In Paris Stoilow was able to benefit from being in a major centre for mathematical research. He was able to attend lectures by Picard, Poincaré, Goursat, Hadamard, Borel and Lebesgue. Later in his career he wrote articles on some of these outstanding French mathematicians (for example Mathematical work of Henri Lebesgue (Romanian) (1942) and Émile Borel and modern mathematical analysis (Romanian) (1956)). Stoilow's thesis advisor was Émile Picard, and in 1914 he submitted his doctoral thesis Sur une classe de fonctions de deux variables definies par les equations lineaires aux derivees partielles Ⓣ. The thesis was a remarkable piece of work which studies the Cauchy problem for initial data containing singularities. However, at this time World War I broke out and it was not possible for Stoilow to defend his thesis until 1916. He did, however, publish his first paper in 1914, namely Sur les intégrales des équations linéaires aux dérivées partielles à deux variables indépendantes Ⓣ. He published two further papers, namely Sur les fonctions quadruplement périodiques Ⓣ (1915) and Sur l'intégration des équations linéaires aux dérivées partielles et la méthode des approximations successives Ⓣ (1916), before publishing his doctoral thesis in 1916. He was occupied with war service during World War I so was not in a position to begin his career until after the war ended.
In 1919 Stoilow was appointed as a lecturer in the Department of Mathematical Analysis at the University of Iasi. He did not spend long in this department before moving in January 1920 to the Department of Higher Algebra. He published three further papers in 1919 including Sur les singularités mobiles des intégrales des équations linéaires aux dérivées partielles et sur leur intégrale générale Ⓣ, and two further papers in 1920. He left Iasi in 1921 when he was appointed as a lecturer in the Department of Analysis at Bucharest University. After two years in Bucharest, he was named Professor of Function Theory and Higher Algebra at Cernauti University. Cernauti is the Romanian name for the city which at that time was in Romania but following World War II became part of the Ukraine and is now called Chernivtsi. Stoilow was Dean of Cernauti University for two periods during the sixteen years he spent there, namely in 1925-26 and again in 1932-39.
Stoilow returned to Bucharest in 1939 when he was appointed Head of the Department of the Theory of Functions at the Polytechnic Institute, succeeding Dimitrie Pompeiu. He served as rector of Bucharest University during 1944-45 and he was Dean of the Faculty of Physics and Mathematics from 1948 to 1951.
The articles  and  both give a slant on Stoilow's political views, and both also give an excellent survey of his mathematical contributions. Andreian Cazacu writes :-
After the 24th August 1944 [Stoilow] wholeheartedly joined the ranks of those working for the democratisation and reconstruction of the country under the leadership of the Romanian Communist Party.However Constantinescu strongly criticises occupation of Romania immediately after the war by the Soviet army :-
... which executed orders of Moscow ... in the direction of the 'russification' of the country (e.g. the citation of a Romanian author instead of a Russian one was sometimes dangerous, since it represented the crime of 'cosmopolitanism').Constantinescu writes:-
I prepared my master's thesis with Stoilow. Finding out my political situation, Stoilow fought with might and main in order to help me.He recounts episodes of Stoilow's political activities in  where he also gives an excellent account of his mathematical work:-
Stoilow was known as a Romanian mathematician with influential work in the field of complex analysis. ... His scientific production was influential in the development of the modern theory of analytic functions, and spanned the period from 1914 through 1972, with 77 titles listed in the paper.The philosophy behind Stoilow's approach to mathematics was summed up by his own statement that his aim was:-
... to deepen what is most essential and characteristic for the phenomenon of analyticity given to us, it may be said, by Nature within which we live.Before he took up his first university appointment in 1919, Stoilow concentrated on the theory of partial differential equations in the complex domain. After this he changed somewhat the direction of his work and began to undertake research on the theory of functions of a real variable and on topology. An example of one of his papers from this period is Sur l'inversion des fonctions continues Ⓣ (1925). From around 1927 he began to work on the topological theory of analytic functions. Three theorems of Stoilow, published in 1928, 1932 and 1935, constitute his main contribution to the topological theory of analytic functions, a field of which he must be considered one of the founders. The fundamental paper he published on this topic was Sur les transformations continues et la topologie des fonctions analytiques Ⓣ (1928) in which he solved a problem proposed by Brouwer to give a topological characterisation of analytic functions. His work was having a major international impact and he was invited to Paris where he gave a series of lectures on his work in February 1931.
In 1936 Stoilow addressed the International Congress of Mathematicians in Oslo. In his lecture he introduced covering surfaces of abstract Riemann surfaces. His paper was deemed of great importance by Ahlfors who included a reference to it in his report on the highlights of the conference. His book Leçons sur les principes topologiques de la théorie des fonctions analytiques Ⓣ, published in the prestigious Collection Borel (Paris, 1937), became a classical reference in the 1940s. In this important work he introduced covering surfaces with the Iversen property and the concept of boundary element. The original edition of this work was republished in 1956. Andreian Cazacu writes :-
Those who study this deeply original book, epoch-making for topology as well as for function theory, are struck by the exceptional variety and richness of the results and the mastery with which the author passes from the concrete intuition of geometrical facts to the most abstract generalisations.Stoilow gave a series of six lectures on Riemann surfaces at the Istituto di Alta Matematica in Rome in April, 1957. These lecturers were published as Sur quelques points de la théorie moderne des surfaces de Riemann Ⓣ (1957) and provide an excellent account of Stoilow's contributions and how they fit around other progress in the same areas over the years.
In the two volumes of Theory of functions of a complex variable (Romanian) (1954, 1958), we see lecture courses which Stoilow gave at the University of Bucharest. After a fairly standard introduction to the general theory, beginning with power series, he goes on, in volume 1, to look at topics such as entire and meromorphic function, doubly periodic functions, conformal mapping on the boundary of a Jordan region, multiple-valued functions, and applications of modular functions to the Picard circle of ideas. The second volume has the following chapter headings: The Dirichlet problem; Local properties of harmonic functions; The Dirichlet problem for multiply-connected domains; The Dirichlet integral and the minimum principle; Green's function, Lindelöf's principle, the principle of harmonic measure; Harmonic measure; Riemann surfaces; Analytic functions on closed Riemann surfaces; Analytic functions on open Riemann surfaces; Regularly and normally exhaustible Riemann surfaces.
In 1936 Stoilow was elected a corresponding member of the Romanian Academy of Sciences and he became a full member in 1945. He became president of the Physics and Mathematics section of the Academy as well as director of the Mathematical Institute of the Academy. He received many honours in State Prizes for his outstanding mathematical contributions and his tireless work in raising the level of scientific research in Romania.
Article by: J J O'Connor and E F Robertson
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