Nikolai Lusin
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Nikolai Nikolayevich Luzin (also spelled Lusin; rus, Никола́й Никола́евич Лу́зин, p=nʲɪkɐˈlaj nʲɪkɐˈlajɪvʲɪtɕ ˈluzʲɪn, a=Ru-Nikilai Nikilayevich Luzin.ogg; 9 December 1883 – 28 February 1950) was a Soviet and Russian
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
known for his work in
descriptive set theory In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" set (mathematics), subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has a ...
and aspects of
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...
with strong connections to
point-set topology In mathematics, general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differ ...
. He was the
eponym An eponym is a noun after which or for which someone or something is, or is believed to be, named. Adjectives derived from the word ''eponym'' include ''eponymous'' and ''eponymic''. Eponyms are commonly used for time periods, places, innovati ...
of Luzitania, a loose group of young Moscow mathematicians of the first half of the 1920s. They adopted his
set-theoretic Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathematics – is mostly ...
orientation, and went on to apply it in other areas of mathematics.


Life

He started studying
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
in 1901 at
Moscow State University Moscow State University (MSU), officially M. V. Lomonosov Moscow State University,. is a public university, public research university in Moscow, Russia. The university includes 15 research institutes, 43 faculties, more than 300 departments, a ...
, where his advisor was
Dmitri Egorov Dmitri Fyodorovich Egorov (; December 22, 1869 – September 10, 1931) was a Russian and Soviet mathematician known for contributions to the areas of differential geometry and mathematical analysis. He was President of the Moscow Mathematical Soc ...
. He graduated in 1905. Luzin underwent great personal turmoil in the years 1905 and 1906, when his materialistic worldview had collapsed and he found himself close to suicide. In 1906 he wrote to
Pavel Florensky Pavel Alexandrovich Florensky (also P. A. Florenskiĭ, Florenskii, Florenskij; ; ; – December 8, 1937) was a Russian Orthodox theologian, priest, philosopher, mathematician, physicist, electrical engineer, inventor, polymath, neomartyr and f ...
, a former fellow mathematics student who was now studying theology: ''You found me a mere child at the University, knowing nothing. I don't know how it happened, but I cannot be satisfied any more with analytic functions and
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
... it happened about a year ago. ... To see the misery of people, to see the torment of life, to wend my way home from a mathematical meeting ... where, shivering in the cold, some women stand waiting in vain for dinner purchased with horror - this is an unbearable sight. It is unbearable, having seen this, to calmly study (in fact to enjoy) science. After that I could not study only mathematics, and I wanted to transfer to the medical school.'' The correspondence between the two men continued for many years and Luzin was greatly influenced by Florensky's religious treatise ''The Pillar and Foundation of Truth'' (1908). From 1910 to 1914 Luzin studied at
Göttingen Göttingen (, ; ; ) is a college town, university city in Lower Saxony, central Germany, the Capital (political), capital of Göttingen (district), the eponymous district. The River Leine runs through it. According to the 2022 German census, t ...
, where he was influenced by
Edmund Landau Edmund Georg Hermann Landau (14 February 1877 – 19 February 1938) was a German mathematician who worked in the fields of number theory and complex analysis. Biography Edmund Landau was born to a Jewish family in Berlin. His father was Leopo ...
. He then returned to Moscow and received his Ph.D. degree in 1915. During the
Russian Civil War The Russian Civil War () was a multi-party civil war in the former Russian Empire sparked by the 1917 overthrowing of the Russian Provisional Government in the October Revolution, as many factions vied to determine Russia's political future. I ...
(1918–1920) Luzin left Moscow for the ''Polytechnical Institute Ivanovo-Voznesensk'' (now called
Ivanovo State University of Chemistry and Technology The Ivanovo State University of Chemistry and Technology () or ISUCT () is a public university located in Ivanovo, the administrative center of Ivanovo Oblast, Russia. Research priorities of the ISUCT are concentrated in chemical technology, che ...
). He returned to Moscow in 1920. In the 1920s Luzin organized a famous research seminar at Moscow State University. His doctoral students included some of the most famous Soviet mathematicians:
Pavel Alexandrov Pavel Sergeyevich Alexandrov (), sometimes romanized ''Paul Alexandroff'' (7 May 1896 – 16 November 1982), was a Soviet mathematician. He wrote roughly three hundred papers, making important contributions to set theory and topology. In topol ...
,
Nina Bari Nina Karlovna Bari (; 19 November 1901 – 15 July 1961) was a Soviet mathematician known for her work on trigonometric series.
,
Aleksandr Khinchin Aleksandr Yakovlevich Khinchin (, ), July 19, 1894 – November 18, 1959, was a Soviet mathematician and one of the most significant contributors to the Soviet school of probability theory. Due to romanization conventions, his name is sometim ...
,
Andrey Kolmogorov Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Soviet ...
, Aleksandr Kronrod,
Mikhail Lavrentyev Mikhail Alekseyevich Lavrentyev (or Lavrentiev, ; November 19, 1900 – October 15, 1980) was a Soviet mathematician and hydrodynamicist. Early years Lavrentyev was born in Kazan, where his father was an instructor at a college (he later became ...
,
Alexey Lyapunov Alexey Andreyevich Lyapunov (; 8 October 1911 – 23 June 1973) was a Soviet mathematician and an early pioneer of computer science. One of the founders of Soviet cybernetics, Lyapunov was member of the Academy of Sciences of the Soviet Union ...
,
Lazar Lyusternik Lazar Aronovich Lyusternik (also Lusternik, Lusternick, Ljusternik; ; 31 December 1899 – 22 July 1981) was a Soviet mathematician. He is famous for his work in topology and differential geometry, to which he applied the variational principle. U ...
,
Pyotr Novikov Pyotr Sergeyevich Novikov (; 15 August 1901, Moscow – 9 January 1975, Moscow) was a Soviet mathematician known for his work in group theory. His son, Sergei Novikov, was also a mathematician. Early life and education Pyotr Sergeyevich Novikov ...
,
Lev Schnirelmann Lev Genrikhovich Schnirelmann (also Shnirelman, Shnirel'man; ; 2 January 1905 – 24 September 1938) was a Soviet mathematician who worked on number theory, topology and differential geometry. Work Schnirelmann sought to prove Goldbach's conjec ...
and
Pavel Urysohn Pavel Samuilovich Urysohn (in Russian: ; 3 February, 1898 – 17 August, 1924) was a Soviet mathematician who is best known for his contributions in dimension theory, and for developing Urysohn's metrization theorem and Urysohn's lemma, both ...
. On 5 January 1927 Luzin was elected as a corresponding member of the
Academy of Sciences of the Soviet Union The Academy of Sciences of the Soviet Union was the highest scientific institution of the Soviet Union from 1925 to 1991. It united the country's leading scientists and was subordinated directly to the Council of Ministers of the Soviet Union (un ...
and became a full member of the Academy of Sciences of the Soviet Union first at the Department of Philosophy and then at the Department of Pure Mathematics (12 January 1929). In 1929 he was elected as a member of the Polish Academy of Sciences and Letters in
Kraków , officially the Royal Capital City of Kraków, is the List of cities and towns in Poland, second-largest and one of the oldest cities in Poland. Situated on the Vistula River in Lesser Poland Voivodeship, the city has a population of 804,237 ...
.


