**Ivan Ivanovich Privalov**'s parents were Evdokia Lvovna and Ivan Andreevich Privalov. Ivan Andreevich was a merchant working in Nizhny Lomov but he also owned a foundry in the major city of Nizhny Novgorod about 180 km to the north. Evdokia Lvovna was from a similar background as Ivan Andreevich coming from a family of merchants. Ivan Ivanovich's early education was in Nizhny Lomov after which he attended the Gymnasium in Novgorod from 1901 to 1909, graduating with a gold medal. He then entered the mathematics department of the Faculty of Physics and Mathematics of the University of Moscow where he was taught by several world-class mathematicians and physicists including K A Andreev, D F Egorov, L K Lakhtin, P P Lazarev, P N Lebedev, B K Mlodzeevskii, and N E Zhukovsky.

Of the mathematicians, Konstantin Alekseevich Andreev was best known for his work on geometry and was Dean of the Faculty during Privalov's undergraduate years, Dimitri Fedorovich Egorov was a leading researcher in differential geometry and integral equations, Leonid Kuzmich Lakhtin was interested in analysis and probability, and Boleslav Kornelievich Mlodzeevskii had been the first to give lectures at Moscow University on set theory and the theory of functions. Petr Nikolaevich Lebedev was an outstanding physicist and Petr Petrovich Lazarev, also an excellent physicist, was one of his former students. Nikolai Egorovich Zhukovsky was the world expert on airfoils and the dynamics of flight, and was Head of the Department of Mechanics. Privalov was most attracted by Egorov's lectures and began to attend his seminar. We should mention that Nikolai Nikolaevich Luzin, although seven years older that Privalov, was a student at the same time. The two became close friends and later collaborated in writing mathematical papers.

During his undergraduate years, Privalov was an active member of the Student Mathematics Circle and served as its President. He was described by a fellow student as "a tall, handsome, thoughtful youth". In 1911 he went to the University of Göttingen (with his friend Luzin) for the summer semester and there he attended lectures by David Hilbert, Felix Klein, and Edmund Landau. He graduated from the University of Moscow in 1913 after being examined on his paper *The reducibility problem in the theory of linear differential equations*. Egorov was very impressed with Privalov's abilities and recommended that he remain at Moscow University to undertake research. He wrote (see for example [9]):-

Mr Privalov has already distinguished himself during his first years at the University by his interest in science, his erudition in the mathematical literature, and his obvious mathematical talent ... He often gave talks at the student Mathematical Circle and in the current year he gave a lecture at the Moscow Mathematical Society. In this lecture he gave an extension of the theorems of Cantor and Fatou on general expansions by orthogonal functions, and with absolute certitude has already shown his capacity for independent mathematical work ... I am convinced that Mr Privalov will develop into an outstanding scientist.

The lecture by Privalov on 18 December 1912 to the Moscow Mathematical Society, which Egorov refers to in this recommendation, was *Properties of expansions in terms of orthogonal functions* and it became his first paper appearing in print in 1914. The examinations for his Master's Degree took place in 1915 and in February of the following year he gave two lectures as part of the examination. He was allowed to choose the topic for one of these lectures - *Picard's Theorem* - while the other - *Summation of Trigonometrical Series* - was on a topic set by the Faculty. After these tests he was given the right to teach as a dozent at the University of Moscow. This was a period of change in the Russian education system and although he was preparing a thesis on the boundary behaviour of analytic functions, the thesis requirement was abolished before he had submitted it. He published the material he had prepared for his thesis as the monograph *The Cauchy Integral* in 1918. This important work built on foundations set up by Pierre Fatou [1]:-

Because "Cauchy Integral" appeared at a time when scientific contacts between Russia and other countries were almost nonexistent, it did not attract attention abroad.

In 1917 Privalov became professor at Saratov University where he remained for five years, then returned to Moscow [7]:-

In1922Privalov returned to Moscow. He was rather tall, with a pale, interesting, nervous face. His manners and speech usually seemed quiet, but when he was excited he spoke faster, and stammered slightly. Privalov smoked a lot, more as time went on.

He was appointed professor and Head of the Department of Function Theory at the Institute of Mathematics and Mechanics of the University of Moscow when the Institute was founded in 1922. He held this post for the rest of his life. He was also appointed as a professor in the Department of Higher Mathematics at the Zhukovsky Air Force Academy and in the Department of Higher Mathematics of the Lomonosov Automechanical Institute. Lazar Aronovich Lyusternik became a research student in the Institute of Mathematics and Mechanics in 1922. In [7] he gives a first hand account of Privalov's life in the Institute:-

At Moscow University Privalov began to give, with great enthusiasm, a course on the theory of functions of a complex variable(before him this course had been given by B K Mlodzeevskii). On the basis of this course he wrote a famous university text-book on complex function theory. Then(in his own words)he got tired of this course and began to give(also with great enthusiasm)a course of analysis, On his arrival Privalov became the oldest scientific worker at the Institute of Mathematics and Mechanics at Moscow University. For the research students of my own and the next generation he regularly conducted examinations(on various topics). So you see him at a long table in the professors' room, smoking, silently listening to a student sitting beside him, sometimes making short notes. Privalov was my opponent for my final Ph.D. thesis.

Lyusternik was also able to paint a picture of Privalov's home life [7]:-

To begin with, Privalov lived in a small house on the site of the Lomonosov Institute in Blagoveshchenskii Lane, and later he moved into a three-roomed flat in a two-story block on the same site. His family consisted of his wife, Anna Ilinichna, a pretty, friendly woman, active and energetic, and a young daughter. The rooms were joined by a number of doors and, according to his friends, Privalov loved to walk to and fro over the whole flat. It was comfortable at the Privalovs(in contrast to the majority of mathematicians' flats at the time, to say nothing of the bachelors' rooms). Privalov was a sociable man, he liked people to come to him, he would meet his guests with a smile, and show them into his study, opening the door with a sweeping gesture. And later the hospitable Anna Ilinichna would invite them for a cup of tea in the dining-room. There a lamp burned cosily under a brown shade. Privalov often accompanied his guests, not only to the tram, but sometimes to their homes.

