88 .

, - . , , , , - -, , , . , -

. , .-, Janiszewski, S. Sur les continus irreductibles entre deux points. J. delEc. Pol. (2) 16, 79-170 (1912).

87 : M . G- M .

88 [356] . . 1924 . , - [] 0. .

60

- - , , - . , - .

1927. ompt. Rendus [50] (. [51],[52]), . , O X ,

X = j .

C(X,O). j . i1 , . . . , ik , i1 , . . . , ik .

j , - C(X,O) X , ( ) . .

, . (Eduard Cheh) 1932. - , . . , , - 89.

20 . . . , . . , . . ,. . , . . , . . , . . , . . .

- , ., , [165], [69], [351], [392]. . 816.

4.9. -: . -, (Joseph Liouville) 1844,

n=0

10n!. -, ( - ), , - . , - , , . 1873. (CharlesHermite) e. 1882 (Ferdinandvon Lindemann) , e, . , ei = 1 , . - . ( - , -, ) , ej

89, , - .

61

( j , Q). ,-, .

: ( ), , , , 6= 1, / Q.

1920 , - - . . , , . . - (. . , 1927, 34:1), - .

[529],

, , 24 1906 . . 1924 . - 1927 . . 1930 . . . . . .. 19291930 . . 1931. ... [ 1966.] - 90

.

, (-, ., , [81]). 1929. [156] (. [157]),

i

n, , n - , (, (1)i = e), - . . . [266]

n. 1933. . (Karl Boehle, Math.Ann., 1933) - . . L Q k. j L Q.

j

., 1934 -

[158], [159]. ( ) . (Theodor Schneider, J. Reine Angew. Math. 172 (1934).). -, 19291933. .

, 1967. (A. Baker)

11 . . . kk ,

j , 6= 0, 1, 1, 1, . . . , k Q.

- - . , 1928-1930., . [633].

a) 1929. : [355]. - ( , , , ) . , .

[356], . , - [347] 1949. -, 199091,

90 1931. - .91W. Klingenberg, Trans. Amer. Math. Soc. 356 (2004), no. 6, 25452556:

In a short note, Lusternik and Schnirelmann [1929] sketched a proof of the fact that actuallythree simple closed geodesics exist on any orientable closed surface of genus 0, i.e., on a surfacegiven by S2, endowed with an arbitrary Riemannian metric. A complete proof of the theorem of

62

, -, , , ( , .[441]).

b) 1929. [426]: - . - ( , ). , , - - . (Otto Toeplitz) 1911. (. [364]). . (Arnold Emch. Amer. J. Math. 35 (1913), Amer. J. Math. 37 (1915)), - . , . : . -, . , .

- 92.

c) : , , - . 1929. , N , 6 N [424]. , , . N , , (800 000).

1937. -

, , , 3315

, 10684616 .

( - -

93. , ,

(Harald Helfgott).

4.10. . [351]:

- ? , , , , ...

Lusternik-Schnirelmann is notoriously complicated. Over the years, many papers have appearedon this subject and only a few are satisfactory; cf. Taimanov [1992].

92 (A,B, C,D) . R8. , , - . , ,AB = BC = CD = DA . , .

93 , , 1074 - . C , , . - , , 103000000 -, , . , - , , ( - -). , , .

63

- - , ;

, . , , -, , . - (, , ), .

:

. . , , , . - , , - , - , , . . , . . -.

, , , . , , , , ,. , , - .

4.11. . - , , -, - . - [65]:

, . . -, . . . - , , , -, . - , , . , , , , - . . . . , , .

[279]:

-- --; . . - : , , - . : . . -, . . ,

64

. . (, -, ). , , .

. . , . . , . - . . ( ) : . . - , . . . . ). . . . , ; (, . . ) 20 . . - . . . , - . ( ). . . . - () ? .

2 , , 2 20. , , , . . , , , , ..., , .

:

1921 . , , . -...