Research work

Luzin's first significant result was a construction of an almost everywhere divergent
trigonometric series In mathematics, trigonometric series are a special class of orthogonal series of the form : A_0 + \sum_^\infty A_n \cos + B_n \sin, where x is the variable and \ and \ are coefficients. It is an infinite version of a trigonometric polynom ...
with monotonic convergence to zero coefficients (1912). This example disproved the
Pierre Fatou Pierre Joseph Louis Fatou (28 February 1878 – 9 August 1929) was a French mathematician and astronomer. He is known for major contributions to several branches of mathematical analysis, analysis. The Fatou lemma and the Fatou set are named aft ...
conjecture and was unexpected to most mathematicians at that time. At approximately the same time, he proved what is now called Lusin's theorem in
real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include co ...
. His Ph.D. thesis titled ''Integral and trigonometric series'' (1915) had a large impact on the subsequent development of the metric theory of functions. A set of problems formulated in this thesis for a long time attracted attention from mathematicians. For example, the first problem in the list, on the convergence of the
Fourier series A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems ...
for a
square-integrable function In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value ...
, came to be called Luzin's conjecture and was solved by
Lennart Carleson Lennart Axel Edvard Carleson (born 18 March 1928) is a Swedish mathematician, known as a leader in the field of harmonic analysis. One of his most noted accomplishments is his proof of Lusin's conjecture. He was awarded the Abel Prize in 200 ...
in 1966 (
Carleson's theorem Carleson's theorem is a fundamental result in mathematical analysis establishing the ( Lebesgue) pointwise almost everywhere convergence of Fourier series of functions, proved by . The name is also often used to refer to the extension of the ...
). In the theory of boundary properties of
analytic functions In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
he proved an important result on the invariance of sets of boundary points under
conformal map In mathematics, a conformal map is a function (mathematics), function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-prese ...
pings (1919). Luzin was one of the founders of
descriptive set theory In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" set (mathematics), subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has a ...
. Together with his student
Mikhail Suslin Mikhail Yakovlevich Suslin (; November 15, 1894 – 21 October 1919, Krasavka) (sometimes transliterated Souslin) was a Russian mathematician who made major contributions to the fields of general topology and descriptive set theory. Biograph ...
, he developed the theory of
analytic set In the mathematical field of descriptive set theory, a subset of a Polish space X is an analytic set if it is a continuous image of a Polish space. These sets were first defined by and his student . Definition There are several equivalent ...
s. He also made contributions to
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
, the theory of differential equations, and
numerical methods Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods t ...
.


