Beholding a bar of metal, it seems an object almost primal in its simplicity. Solid, reliable, the stuff of which cities are made. Peek beneath the surface, however, and what you see is a miniature universe of swirling complexities, of tightly packed atoms and delocalised electrons clambering hither and thither with no set home, behaving in ways that can only be described statistically. That resting slab of matter is a mess mathematically, and when you put it under stress, a whole new range of bedevilments opens up.

Fortunately, in the early twentieth century, a mathematician happened along who was able to significantly tame the chaos of stressed metals, publishing the equations for metal deformation under stress that still bear her name only two years before Nazi persecution compelled her to flee her homeland. Hilda Geiringer (1893–1973) was born in Vienna on 28 September 1893, to a middle-class Viennese Jewish family. Her mother, Martha Wertheimer, was Austrian, and her father, Ludwig, a textile manufacturer, was either Hungarian or Slovakian, and both of them believed powerfully in women’s education, sending Hilda to the University of Vienna to study under Wilhelm Wirtinger after she showed prodigious mathematical talent in high school.

Wirtinger was a founding figure in spectral theory with broad interests in mathematics who had a knack for attracting some of the century’s greatest minds as his students, including Erwin Schrödinger, Kurt Gödel and Olga Taussky-Todd. Geiringer studied at the University of Vienna from 1913 to 1917, earning her doctorate in 1918 on the strength of her dissertation, ‘Trigonometrische Doppelreihen’, which dealt with double trigonometric series. In 1807, Jean-Baptiste Joseph Fourier had discovered how to represent any smooth function as a series of sines and cosines, which remains one of our most beloved of mathematical tools. Any time we can take an ill-behaved function and build it out of a set of nice, well-understood, well-behaving functions like sines and cosines, we tend to be very happy in mathematics, and Fourier’s series (which we now call Fourier Series) helped him in 1822 to solve the long-standing problem of how heat distributes itself through an object over time.

Geiringer’s work in 1918 was on *double *trigonometric series, which involves what happens when, instead of adding single sines and cosines together, as Fourier did to construct representations of functions, you investigate what happens when you add *products* of sines or cosines together. So, instead of something like a1 sin(x) + b1 cos (x) + a2 sin (2x) + b2 cos (2x) + …, Geiringer looked at quantities like a1 sin(x)sin(y) + a2 sin (2x) sin(y) + a3 sin(x)sin(2y) + …, which had been studied previously by Martin Krause and G.H. Hardy. Geiringer provided important criteria for when double trigonometric series do and don’t converge, which proved foundational to the evolution of Fourier Series in the twentieth century.

For the two years following her dissertation, Geiringer worked as an assistant editor at the *Jahrbuch über Fortschritte der Mathematik*, before moving on to the University of Berlin in 1921 as research assistant to Richard von Mises, who did research in statistics and probability and their application to problems in fluid mechanics and aerodynamics, among other mathematical interests. With that position, she became the first woman with an academic appointment in mathematics at the University of Berlin, rising in 1927 to the position of *privatdozent* in applied mathematics. She married fellow mathematician Felix Pollaczek in 1921, and gave birth to her only child, Magda, in 1922. Pollaczek and Geiringer separated in 1925, leaving Geiringer to raise Magda on her own while continuing her teaching and research responsibilities, the couple ultimately divorcing in 1932.

Professionally, Geiringer’s time at the University of Berlin resulted in arguably her most significant discovery, contained in her 1931 paper, ‘Beitrag zum vollständigen ebenen Plastizitätsproblem’. This was the paper that presented a new, and far improved, way of modelling the behaviour of metals under stress, that ultimately led to the development of slip-line theory, which still stands as an important way to visualise and compute what will happen when metals are introduced to different pressures.