, , , : .

:

- -, . , , . - : - , , . . ( .) , , , .

-. 1967 , [165] ( ) 1946, , - [69], 1948. ,

65

. , , -, , . , , , [350]-[351], . , - , - . , , , - . (-) .

1. ( 3- ).2. ( ).3. ( ).4. (, , 2- ; -

).5. (, ).6. (, 3- ,

, ).7. ( , ).8. ( -

).9. 1 ( ).10. 2 ( , , -

).11. (, ).12. ( ).13. ( -

, , ).14. ( , ).15. 1 (, , 1- ).16. 2 ( , , . . -

).

( ). ?

- - , , - . .... , . . .

4.12. -. - . , , .

- . . [417]:

... ? , , . : , . , . -. , , , . : , . - . ....

. . [91], -:

. - , ... . ...

66

, , - . , 1921 . 1920/21 , ( - ). -, - , . , . - ( ). . , , , ...

? , . , - . ...

() . -, . -, . . , - . . . , - ...

, . , , . 1921/22 . - , , . . - . - . -. .. ( ) ...

: , - -. , - . , . -, , ...

, . - , . -, , , ,

67

. - . , . ?.......

: - [ ]

, , - : . .

, ...

: , . . [,

].... , . ? -

, - , , -. , . - ...

, . , . - ... . , ...

, , [351]:

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, : - , , - ? , , - , , , . , . : -, , . - . , - ? , , - - - , , - , ....

68

, . . :

- . , - , - , -, -. ...

, , - , , - . , , -, , , - - , .

4.13. - . -, [242]:

. . ?

. -....

, - . , - . . . ( , ): -, . . - , , . . -, . - ( - ), (, , ).

[351]:

. - ..... - , , - . - . , ,

69

. - ...

[279]:

. . - , - , , . , . . , ( -) - . . . : - . . . . . , - ....

, , - , , (, -, , ). . . , 2030%. , ( ) . , , ; , -. ( -) , . : , -. , 200300. ( , ), - () ...

- - - , .

4.14. . . , , , Compt. Rendus ( ). - ( 20) Fundamenta Mathematicae, 1920., , , ,, . . . , 1982, [480]:

, . -, , , , -, . .

70

, , - . .

4.15. . -, 1970., [353]:

. .: , . , - . , , , , . . .

. .[ ]: .. .: . . . . -. -

, ... -... -. , - . , ... 14- 24- . . , - . - , ... , -... , ... - - , . , . , -, , , ... , , - - . .

5.

, , , ... , , , , , . ,, , ... , , ... , , , , .

1971., [61].

, - 1936 . , , .

5.1. . ([351]. - ( [354]), . [353].

71

, , . . . , ( , , . .94 , , .) ( ), , , , , , , , ...

:

: , ? ? , , . : . , , . .( .)95

94. . . . 95 -

. , .

72

:

20- - . . . - : - - 96

.11.13., , -; , . . - ( -: . . ... )

5.2. .

. , , , , , , . , , . , - .

, .. -, . , - .. 1914 , - 1922 . , , -. , . 1914 .. 4-5 , 1922 , .., , . - , , - . , -, , , , - , - ,

1926 . [344]. . 1924 . - , 1925 (R. Courant) . (J. le Roux, 1914; R. G. D. Richardson, 1917), -, , , . [104].

96 :. , H. H. , , , , -, , H. H. ,

, . , , , , -, H. H. . , , .

73

.

.. ; , , , . .. , - .

.............. 1919 . - . . ; , . - ....

, ( -), . , - . , , 1936:

. ... , . . , -, , .

, . , .

:

. . . . . - . , . . -, , . . . . . . , - . . 1921 . - . . . . , . . . -97. . .

1925 . . -. 1918 1924 . - , 1925 1930 . , , , . 1925. 1926 . 1927. (27.04-4.05) . .

97 [67] . . , [276], .