Letter to Vygodsky

In a letter to M. Ya. Vygodsky dating from 1932, Luzin expresses sympathy with Vygodsky's
infinitesimal In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referred to the " ...
approach to developing calculus. He mocks accusations of bourgeois decadence against Vygodsky's textbook, and relates his own youthful experience with what he felt were unnecessary formal complications of the traditional development of analysis. Typical is his youthful reaction to his teachers' insistence that the derivative is a limit: "They won't fool me: it's simply the ratio of infinitesimals, nothing else." A recent study notes that Luzin's letter contained remarkable anticipations of modern calculus with infinitesimals.


Luzin affair of 1936

On 21 November 1930, the declaration of the "initiative group" of the
Moscow Mathematical Society The Moscow Mathematical Society (MMS) is a society of Moscow mathematicians aimed at the development of mathematics in Russia. It was created in 1864, and Victor Vassiliev is the current president. History The first meeting of the society w ...
which consisted of Luzin's former students Lazar Lyusternik and
Lev Schnirelmann Lev Genrikhovich Schnirelmann (also Shnirelman, Shnirel'man; ; 2 January 1905 – 24 September 1938) was a Soviet mathematician who worked on number theory, topology and differential geometry. Work Schnirelmann sought to prove Goldbach's conjec ...
along with
Alexander Gelfond Alexander Osipovich Gelfond (; 24October 19067November 1968) was a Soviet mathematician. Gelfond's theorem, also known as the Gelfond–Schneider theorem, is named after him. Biography Alexander Gelfond was born in Saint Petersburg, Russian ...
and
Lev Pontryagin Lev Semyonovich Pontryagin (, also written Pontriagin or Pontrjagin, first name sometimes anglicized as Leon) (3 September 1908 – 3 May 1988) was a Soviet mathematician. Completely blind from the age of 14, he made major discoveries in a numbe ...
claimed that “there appeared active counter-revolutionaries among mathematicians”. Some of these mathematicians were pointed out, including the advisor of Luzin,
Dmitri Egorov Dmitri Fyodorovich Egorov (; December 22, 1869 – September 10, 1931) was a Russian and Soviet mathematician known for contributions to the areas of differential geometry and mathematical analysis. He was President of the Moscow Mathematical Soc ...
. In September 1930, Egorov was arrested on the basis of his religious beliefs. He then left the position of director of the Moscow Mathematical Society and was replaced by
Ernst Kolman Ernst Kolman or Arnošt Yaromirovich Kolman (; 6 December 1892 – 22 January 1979) was a Marxist philosopher, who renounced his former activities as an ideological enforcer in Soviet science. At the age of 84 he sought asylum in Sweden and publish ...
. As a result, Luzin left the Moscow Mathematical Society and Moscow State University. Egorov died on 10 September 1931, after a
hunger strike A hunger strike is a method of non-violent resistance where participants fasting, fast as an act of political protest, usually with the objective of achieving a specific goal, such as a policy change. Hunger strikers that do not take fluids are ...
initiated in prison. In 1931, Kolman brought the first complaint against Luzin. In 1936 the
Great Purge The Great Purge, or the Great Terror (), also known as the Year of '37 () and the Yezhovshchina ( , ), was a political purge in the Soviet Union that took place from 1936 to 1938. After the Assassination of Sergei Kirov, assassination of ...
began. Millions of people were arrested or executed, including leading members of the intelligentsia. In July–August of that year, Luzin was criticized in ''
Pravda ''Pravda'' ( rus, Правда, p=ˈpravdə, a=Ru-правда.ogg, 'Truth') is a Russian broadsheet newspaper, and was the official newspaper of the Central Committee of the Communist Party of the Soviet Union, when it was one of the most in ...
'' in a series of anonymous articles whose authorship later was attributed to Kolman. It was alleged that Luzin published “would-be scientific papers”, “felt no shame in declaring the discoveries of his students to be his own achievements”, and stood close to the ideology of the “
black hundreds The Black Hundreds were reactionary, monarchist, and ultra-nationalist groups in Russia in the early 20th century. They were staunch supporters of the House of Romanov, and opposed any retreat from the autocracy of the reigning monarch. Their na ...
”, orthodoxy, and
monarchy A monarchy is a form of government in which a person, the monarch, reigns as head of state for the rest of their life, or until abdication. The extent of the authority of the monarch may vary from restricted and largely symbolic (constitutio ...
“fascist-type modernized but slightly.” One of the complaints was that he published his major results in foreign journals. The article triggered a special hearing on Luzin's case by the Commission of the Academy of Sciences of the Soviet Union, where the allegations were reviewed and formalized. At the hearing,
Alexandrov Alexandrov (masculine, also written Alexandrow) or Alexandrova (feminine) may refer to: * Alexandrov (surname) (including ''Alexandrova''), a Slavic last name * Alexandrov, Vladimir Oblast, Russia * Alexandrov (inhabited locality), several inhabite ...
, Lyusternik, Khinchin,
