Nikolai Luzin

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Nikolai Luzin
Born (1883-12-09)9 December 1883
Irkutsk, Russian Empire
Died 28 January 1950(1950-01-28) (aged 66)
Moscow, Soviet Union
Citizenship Russian Empire
Soviet Union
Nationality Russia
Fields Mathematician
Institutions Moscow State University
Steklov Mathematical Institute
Polytechnical Institute Ivanovo-Voznesensk
Alma mater Moscow State University
Doctoral advisor Dmitri Egorov
Doctoral students Pavel Aleksandrov
Nina Bari
Aleksandr Khinchin
Andrey Kolmogorov
Alexander Kronrod
Mikhail Lavrentyev
Alexey Lyapunov
Lazar Lyusternik
Pyotr Novikov
Lev Schnirelmann
Pavel Urysohn
Known for contribution to descriptive set theory, mathematical analysis, point-set topology; Luzin's theorem, Lusin spaces, Luzin sets;

Nikolai Nikolaevich Luzin (also spelled Lusin; Russian: Никола́й Никола́евич Лу́зин; IPA: [nʲɪkɐˈlaj nʲɪkɐˈlaɪvʲɪtɕ ˈluzʲɪn]; 9 December 1883 – 28 January 1950) was a Soviet/Russian mathematician known for his work in descriptive set theory and aspects of mathematical analysis with strong connections to point-set topology. He was the eponym of Luzitania, a loose group of young Moscow mathematicians of the first half of the 1920s. They adopted his set-theoretic orientation, and went on to apply it in other areas of mathematics.


He started studying mathematics in 1901 at Moscow University, where his advisor was Dimitri Egorov. Luzin underwent great personal turmoil in the years 1905 and 1906. He wrote to Pavel Florensky that: 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 ... 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. ... I have been here about five months, but have only recently begun to study.[1] From 1910 to 1914 he studied at Göttingen, where he was influenced by Edmund Landau. He then returned to Moscow and received his Ph.D. degree in 1915. During the Russian Civil War (1918–1920) Luzin left Moscow for the Polytechnical Institute Ivanovo-Voznesensk (now called Ivanovo State University of Chemistry and Technology). He returned to Moscow in 1920. On 5 January 1927 Luzin was elected as a corresponding member of the USSR Academy of Sciences and became a full member of the USSR Academy of Sciences first at the Department of Philosophy and then at the Department of Pure Mathematics (12 January 1929).

In the 1920s Luzin organized a famous research seminar at Moscow University. His doctoral students included some of the most famous Soviet mathematicians: Pavel Aleksandrov, Nina Bari, Aleksandr Khinchin, Andrey Kolmogorov, Alexander Kronrod, Mikhail Lavrentyev, Alexey Lyapunov, Lazar Lyusternik, Pyotr Novikov, Lev Schnirelmann and Pavel Urysohn.

Research work[edit]

Luzin's first significant result was a construction of an almost everywhere divergent trigonometric series with monotonic convergence to zero coefficients (1912). This example disproved the Pierre Fatou 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.

His Ph.D. thesis entitled 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 for a square-integrable function, was solved by Lennart Carleson[2] in 1966 (Carleson's theorem).

In the theory of boundary properties of analytic functions he proved an important result on the invariance of sets of boundary points under conformal mappings (1919).

Luzin was one of the founders of descriptive set theory.[3] Together with his student Mikhail Yakovlevich Suslin, he developed the theory of analytic sets.

He also made contributions to complex analysis, the theory of differential equations, and numerical methods.[4]

Letter to Vygodsky[edit]

In a letter to Vygodsky dating from 1932, Luzin expresses sympathy with Vygodsky's infinitesimal 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.[5]

Luzin affair of 1936[edit]

On 21 November 1930, the declaration of the “initiative group” of the Moscow Mathematical Society which consisted of Luzin's former students Lazar Lyusternik and Lev Shnirelman along with Alexander Gelfond and Lev Pontryagin claimed that “there appeared active counter-revolutionaries among mathematicians”. [6] Some of these mathematicians were pointed out, including the advisor of Luzin, Dmitri Egorov. In September 1930, Egorov was arrested on the basis of his religious beliefs. After arrest, he left the position of the director of the Moscow Mathematical Society. The new director was Ernst Kolman. As a result, Luzin left the Moscow Mathematical Society and Moscow State University. Egorov died on 10 September 1931, after a hunger strike initiated in prison. In 1931, Kolman made the first complaint against Luzin.