74

. 1927. , - . 1928 . , 1928 . [] - , , , Lecons sur les ensembles analytiques. . .

-, -. - ; -, , , , . . . - .

-. -, , , . , , , - , . , - . . . .

, , 1924 1930 . . , , , , . - , , .

5.3. .

, - .

, , 19251928 . : 1) , . . , - , . . - . . - . . . , . . , - (. . , . . ). - . . , - . .

, - . , . .

75

.

5.4. . -

.98 - . , -, . . . , .

............................................. , -

-, ?

. , - , . . , , , - , : - - , ?

. 1922 . H. H. : - , .

. : - . .

............................ ,

. -.

:

, -, . -. . , - .

:

, , , -

, ,

98 (1881-1958), 19341941, 1943. . 1906, ( ).. . . . . 19241929. , , .

76

.

5.5. . [480], 1982. 2005.

( ). . ... , , -

. ?. .[ ]: , ...: , , . , ,

. . , ... , , , , [?] , ... , , - , -, . , , -, , , , ; , - , , , , , , , . , , ( - ) , . , , . , . , , , , - , - , , , , , - , , , . , , . : , - . . - , , , -, . - , , , , , , , , , , .

, , , , .

, , ... : , , ,

77

- . , -, . . , , , , - , , ,, , , .

.

5.6. . 1936, , , . , 50 . , ( . 8.14), .

. ( ), ( ), - , . , . - ...

1925-1928, . ...

- . - , .

6.

, : , .

[310], 1927.

6.1. . - 1924 . . 1924 . X , R, R \Q. Xn - . : Xk+l Xk. , , - . , , F.

78

( ) - , -, .

, - . [309].

[305], 1925, [306], 1926. , (, , . -

, ) .

6.2. . 1940 , -- [175]. 1963 . , - . , , , .

, - - [187], 1914, 1927, - . 1927 1963 ?

6.3. , 19251930. - , 1930, - , , . ,, . , :

, : , .... - , , - , , , - , ,

[305], 1925, :

, , , - : , [ ], , , (- ) ..., . ...

, , - .

:

79

, , CA- [ - ], , - [ , , ].

, - ( . . . [455]).

, , , - -. : 1951 , CA- . - .

, . . . [455] . . :

. . . , :, , , . . . . .

6.4. -. , - . - :

. - : , ., , . . -. - ...

:

, - . . -, . - -. : , - . ,, -, . : , ? ? ! ! , , . .

, , 1921 , . -

80

, , , .

, ( 1928 )

. -: Zmel, , , continiiuma , - , , - , - continuuma. . , - , - - . , - , . - , , , , - , -. : - 99 . , : - . - , . .

, 1935 [319]:

, . - , -

20 = 21

, , , , - . .

99

81

, . , - Hilberta , .

- , -. , , . , - E. Borel, : - , - : ( , 3,1914).

, - -, , - .

6.5. . , 20 , , , . 1916-1922 - - , , . -, , . , 30 - . , . . . 40 .

6.6. ? :

, B, - . , - , - , [ -], . - , . , ....

- , . . . - , . : - ,

82

, . - , .....

- , -. , .

- . ?

, 1974:

, , . . - , , . . . , , - . , , -, , , - . , , . , . . , , : , , , - ...................

. . , , , , . , .

, , , [216] :

- , . . - . , - , .

. - :

-, - , , , , , . , , - ,

83

-....

: - ; , , .

... - , . , , , . ( - ), , , -. , , , ; ( ) . , , . , - ( , , ). - ! - , ...

:

. , [ - ].

. . , - -. , , . - , , .

, - XX , , , -, . [Hermann Weyl], . , - , , .

:

, , , - ; . , - .

, , - : ....

84

- , , ; , .

... - ... - - ... , , - , , - .

, ( 100). ( ), , ( , ). , 101. , , .

6.7. . . , (. [105]) , , .

. [314], :

. . . , , , , , , , , , , .. . .