Kolmogorov Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Soviet ...
and some other students of Luzin accused him of plagiarism from
Pyotr Novikov Pyotr Sergeyevich Novikov (; 15 August 1901, Moscow – 9 January 1975, Moscow) was a Soviet mathematician known for his work in group theory. His son, Sergei Novikov, was also a mathematician. Early life and education Pyotr Sergeyevich Novikov ...
and
Mikhail Suslin Mikhail Yakovlevich Suslin (; November 15, 1894 – 21 October 1919, Krasavka) (sometimes transliterated Souslin) was a Russian mathematician who made major contributions to the fields of general topology and descriptive set theory. Biograph ...
and various forms of misconduct, which included denying promotions to Kolmogorov and Khinchin. According to some researchers, Alexandrov and Kolmogorov had been involved in a homosexual relationship in the 1930s, a fact the police used to pressure them into testifying against their former teacher.
Sergei Sobolev Prof Sergei Lvovich Sobolev, FRSE (; 6 October 1908 – 3 January 1989) was a Soviet Union, Soviet mathematician working in mathematical analysis and partial differential equations. Sobolev introduced notions that are now fundamental for severa ...
,
Gleb Krzhizhanovsky Gleb Maksimilianovich Krzhizhanovsky (; 24 January Old_Style_and_New_Style_dates.html" ;"title="nowiki/>Old Style and New Style dates">O.S. 12 January1872 – 31 March 1959) was a Soviet Union">Soviet scientist, statesman, revolutionary, Old Bol ...
and
Otto Schmidt Otto Yulyevich Shmidt (born Otto Friedrich Julius Schmidt; – 7 September 1956), better known as Otto Schmidt, was a Soviet scientist, mathematician, astronomer, geophysicist, statesman, and academician. Biography He was born in the town of ...
incriminated Luzin with charges of disloyalty to Soviet power. The methods of political insinuations and slander had been used against the old Muscovite professorship already several years before the article in ''Pravda''. The hearings were completed in five sessions between July 7, 1936, and July 15, 1936, and people testifying, as well as the nature of accusations, changed from one session to another. In the initial session, the accusations were separated into accusations of scientific misconduct, which included plagiarism; accusations of professional misconduct, which mostly involved accusations of nepotism in promotions and reviews; and political accusations, which were the most serious. The initial review on July 7, which most prominently featured Alexandrov and Kolmogorov, concluded in a warning to Luzin regarding plagiarism while stressing the overall importance of his work, cleared him politically, yet recommended to relieve him of administrative duties. However, this outcome did not seem to satisfy the instigators of the case, so that from the second hearing on, the nature of accusations shifted: now the primary focus was the fact that Luzin published his papers extensively in France rather than in Soviet journals, and his pre-Soviet sympathies were brought to the forefront. The special hearing of the Commission of the Academy of Sciences of the Soviet Union endorsed all accusations of Luzin as an "enemy under the mask of a Soviet citizen." Although the Commission convicted Luzin, he was neither expelled from the academy nor arrested, but his department in the
Steklov Institute Steklov Institute of Mathematics or Steklov Mathematical Institute () is a premier research institute based in Moscow, specialized in mathematics, and a part of the Russian Academy of Sciences. The institute is named after Vladimir Andreevich Ste ...
was closed and he lost all his official positions. There has been some speculation about why his punishment was so much milder than that of most other people condemned at that time, but the reason for this does not seem to be known for certain.
Historian of mathematics The history of mathematics deals with the origin of discoveries in mathematics and the History of mathematical notation, mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples ...
Adolph P. Yushkevich Adolph-Andrei Pavlovich Yushkevich (; 15 July 1906 – 17 July 1993) was a Soviet historian of mathematics, leading expert in medieval mathematics of the East and the work of Leonhard Euler. He is a winner of George Sarton Medal by the History ...
speculated that at the time,
Stalin Joseph Vissarionovich Stalin (born Dzhugashvili; 5 March 1953) was a Soviet politician and revolutionary who led the Soviet Union from 1924 until Death and state funeral of Joseph Stalin, his death in 1953. He held power as General Secret ...
was more concerned with the forthcoming Moscow trials of
Lev Kamenev Lev Borisovich Kamenev. ( Rozenfeld; – 25 August 1936) was a Russian revolutionary and Soviet politician. A prominent Old Bolsheviks, Old Bolshevik, Kamenev was a leading figure in the early Soviet government and served as a Deputy Premier ...
,
Grigory Zinoviev Grigory Yevseyevich Zinoviev (born Ovsei-Gershon Aronovich Radomyslsky; – 25 August 1936) was a Russian revolutionary and Soviet politician. A prominent Old Bolsheviks, Old Bolshevik, Zinoviev was a close associate of Vladimir Lenin prior to ...
and others, whereas the eventual fate of Luzin was of a little interest to him. The 1936 decision of the Academy of Sciences was not canceled after Stalin's death. The decision was finally reversed on January 17, 2012.


