The nursery of a house in the Russian province of Vitebsk ran short of wallpaper during a refurbishment, and one room was papered instead with sheets of lithographed lecture notes on differential and integral calculus that had belonged to the owner as a student. The child who slept in that room spent hours staring at the symbols, trying to work out the order the sheets should go in.
Sofia Kovalevskaya told the story herself, and said that when she later encountered the formulas properly, they had the strange familiarity of something known long before. She became the first woman in Europe to receive a doctorate in mathematics, and, fifteen years later, the first to hold a full professorship in the subject anywhere on the continent.
Marriage as a Travel Document
Russian universities did not admit women, and a Russian woman could not obtain a passport to study abroad without the permission of her father or her husband. Her father refused.
The standard solution among educated young Russian women of the 1860s was the fictitious marriage, and Sofia Korvin Krukovskaya entered one in 1868 with Vladimir Kovalevsky, a young palaeontologist with radical sympathies. It was intended as an arrangement of convenience for both of them, and it allowed her to leave for Germany.
She went to Heidelberg, where she was permitted to attend lectures only with the individual consent of each professor, then to Berlin in 1870, where the university refused her admission outright.
Weierstrass
Karl Weierstrass, the most demanding analyst in Europe, was approached and had no wish to take her on. To be rid of her politely, he gave her a set of problems he considered impossibly hard and told her to bring back solutions.
She returned with all of them solved, and with methods he had not expected. He taught her privately for four years, twice a week, repeating for her alone the material of his university lectures, and campaigned unsuccessfully to have her admitted formally. Their correspondence, which ran for twenty years, is one of the more affecting documents in the history of mathematics, and he burned her side of it after her death.
In 1874 he arranged for the University of Göttingen to award her a doctorate in absentia, on the strength of three papers, without the usual oral examination, which she was not permitted to sit. Any one of the three would have been sufficient.
- Partial differential equations. The result now known as the Cauchy Kovalevskaya theorem, which establishes conditions under which such an equation has a unique analytic solution near a given surface. It is a foundational result and is still taught under both names
- The rings of Saturn, extending Laplace’s analysis to the stability of a rotating ring of fluid
- Abelian integrals, a technical paper reducing a class of them to simpler elliptic forms
Six Years Without Mathematics
She returned to Russia with a doctorate from a German university, the highest qualification then available in the subject, and no institution in Europe would give her a post. The only teaching position she was offered was arithmetic to girls in an elementary school, and she remarked that she was unfortunately weak at the multiplication table.
For about six years she did almost no mathematics. She and Vladimir turned their arrangement into a real marriage and had a daughter. They speculated in property in a Moscow building boom, went badly wrong, and Vladimir became entangled in a fraudulent oil company. She wrote theatre reviews and popular science pieces for newspapers.
In 1883 Vladimir, facing prosecution and ruin, took his own life. She was in Paris, and shut herself in a room for several days.
Stockholm
The rescue came from Gösta Mittag Leffler, a former student of Weierstrass who had recently helped to establish the new university in Stockholm, and who had been trying for years to find her a position.
He got her appointed as a privatdocent in 1883, on a trial basis and without salary, paid from students’ fees. She learned Swedish quickly enough to lecture in it within a year. In 1889 she was made a full professor with tenure, the first woman in Europe to hold such a chair in mathematics.
It was not comfortable. The appointment was opposed in the Swedish press, and August Strindberg wrote a widely quoted piece arguing that a female professor of mathematics was a monstrosity. She was also the editor of Mittag Leffler’s new journal Acta Mathematica, which she used to build connections between the Russian, German and French mathematical communities.
The Spinning Top
Her best known result came in 1888, in response to a prize competition set by the French Academy of Sciences on the motion of a rigid body about a fixed point.
The problem is the mathematics of a spinning top. Euler had solved the case of a body with no gravity acting, and Lagrange the case of a symmetrical top spinning about its axis. Everyone assumed those were the only cases that could be solved exactly, and dozens of mathematicians had tried and failed to find another.
Kovalevskaya found a third: an asymmetric body with a particular relationship between its moments of inertia and the position of its centre of mass. The Kovalevskaya top is integrable, and its solution requires hyperelliptic functions of a kind she had learned from Weierstrass.
It was later proved that these three are the only integrable cases, which makes her result the last one available rather than merely another one.
The entries were judged anonymously. Hers was judged so far superior that the Academy increased the prize from three thousand francs to five thousand. When the envelope was opened, the winner was a woman, and the Academy did not follow the prize with membership.
The Other Half of Her Work
She wrote fiction and memoir seriously, and considered the two activities to be closer than her colleagues did. Her account of growing up in a Russian gentry household, published as A Russian Childhood, is still read as literature. Her novel Nihilist Girl drew on the radical circles she had known, including her sister Anna, who took part in the Paris Commune and married a Communard.
Asked to explain the combination, she said that a mathematician must be a poet in soul, and that it is impossible to be a mathematician without being one.
She died in January 1891 of pneumonia following influenza, at forty one, after a winter journey back from Genoa. She had been appointed to a chair fourteen months earlier.
The Pattern She Belongs To
The obstacles were institutional rather than intellectual, and they were the same ones that had stopped Sophie Germain, who corresponded with Gauss under a male pseudonym and never held any position at all, and Émilie du Châtelet, who produced the standard French Principia and could not enter the café where the mathematicians met.
What Kovalevskaya added to that sequence was the precedent. Germain worked outside the institutions and du Châtelet worked around them. Kovalevskaya got inside one, was paid by it, examined its students and edited its journal, which made the argument that women could not do such work considerably harder to make with a straight face.
Emmy Noether, who produced the theorem connecting symmetry to conservation laws, still had to lecture at Göttingen for four years under Hilbert’s name before she was allowed to hold a position of her own. That was 1919, thirty years after Stockholm.
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