One might expect that Nikolai would have shown a special talent for mathematics at the Gymnasium, but this was far from the case ( and ):-
This was because the system of instruction ... was based on mechanical memory: it was required to learn the theorems by heart and to reproduce their proofs exactly. For Luzin this was torture. His progress in mathematics at the Gymnasium became worse and worse, so that his father was obliged to engage a tutor ...Fortunately the tutor was a talented young man who quickly discovered that, despite Luzin's poor performance in mathematics, he could solve hard problems but often using a novel method that the tutor had never seen before. Soon the tutor had shown Luzin that mathematics was not a subject where one had to learn long lists of facts, but a topic where creativity and imagination played a major role.
In 1901 Luzin left the Gymnasium and at this time his father sold his business and the family moved to Moscow. There Luzin entered the Faculty of Physics and Mathematics at Moscow University intending to train to become an engineer. At first Luzin lived in the new family home in Moscow, but Luzin's father began to gamble on the stock exchange with the money he had made from the sale of his business. The family soon hit hard times as Luzin's father lost all their savings and the family had to leave their home. Luzin, together with a friend, moved into a room owned by the widow of a doctor. His friend soon became involved with the Revolution and was forced into hiding. Luzin stayed on by himself in the room but he clearly got on well with the owners since he later, in 1908, married the widow's daughter.
At Moscow University Luzin studied under Bugaev, learning from him the theory of functions which was to influence greatly the direction his research would eventually take. However he was only an average student who seemed to show little flair for mathematics. However, although Luzin appeared to lack talent in mathematics, one of his teachers Egorov spotted his great talent, invited him to his home, and began to set him hard problems.
There was a mathematics student at the university, Pavel Florensky, who experienced a crisis after graduating and turned to religion and the study of theology. This had a major effect on Luzin, who was a close friend of Florensky, as we shall describe below.
After graduating in the autumn of 1905 Luzin seemed unsure whether to devote himself to mathematics. In fact Luzin's crisis had hit him in the spring of 1905 and, on 1 May 1906, Luzin wrote to Florensky from Paris where Egorov had sent him five months earlier in an attempt to get him through the crisis (see ):-
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.Luzin was not only upset by seeing the prostitutes, he also says in the letter how he had been affected by the 'terrible days' of the 1905 Revolution. There are letters from Egorov at this time pleading with Luzin not to give up mathematics. After returning to Russia, Luzin studied medicine and theology as well as mathematics. However in April 1908 he wrote of the joy he was finding in number theory (see ):-
It is a mysterious area that envelops me deeper and deeper.In the same letter he says that he has just married and:-
... my wife is also very interested and shares my commitment to the search for the profound truths of life.Largely Luzin's crisis seems to have been solved by Florensky to whom Luzin wrote in July 1908:-
Two times I was very close to suicide - then I came ... looking to talk with you, and both times I felt as if I had leaned on a pillar and with this feeling of support I returned home ... I owe my interest in life to you...His interest in mathematics slowly returned but it was not until 1909 that Luzin seems to have finally committed himself completely to mathematics. Under Egorov's supervision he worked on his master's thesis. In 1910 he was appointed as assistant lecturer in Pure Mathematics at Moscow University. He worked for a year with Egorov and they went on to publish joint papers on function theory which mark the beginnings of the Moscow school of function theory.
In 1910 Luzin travelled abroad visiting Göttingen where he was influenced by Edmund Landau. He returned to Moscow in 1914 and he completed his thesis The integral and trigonometric series which he submitted in 1915. After his oral examination he was awarded a doctorate, despite having submitted his thesis for the Master's Degree. Egorov was extraordinarily impressed by the work and had pressed for the award of the doctorate, but it was written in a style quite different from the accepted Russian style of the time. Some of the results were not rigorously proved but were justified using phrases such as 'it seems to me' and 'I am convinced'. Other mathematicians were not so impressed at the time, for example Steklov wrote comments in the margin such as 'it seems to him, but it doesn't seem to me' and 'Göttingen chatter'.
However, the work was of fundamental importance as is stated in  and :-
The influence of Luzin's dissertation on the future development of the theory of functions cannot be overestimated. Its fundamental results, deep methods of investigation and fundamental statements of problems put it into the ranks of works with which it is difficult to compare any dissertation or monograph of the time.In 1914 Luzin and his wife separated for a short time and again Florensky seems to have helped them through the difficult time. He wrote to Luzin's wife (see ):-
Nikolai Nikolaevich is a very sweet and fine person; but in personal relationships he is not at all mature, especially in intuitively perceiving the hidden currents of life. ... You will have to take the relationship in hand and create a family tone, simplicity. Instead, as I perceive it ... you have established the tone of an acquaintanceship rather than a family.Florensky seems to have given good advice since Luzin and his wife returned to a successful marriage.
