As a topic for his doctorate Zaremba looked to build on ideas introduced by Riemann in 1861. His doctoral thesis Sur un problème concernant l'état calorifique d'un corp homogène indéfini Ⓣ was presented in 1889. Zaremba made many contacts with mathematicians of the French school at this time which would provide him with international collaborators after returning to Poland. In particular he collaborated with Painlevé and Goursat.
For eleven years he taught in schools in France, during which time he concentrated hard on his research. The fact that he published his results in French mathematical journals meant that his work became well known and highly respected by leading French mathematicians such as Poincaré and Hadamard.
Zaremba returned to Poland in 1900 where he was appointed to a chair in the Jagiellonian University in Kraków. In the following years he achieved much in teaching, writing textbooks, and organising the progress of mathematics in Kraków. Stanisław Golab, a differential geometer, wrote on the history of mathematics in Poland. He described Zaremba's teaching style (see ):-
[Zaremba's] teaching was characterised by absolute rigour and an insistence on an exposition of a subject's subtleties. His lecturing style employed long and convoluted sentences, whose logical progression became clear only after closer scrutiny. He enjoyed working on and solving difficult problems that bogged down other researchers. Always taking a philosophical view of a problem, Zaremba combined physical intuition with enormous erudition, a method that enabled him to connect seemingly unrelated problems.When the Mathematical Society of Kraków was set up in 1919, Zaremba chaired the inaugural meeting and was elected as the first President of the Society. This Society went on to became the Polish Mathematical Society in 1920. For many years he served the Society as editor of the Annals of the Polish Mathematical Society.
From very unpromising times up to World War I, with the recreation of the Polish nation at the end of that war, Polish mathematics entered a golden age. Zaremba played a crucial role in this transformation. Slebodzinski, one of the mathematicians to work for the re-establishment of Polish mathematics after the Nazi destruction of the World War II, stressed the importance of the role of Zaremba in the creation of the golden age between the wars (see for example ):-
With the appearance of these two scholars [Zaremba and Zorawski], Polish mathematics ceased to consume exclusively other people's thoughts and results and from that moment onwards began to participate actively in the development of its own science. The political circumstances of the period were such that, for a decade or more, Stanisław Zaremba and Kazimierz Zorawski were the only representatives of Polish mathematics in contact with foreign countries.Much of Zaremba's research work was in partial differential equations and potential theory. He also made major contributions to mathematical physics and to crystallography.
He made important contributions to the study of viscoelastic materials around 1905. He showed how to make tensorial definitions of stress rate that were invariant to spin and thus were suitable for use in relations between the stress history and the deformation history of a material. He studied elliptic equations and in particular contributed to the Dirichlet principle. In  his contribution is described as follows:-
In the work of the eminent Polish mathematician Stanisław Zaremba (1863 - 1942), the problem of an axiomatic development of classical mechanics plays an important role, as is well known, this problem constitutes part of Hilbert's Sixth Problem. Starting with the works of G Hamel, this question has been studied by many specialists in mechanics, mathematics and logic.In  the authors describe Zaremba's axiomatic justification of the notion of time in classical mechanics which he worked on during the period from 1933 to 1940. I have spoken to someone who was a student at the Jagiellonian University in Kraków during this last period of Zaremba's life. By this time Zaremba was essentially retired from normal teaching duties but still came to give special lectures and was often seen by the students who held him in great respect and in some awe. He had a reputation as a hard examiner, someone who would expect a lot of his students and who thought up hard problems to spring on them in oral examinations. There was another side to him, however, for despite this reputation as a hard examiner, Zaremba showed great kindness and understanding to students who approached the oral examination in fear and trembling.
Lebesgue, someone who seldom heaped praise on his colleagues, paid tribute to him in 1930 when Zaremba received an honorary degree from the Jagiellonian University in Kraków (see for example  or ):-
Zaremba's scientific activity influenced so many research areas that his name cannot be unknown to anyone interested in mathematics. However, it seems that the power of the methods he created, and the originality of his imagination, can be appreciated best by those who work in the area of mathematical physics. There he showed his style and his name is imprinted forever.On the same occasion in 1930, Hadamard also described Zaremba's contributions (see for example ):-
One cannot help mentioning the ideas which he inspired in the domain of research pertaining to those fields to which French science of the present century has devoted the most effort. The profound generalisation due to him has recently transformed the foundations of potential theory and immediately became the starting point of research by young mathematicians of the French school. That generalisation, in a degree truly unexpected in that field, is marked by that simplicity and elegance which characterise ideas pertinently and profoundly grasping the nature of things. And as for my speciality, why, how could I forget the splendid results in the domain of mixed boundary problems and of harmonic functions, as well as of hyperbolic equations, research by means of which he opened a new path along which contemporary knowledge will proceed in the near future.Zaremba received many honours. In addition to the honorary degree from the Jagiellonian University mentioned above, he received honorary degrees from Caen and Poznan. He was elected to the Soviet Academy in 1925. Kuratowski writes in :-
Stanisław Zaremba is the pride of Polish science.
Article by: J J O'Connor and E F Robertson
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