Privalov regularly lectured to the Moscow Mathematical Society following his appointment in 1922. The titles of his lectures over the years 1922-25 were: *A generalization of a theorem of Fatou* (1922), *Properties of the coefficients of a Taylor series* (1923), *The uniform convergence of sequences of analytic functions which give a 'schlicht' conformal mapping* (1923), *The uniqueness of an analytic function* (1923), *A generalization of Vitali's theorem on sequences of analytic functions* (1924), *The convergence of conjugate trigonometric series* (1924), *The convergence of sequences of analytic functions* (1924), *Concerning a condition of Blaschke* (1925), *A new definition of a harmonic function* (1925), *Harmonic functions* (1925). Privalov was appointed secretary of the Moscow Mathematical Society and later served as a vice-president. Lyusternik writes in [7]:-

Privalov's closest friends were V V Stepanov and V I Veniaminov. Later Privalov kept up friendly relations with his former pupils, especially M A Kreines and S A Galpern. In the thirties, after a session of the Mathematical Society or a long seminar conducted by Privalov on the theory of functions, the mathematicians would gather in a small restaurant at the end of Tverskoi; Privalov was invariably present, sitting with a glass of brandy, and interjecting brief remarks into the excited mathematical conversation.

Privalov, often in collaboration with Luzin, studied analytic functions in the vicinity of singular points by means of measure theory and Lebesgue integrals. He also obtained important results on conformal mappings showing that angles were preserved on the boundary almost everywhere. In 1934 he studied subharmonic functions, building on the work of Riesz. He published the monograph *Subharmonic Functions* in 1937 which gave the general theory of these functions and contained many results from his papers published between 1934 and 1937. Sergey Ivanovich Vavilov, a leading Russian physicist, wrote about Privalov's research contributions (see for example [9]):-

Professor Privalov is among the most distinguished mathematicians of the Soviet Union. Important research in the theory of functions is due to him. In this domain he has obtained profound results on the theory of trigonometric series, on integrals of Cauchy type, on boundary value problems, in the study of properties of analytic functions inside domains and on the theory of subharmonic functions.

Sergei Alekseevich Chaplygin and Nikolai Nikolaevich Luzin explained Privalov's approach to research (see for example [9]):-

When reviewing the scientific activity of I I Privalov one should note above all the wide range of questions in mathematical analysis, including the theory of functions of a complex variable, and also, the strength of his results. Privalov never studied a particular problem for the sake of the problem itself. He always placed the problem in connection with all known methods developed by mathematical analysis, constantly trying to extend the very methods of mathematical analysis, to improve them and to deepen their strength as much as possible. Very characteristic of Privalov, in this connection, is the theory of subharmonic functions, which he created and in which he tried to combine, improve and deepen the most important methods of the theory of functions of a complex variable.

Pavel Sergeevich Aleksandrov explained why Privalov had very few students (see for example [9]):-

A universal character in both mathematical and general cultural interests gave exceptional scope and rare breadth to Hausdorff's university lectures. But probably because of this he did not have his own students: he demanded too much of them and he himself knew too much. This characteristic, incidentally, was also possessed by our great Soviet mathematician I I Privalov, who for this reason did not have many students: not everyone could endure his extraordinary exactitude. The knowledge of the scientist himself apparently subdued many beginning young people ... It seemed that it was impossible to imitate him, and students are not able to avoid imitating their teacher to some extent.

Privalov wrote a number of research monographs, some of which we have already mentioned above. We must also note the important *Subharmonic Functions* (1937), and *Boundary Properties of Single-Valued Analytic Functions* (1941) Wolf Frantisek begins his review of this 1941 publication as follows:-

This book of336pages deals with a small and most difficult section of the theory of complex variables to which the author has made very substantial contributions.

It was not only as a researcher that Privalov excelled for he also had an outstanding reputation as a teacher. An obituary written in 1941 contained this appreciation of his teaching:-

An outstanding scientist, he was also a remarkable teacher. His brilliant lectures, rich in scientific content and remarkable in their presentation, are remembered by his numerous students, by mathematicians and by engineers, now holding important scientific, military, and technical posts.

His teaching skills extended beyond his lectures for he wrote many popular textbooks. A course of lectures that he gave in Saratov gave rise to *Analytic geometry in the plane* (1918). He followed this by *Elements of mathematical analysis* (1924) and *Introduction to the theory of functions of a complex variable* (1927). The popularity of this last mentioned text is shown by the fact that a 12^{th} edition appeared in 1977. R P Boas, Jr. reviewed the book:-

This is a carefully written and detailed textbook which covers all the material customary in an introductory course and goes as far as the strong form of Cauchy's theorem, Picard's theorem via Bloch's theorem, conformal mapping on the boundary, and classical results on univalent functions.

His textbook *Analytical geometry* (1927) was another that proved extremely popular with a 12^{th} edition appearing in 1939 - editions had appeared at the rate of one per year for twelve years. A 13^{th} edition was published in 1966, twenty-five years after Privalov's death. Later textbook were: *Fourier series* (1930); *Course of differential calculus* (1934); *Course of integral calculus* (1934); *Integral equations* (1935); *Foundation of the analysis of infinitesimals, textbook for self-education* (1935); and *Elements of the theory of elliptic functions* (1939).

**Article by:** *J J O'Connor* and *E F Robertson*

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