In 1917 Luzin was appointed as Professor of Pure Mathematics at Moscow University just before the Revolution. The Revolution caused Luzin to rethink some of the same thoughts as he had done at the time of his crisis and again he exchanged letters with Florensky. By this stage, however, his mathematical career was extremely successful and the second crisis did not materialise.
Over the next ten years Luzin and Egorov built up an impressive research group at the University of Moscow which the students called 'Luzitania'. The first students included P S Aleksandrov, M Ya Suslin, D E Menshov and A Ya Khinchin. The next students included P S Urysohn, A N Kolmogorov, N K Bari, rnik and N G Shnirelman. In 1923 P S Novikov and L V Keldysh joined the group.
Another of the members of the Luzitania research group at this time was Lavrent'ev. In fact Lavrent'ev draws the following picture of the group:-
Whereas Egorov was reserved and formal, Luzin was extroverted and theatrical, inspiring real devotion among these students and young colleagues. ... There was intense camaraderie ... inspired by Luzin.Luzin's main contributions are in the area of foundations of mathematics and measure theory. He also made significant contributions to descriptive set topology. In the theory of boundary properties of analytic functions he proved an important result in 1919 on the invariance of sets of boundary points under conformal mappings. He also studied, together with Privalov, boundary uniqueness properties of analytic functions.
From 1917 onwards, Luzin studied descriptive set theory. He stated the fundamental problem ( and ):-
The aim of set theory is a question of great importance: can we regard a line atomistically as a set of points: incidentally this question is not new, but goes back to the Greeks.Much of Luzin's work on set theory involved the study of effective sets, that is sets which can be constructed without the axiom of choice. Keldysh describes this work in  and :-
... Luzin proceeded from the point of view of the French school (Borel, Lebesgue), which greatly influenced him. But whereas the French had analysed set-theoretical constructions carried out with the help of the Axiom of Choice, Luzin went considerably further and considered difficulties arising within the theory of effective sets. The study of effective sets that he embarked upon was pursued intensively for more than two decades and led to the solution of many important problems of set theory ...Luzin's school was at its peak during the years 1922 to 1926, but then Luzin concentrated on writing his second monograph on the theory of functions and spent less time with the young mathematicians in the school. Many of these mathematicians turned to other topics such as topology, differential equations, and functions of a complex variable.
In 1927 Luzin was elected as a member of the USSR Academy of Sciences. Two years later he became a full member of first the Department of Philosophy, then to the Department of Pure Mathematics. He worked from this time until his death in the USSR Academy of Sciences. From 1935 he headed the Department of the Theory of Functions of Real Variables at the Steklov Institute.
In 1931 Luzin himself turned to a new area when he began to study differential equations and their application to geometry and to control theory. His work in this area led him to study the bending of surfaces which is described in  and :-
The bending of a surface on a principal base is a continuous bending of a surface under which the conjugacy of the net of certain curves on the surface is preserved. ... Finikov had derived differential equations that determine all principal on a given surface, and Byushgens had obtained differential equations that determine surfaces which have a given linear element and admit a bending on a principal base. However, the question of solubility of these equations, in general, remained unclear. ... no example was found in which the equations ... were insoluble ... up to 1938, when Luzin, by means of a subtle analysis of these equations, established that the existence of a principal base is rather rare.It has been drawn to our attention by , that in 1936, Luzin was the victim of a violent political campaign organized by the Soviet authorities through the newspaper Pravda. He was accused of anti-Soviet propaganda and sabotage by publishing all his important results abroad and only minor papers in Soviet journals. The aim was obviously to get rid of Luzin as a representative of the old pre-Soviet mathematical school of Moscow: his master, Egorov, had been himself the victim of such a campaign in 1930 (based on his religious sympathies) and died shortly after in 1931 in despair and misery. A contemporary record of the "Luzin affair" has been miraculously preserved and recently edited in Moscow by Demidov and Levchin , . It shows that Luzin had had a narrow escape from a tragic fate as the Soviet authorities may have feared the international consequences of a too strong attack on a scientist so famous abroad. The main visible consequence of the Luzin affair was that, from this precise moment, Soviet mathematicians began to publish almost exclusively in Soviet journals and in Russian.
Luzin always had an interest in the history of mathematics and late in his career he wrote important articles on Newton and on Euler.
As a teacher his remarkable talents are described by Kuznetsov ( or ):-
His presentation was always very elegant and at first sight apparently unnecessarily simple - the result of his great pedagogic talent. The solution of the large problems that he undertook is distinguished by their subtlety, elegance, and simplicity of presentation.Keldysh and Novikov wrote in :-
Thanks to his exceptional intuition and his ability to see deeply into the heart of a question, Luzin frequently predicted mathematical facts whose proof turned out to be possible only after many years and required the creation of completely new mathematical methods. He was one of the outstanding mathematicians and thinkers of our time ...
Article by: J J O'Connor and E F Robertson