Honors

In 1976, Martian crater '' Luzin'' was named in his honor.


Bibliography

*Luzin, N. N. (1931). ''Two letters by N. N. Luzin to M. Ya. Vygodskii. With an introduction by S. S. Demidov.'' Translated from the 1997 Russian original by Abe Shenitzer.
The American Mathematical Monthly ''The American Mathematical Monthly'' is a peer-reviewed scientific journal of mathematics. It was established by Benjamin Finkel in 1894 and is published by Taylor & Francis on behalf of the Mathematical Association of America. It is an exposito ...
107 (2000), no. 1, 64–82.


See also

* Denjoy–Luzin theorem * Denjoy–Luzin–Saks theorem * Lusin's theorem * Luzin space


References


Further reading

* The Case of Academician Nikolai Nikolayevich Luzin / eds. Sergei S. Demidov, Boris V. Levshin; trans. Roger Cooke. — American Mathematical Society, 2016. — 416 p. — (History of Mathematics, 43). — . — . * Laugwitz, Detlef: Comments on the paper: "Two letters by N. N. Luzin to M. Ya. Vygodskiĭ". Amer. Math. Monthly 107 (2000), no. 1, 64–82.


External links

* * * Lorentz G.G.
Mathematics and Politics in the Soviet Union from 1928 to 1953
* Kutateladze S.S.
The Tragedy of Mathematics in Russia
* Kutateladze S.S.
Roots of Luzin's Case
{{DEFAULTSORT:Luzin, Nikolai 1883 births 1950 deaths 20th-century Russian mathematicians People from Irkutsk Academic staff of Moscow State University Full Members of the USSR Academy of Sciences Moscow State University alumni Recipients of the Order of the Red Banner of Labour Set theorists Soviet mathematicians Burials at Vvedenskoye Cemetery Russian scientists