What we want to be able to do is to say: when you apply this stress to this metal, this is how it will respond. How they do this in slip-line theory is by creating a net of alpha and beta lines that criss-cross the metal surface in question. The alpha lines show you the directions of maximum stress on the object, and generally speaking, the more bendy they are, the greater the average stress. Here’s an expertly rendered example of one such net that was not at all drawn on a napkin:

Previously, the Hencky equations had provided a clever way of figuring out the hydrostatic stress at any point on an alpha line, provided that you knew the stress at any other point on that line, but what they didn’t do was provide a relatively pleasant way of calculating the velocity field along the surface, i.e. of telling you where the atoms are going to *move *and how violently they will do so. The Geiringer equations do this, relating v, the velocity of a particle at a given point, to the direction of maximum stress there. These equations, while not always simple to solve for a given situation, at the very least represented a marked improvement over the prevailing methods of the time.

It was at this moment, with Geiringer at the top of her mathematical form, that German political developments derailed her career. Hitler’s rise to power in 1933 meant that Jewish intellectuals like Geiringer had to scramble for new academic positions outside of Hitler’s reach. In 1933, Geiringer fled to Brussels, and ultimately established herself in Turkey in 1934, where the government of Kemal Atatürk was determined to improve the state of Turkish education by importing two hundred world-class European professors, among them Geiringer and von Mises. Fortunately, mathematicians require less specialised equipment than, say, a physicist or a chemist, and Geiringer was able to carry on her work, publishing eighteen papers during her four-year Turkish sojourn, including interesting forays into the application of probability to biological problems that extended G.H. Hardy’s early work in that field.

The death of Atatürk in 1938 also meant the death of his grand educational initiative, and the imported professors were duly informed that their five-year contracts were not going to be renewed. With Austria consumed in the Anschluss of 1938, and all signs pointing to the full-scale European war that would break out in 1939, Geiringer’s best option for a life of peace and research appeared to be the United States, where she and von Mises emigrated in 1939.

The flight of European intellectuals to the United States in the 1930s and 1940s was a boon to American research and university development, but it also meant, in the short term, too few positions available with too many destitute but brilliant professors applying for them. For the rest of her life, Geiringer was unable to achieve a position in the United States commensurate with her abilities. From 1939 to 1944, she taught at Bryn Mawr for low pay. In 1943, she married von Mises, who was teaching at Harvard at the time, but as she was unable to find a university position in Boston, the newly married couple had to live in different states if they wanted to both keep working.

In 1944, Geiringer became the mathematics department chair at Wheaton College, in Illinois, which sounds like a step up until you realise that Wheaton’s mathematics department at the time consisted of only two people. Her publication rate slowed, her most significant work in terms of long-term impact being a mimeographed series of lectures on the Geometrical Foundations of Mechanics she delivered in the summer of 1942 at Brown University, which saw wide distribution. In 1953, von Mises died and, as often happens in the history of women in science, Geiringer’s life became largely consumed in the task of editing her late husband’s works, for which task Harvard made her, at long last, a Research Fellow. In 1958, she completed and published von Mises’s *Mathematical Theory of Compressible Fluid Flow*, which led to a collaboration with Alfred Freudenthal that produced her last paper prior to her official retirement, ‘The mathematical theories of the inelastic continuum’, which represented a continuation of her interest in, and fundamental contributions to, the study of material plasticity.

Geiringer retired in 1959, but continued editing the work of von Mises, publishing his *Mathematical Theory of Probability and Statistics *in 1964 with updated material pulling from the last decade of mathematical advances. Her final decade was spent deepening not only her love of mathematics but her larger interests in literature and music. She died in 1973 of influenzal pneumonia while visiting her brother in California.

FURTHER READING: There isn’t a full work on Hilda Geiringer, but you can find bits and pieces of her story here and there. Renate Strohmeier’s *Lexikon der Naturwissenschaftlerinnen *(1998) has a nice little column on her, and of course *Women of Mathematics: A Biobibliographic Sourcebook* has a bit more. If you want to get started in the awesome world of Fourier Theory, the book I usually recommend is Stein and Shakarachi’s *Fourier Analysis: An Introduction *(2003), which is nice and clearly laid out.

If you'd like to read more about women mathematicians like this one, check out my History of Women in Mathematics, launching in October 2023 from Pen and Sword Books!

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