After receiving her first degree, Geiringer continued her study of mathematics at Vienna, working under Wirtinger for her doctorate. This was awarded in 1917 for a thesis on Fourier series in two variables. She spent the following two years as Leon Lichtenstein's assistant editing the mathematics review journal Jahrbuchs über die Fortschritte der Mathematik .
In 1921 Geiringer moved to Berlin where she was employed as an assistant to von Mises in the Institute of Applied Mathematics. In this same year she married Félix Pollaczek who, like Geiringer, had been born in Vienna into a Jewish family but had studied in Berlin. He obtained his doctorate in 1922 working under Schur and went on to work for the Reichspost in Berlin applying mathematical methods to telephone connections. Hilda and Félix had a child, Magda Pollaczek, in 1922 but their marriage broke up. After the divorce Geiringer continued working for von Mises and at the same time brought up her child on her own.
Although trained as a pure mathematician, Geiringer moved towards applied mathematics to fit in with the work being undertaken at the Institute of Applied Mathematics. Her work at this time was on statistics, in particular probability theory, and also on the mathematical theory of plasticity. She submitted a thesis for her habilitation to the University of Berlin but it was not immediately accepted. Siegmund-Schultze writes in :-
The controversy surrounding Hilda Geiringer's application for Habilitation at the University of Berlin (1925 - 1927) sheds some light on the struggle of 'applied mathematics' for cognitive and institutional independence. The controversy and Geiringer's unpublished reminiscences reveal the decisive influence of Richard von Mises ... on both her career and the course of applied mathematics at the University of Berlin. ... The debate over Geiringer's theses for Habilitation opens up a chapter of the history of mathematical statistics, namely, expansions of a discrete distribution with an infinite number of values in a series in successive derivatives of the Poisson distribution with respect to the parameter.On 30 January 1933 Hitler came to power and on 7 April 1933 the Civil Service Law provided the means of removing Jewish teachers from the universities, and of course remove those of Jewish descent from other roles. All civil servants who were not of Aryan descent (having one grandparent of the Jewish religion made someone non-Aryan) were to be retired. Under this law, Geiringer lost the right to teach at the university in December 1933. In fact she had been proposed for appointment to the position of extraordinary professor in 1933 but the proposal had been put on hold after the Civil Service Law came into effect in April of that year.
Geiringer left Germany after she was dismissed from the University of Berlin and, together with her child, she went to Brussels. There she was appointed to the Institute of Mechanics and worked on the theory of vibrations. Von Mises, though a convert to Catholicism from Judaism, left Germany at the very end of 1933 to take up a newly founded chair of mathematics in Istanbul and, in 1934, Geiringer followed him to Istanbul. There she was appointed as professor of mathematics and Richards writes in :-
In Turkey, Geiringer was part of a larger German community that was seeking refuge from Hitler's regime. Despite the obvious difficulties associated with this exile - for example, she had to learn Turkish in order to give her lectures - Geiringer continued to pursue her mathematical interests, particularly in plasticity.In 1938 Kemal Atatürk died and those in Turkey who had fled from the Nazis feared that their safe haven would become unsafe. In 1939 von Mises left Turkey for the United States. Geiringer feared that she might not find it so easy to obtain an entrance permit to the United States and she wrote to von Mises from Istanbul:-
Is there no way to marry pro cura? Here an emigrant who has a resident's permit has married his 'bride' and she was then allowed to come to him straight from Vienna.Her fears of not getting an entrance permit were unfounded, however, and together with her daughter she went to Bryn Mawr College where she was appointed as a lecturer. Again of course, Geiringer had to learn another language in order to teach. She also had to adjust to what she referred to as "the American form of teaching" and in this she was greatly assisted by Anna Wheeler.
In addition to her lecturing duties at Bryn Mawr College, Geiringer undertook classified work for the National Research Council as part of the war effort. During 1942 she gave an advanced course in mechanics at Brown University, with the aim of raising the American standards of education to the level that had been attained in Germany. She wrote up her outstanding series of lectures on the geometrical foundations of mechanics and, although they were never properly published, these were widely used in the United States for many years.
One has to understand the problems that there were in the United States at this time as they tried, in general very successfully, to integrate many leading scientists fleeing from the Nazis into their system. Neyman, who himself had emigrated to the United States, wrote a report on Geiringer in April 1940, shortly after she arrived from Turkey. This is very fair in explaining where Geiringer fitted into the spectrum of professors of mathematics:-
Whether she is to be considered outstanding in ability or not depends on the standards of comparison. Among the present day mathematicians there are few whose names will remain in the history of mathematics ... As for the newcomers to this country, I have not the slightest doubt that von Mises is one of the men of such calibre. ... There will perhaps be a dozen or perhaps a score of such persons all over the world. ... and Mrs Geiringer does not belong in this category. But it may be reasonable to take another standard, that of a university professor of probability and statistics, perhaps an author of the now numerous books on statistical methods. In comparison with many of these people Mrs Geiringer is an outstanding person and I think it would be in the interests of American science and instruction to keep her in some university.Geiringer married von Mises in 1943 and the following year she left her lecturing post at Bryn Mawr College to be nearer to him. She accepted a post as professor and chairman of Wheaton College in Norton, Massachusetts. During the week she taught at the College, travelling to Cambridge every weekend to be with von Mises who worked at Harvard at this time.
For many reasons this was not a good arrangement. There were only two members of the mathematics faculty at Wheaton College and Geiringer longed for a situation where she was among mathematicians who were carrying out research. She made many applications for other posts but these failed due to fairly open discrimination against women. As Richards writes in  one response she received was quite typical:-
I am sure that our President would not approve of a woman. We have some women on our staff, so it is not merely prejudice against women, yet it is partly that, for we do not want to bring in more if we can get men.For Geiringer who had been so discriminated against in Germany because of her Jewish background, to now be discriminated against because she was a woman must have been a difficult blow. However, she took it all remarkably calmly, believing that if she could do something for future generations of women then she would have achieved something positive. She also never gave up her research while at Wheaton College; in 1953 she wrote:-
I have to work scientifically, besides my college work. This is a necessity for me; I never stopped it since my student days, it is the deepest need of my life.In 1953 von Mises died and the following year Geiringer, although retaining her job at Wheaton College, began to work at Harvard on editing von Mises' works. The year 1956 saw the University of Berlin elect her professor emeritus on full salary. In 1959 she formally retired from Wheaton College and the following year the College honoured her with the award of an honorary Doctorate of Science.
Article by: J J O'Connor and E F Robertson