In 1936 the Great Purge began. Unknown masses of people were arrested and/or executed, including leading members of the intelligentsia.[7] In July–August of that year, Luzin was criticized in Pravda in a series of anonymous articles whose authorship later was attributed to Kolman.[8] The attack on Luzin was supported by some of his students and was instigated by a letter of Pavel Aleksandrov.[9] 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”, stood close to the ideology of the “black hundreds”, orthodoxy, and monarchy “fascist-type modernized but slightly.” [10] Luzin was tried at a special hearing of the Commission of the Academy of Sciences of the USSR, which endorsed all accusations of Luzin as an "enemy under the mask of a Soviet citizen." [10] One of the complaints was that he published his major results in foreign journals. Aleksandrov, Kolmogorov and some other students of Luzin accused him of plagiarism and various forms of misconduct.[11] Sergei Sobolev and Otto Schmidt incriminated him with charges of disloyalty to Soviet power. The methods of political insinuations and slander were used against the old Muscovite professorship many years before the article in Pravda. Although the Commission convicted Luzin, he was neither expelled from the Academy nor arrested, but his department in the Steklov Institute 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 A.P. Yushkevich speculated that at the time, Stalin was more concerned with the forthcoming Moscow Trials of Lev Kamenev, Grigory Zinoviev and others, whereas the eventual fate of Luzin was of a little interest to him.[12] The 1936 decision of the Academy of Sciences was not canceled immediately after Stalin's death.[10] [13] The decision was finally reversed on January 17, 2012.[14][15]

See also[edit]


  1. ^ C. E. Ford, The influence of P A Florensky on N N Luzin, Historia Mathematica 25 (1998), 332-339, obtained from MacTutor History of Mathematics Archive.
  2. ^ Carleson L. (1966). "On convergence and growth of partial sums of Fourier series". Acta Math. 116: 135–157. doi:10.1007/BF02392815. 
  3. ^ Lusin Nicolas (1930). Leçons sur les Ensembles Analytiques et leurs Applications. With a preface by Henri Lebesgue and a note by Waclaw Sierpinski. Paris: Gauthier-Villars. p. 328. 
  4. ^ For example, his work

    Лузин, Н. Н. (1931). "О методе академика А. Н. Крылова составления векового уравнения". Известия Академии наук СССР. VII серия 7: 903–958. JFM 57.1455.01. 

    is devoted to the Krylov subspace method
  5. ^ Katz, Mikhail; Tall, David (2011), Tension between Intuitive Infinitesimals and Formal Mathematical Analysis, Bharath Sriraman, Editor. Crossroads in the History of Mathematics and Mathematics Education. The Montana Mathematics Enthusiast Monographs in Mathematics Education 12, Information Age Publishing, Inc., Charlotte, NC, arXiv:1110.5747 
  6. ^ Bogolyubov A. N. and Rozhenko N.M. "The Experiment of ‘Implanting’ Dialectics into Mathematics from the End of the 1920s to the Beginning of the 1930s.". Problems of Philosophy. No.9, 1991 (in Russian): 32–43. 
  7. ^ See Great Purge article
  8. ^ Levin, A. E. (1990). "Anatomy of a public campaign: "Academician Luzin`s case" in Soviet political history.". Slavic Review (Slavic Review, Vol. 49, No. 1) 49 (1): 90–108. doi:10.2307/2500418. JSTOR 2500418. 
  9. ^ S.P. Novikov My Stories
  10. ^ a b c Demidov, S. S.; Levshin, B. V. (eds.) (1999). Delo akademika Nikolaya Nikolaevicha Luzina. (Russian) (The case of Academician Nikolai Nikolaevich Luzin). St. Petersburg: Russkii Khristianskii Gumanitarnyi Institut. ISBN 5-88812-103-7. MR 1790419. 
  11. ^ Graham L., Kantor J.-M. Naming Infinity: A True Story of Religious Mysticism and Mathematical Creativity. Belknap Press, Cambridge and London (2009).
  12. ^ A.P. Yushkevich, The Lusin Affair (in Russian).
  13. ^ Demidov, Sergei S.; Ford, Charles E. (1996). N. N. Luzin and the affair of the "National Fascist Center". San Diego, CA: Academic Press. pp. 137–148. ISBN 5-88812-103-7. MR 1388788. 
  14. ^ "Resolution No.8 of the Presidium of the Russian Academy of Sciences" (in Russian). RAS. 
  15. ^ S.S. Kutateladze, An Epilog to the Luzin Case, Siberian Electronic Mathematical Reports, Vol. 10 (2013),A.1-6.


  • 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. American Mathematical Monthly 107 (2000), no. 1, 64–82. JSTOR 2589382
  • 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. here

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