Nikolai Nikolaevich Luzin (also spelled Lusin; Russian: Николай Николаевич Лузин; 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.

Life

He started studying mathematics in 1901 at Moscow State University, where his advisor was Dimitri Egorov. 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, 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 ... 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, 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.

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 Aleksandrov, Nina Bari, Aleksandr Khinchin, Andrey Kolmogorov, Alexander Kronrod, Mikhail Lavrentyev, Alexey Lyapunov, Lazar Lyusternik, Pyotr Novikov, Lev Schnirelmann and Pavel Urysohn.

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 1929 he was elected as a member of the Polish Academy of Sciences and Letters in Kraków.

Research work

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 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 for a square-integrable function, was solved by Lennart Carleson 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. 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.

In a letter to M. Ya. 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.

Luzin affair of 1936

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 Schnirelmann along with Alexander Gelfond and Lev Pontryagin claimed that “there appeared active counter-revolutionaries among mathematicians”. 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. He then left the position of director of the Moscow Mathematical Society and was replaced by 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 brought 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. In July–August of that year, Luzin was criticized in Pravda in a series of anonymous articles whose authorship later was attributed to Kolman. The attack on Luzin was supported by some of his students and was instigated by a letter of Pavel Aleksandrov. 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.” 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. (According to some researchers, Aleksandrov 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 and Otto Schmidt 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.

In 1936 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." 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.

The 1936 decision of the Academy of Sciences was not canceled after Stalin's death. The decision was finally reversed on January 17, 2012.