, , - , . , , , :

, , ,

1,2, 3, 4, 5, 6,. . .

.

100 . .

101 , .

85

, - .

, [Willem De Sitter] - [Georges Lemaitre]. , - , ? , , , , - , , . - , 1. - . XVII. , .

, . . , . . .

-, : , , , , - , - , .

., 14 1933.

1933. . 14.16, - , .

, , . , .

, ., , . . [400] - . , 10 - . . -, . [384], ( ) . , . . ( ) , - , (, -, ).

7.

, - Graham, Kantor Naminginfinity [178], 2009 (

86

, , --, ). , - 2030 , .

7.1. , , . - , .

[288] 1930. , -:

- , . ; , , - . .

- : , : , , ? , .... -, , . , ; , . , ; - , , - , , , , . -, , , , .

. -, - , ; , , , - . , , , .

? - , , , - ? , - , , , - .

, , - , , . , , , ; ,

87

. - , , , . , , , - .

( - 1906-1907 ):

1/2 - , , - , , - , , , , .

, 102.

7.2. -. , 1929 . ( )., , - , -. -, . . . :

. - . , , . . .

, , , -, , , , .

. ?

. [] , , - .

102 . . , 26.8.1939 [131]: , - , ., , , , -, , , , , , , .. , ... , .

-. - - - . , .

88

, 1929. () . . , - .

,, . . , - , . . , . . -.

7.3. . [Loren Graham] [Michel Kantor], : , 1890-1930 . , -.

, , :

; , , .

; . , .

, , - ( ). - , , , , , , . () . 1908-1909 , . 1913 . - . , , ( , ) .

, [176]. , .

, , - , . -, , , : - , - . , 103 - , .

:

, - . . . - 104.

103 , .

104 , ( . 12.8) , - -. 12.8 , . . - , - .

89

( , - , , .) , -. [.., , ] - , , ...

-105, , - - ... , - , , , , , 106.

-, , . . 1920- . , . , -. , . , , -, , ...

, . - . , - . ....

, - . , - . -, , -107. , , , . , ...

105, , - .

106, . 1900-01 . - , . 1902-1903. , (, , ), . .

107 , , , .

90

, - , (). , - ( ). -, , ...

: - , ? - , , : . . , - , , ...

, , . , - , , - - . , , , , .

, . , . . ,. . [162], . . [89], . . [195] ( - 108), . [431]. - , , . . - .

7.4. . , , , -. .

a) (, , ). , , , ( ) . , 109.

108 , , , -. , , , -, ... , , , . -, ... . - .

109 1904. [142]. - - . , , , (- ). ( ).

91

( ).

. 3 1908 [112]:

- : ., - , , 110. , ?

. . :

[ ] ... , , ? - ?! [ ] , , ... [] , ....

, .

, - , 1904 , . , , .

- , . - .

,, 111....

, - . , , - , . [89]. , 1902., , [367].

110 , , , 1922.

111 [219] , 1 ., , , . , -, 1927 ( ) , . , -. :

=

1 v2/c2

, , ., . - (, , ): . . . , , ,

, 20- , (, , , . . )., !

92

, , , . .

. . . ( 20 ):

. . . . . ... . . - . , . . , , - . - ...

. . , . , , , .

27.2(12.3)1906:

... . , , . - , - !

) . , , . Comptes Rendus [126], [127] 1911 1912 ( - ), - ( ). [269].

. 20-30 () ( ). - ( - , ; .; - ).

) . 1920 -. ( ).

( ) - .

, 2000 , , (, , -). . , , . - . ... - . .

93

( ) , , , . , , -, . ( , ), ( ) . - .

. , ( , , - ). - .

1) - [442] ( . . ) - , , , 112. ( ) , , , .

2) 1930 ( ; - , ). . . , [109]), ...

) . : , , , , -, , , , 113. , . - . . . . -. , , - . , , .

-, (Kondo), . , .

)