As Chalkdust readers, we know of more mathematicians than the average person, but if someone were to say “name a female mathematician”, the first answer would still usually be Ada Lovelace. This is why I want to talk about someone you might not have heard of who has opened many doors for women in academia: Sofya Kovalevskaya.
Sofya Vasilyevna Kovalevskaya was born in Moscow on 15 January 1850. Her father was in the Imperial Russian army, before retiring when she was 8 years old, and was minor nobility. She had scientists and mathematicians on both sides of her family and due to her family being relatively wealthy, she had a better education than most, being tutored in science, maths and languages. At age 11, she had pages of old lecture notes on differential and integral analysis from her father’s student days on her bedroom walls. She wrote in her autobiography that these “acted on my imagination, instilling in me a reverence for mathematics as an exalted and mysterious science which opens up to its initiates a new world of wonders, inaccessible to ordinary mortals.”
When she was 16, Nicholas Tyrtov—a neighbour who was a physics professor—accidentally left one of his textbooks at her home. When he came back to pick it up, she explained to him how she understood some of the trigonometric functions in the book, none of which she had seen in her studies before. This led him to convince her father to allow her to study further in mathematics. Hence, the family spent their winters in St Petersburg so that she could be tutored by AN Strannoliubsky, who was considered the best Russian mathematics teacher of the time and a well-known supporter of higher education for women. His time with Sofya just solidified his beliefs.
Unfortunately, after finishing her studies with Strannoliubsky, she encountered a big problem: women weren’t allowed to attend university in Russia. Studying abroad wasn’t an option either, as women couldn’t live outside their family home without written permission from their father or husband, and her father wouldn’t give her this. Meanwhile, her younger sister Anna wanted to leave for Europe as well to study writing, as she had been exchanging letters with Fyodor Dostoevsky who had published one of her stories in his literary journal. So, Sofya and her sister came up with a plan: when she was 18, Sofya contracted a marriage with Vladimir Kovalevskij, a young palaeontology student and publisher who was the first to translate Darwin’s work to Russian. A year later they left Russia with Anna being allowed to leave under her guardianship, Sofya acting as her chaperone. Once out of Russia, they parted ways, with Sofya and Vladimir first going to Vienna, where Sofya was told she could attend physics lectures but not mathematics. Soon, Sofya moved to Heidelberg to study mathematics and natural sciences, only to discover that women couldn’t matriculate at the University of Heidelberg. After much persuasion, the university allowed her to attend lectures provided she could get permission from each lecturer. Here, she was able to attend physics lectures with Gustav Kirchhoff and mathematics lectures with Paul du Bois-Reymond and Leo Königsberger. Her resilience set a precedent, and another female friend of hers was also able to attend these lectures.
A year later, she moved to Berlin hoping that she could be a student of Weierstrass, who was then considered one of the most noted mathematicians in the world, even though the University of Berlin wouldn’t allow her to be a student. She decided to appeal to Weierstrass personally, who was against the idea of women studying in university but decided to give her tests to assess her abilities after the recommendation letters from Königsberger (who used to be his student) and Du Bois-Reymond. He was so impressed by her mathematical skills that he decided to take her on as a private student since, unlike Heidelberg, the University of Berlin didn’t allow her to attend classes even unofficially. He taught her the exact lectures he gave at the university but he also discussed his latest work and theories with her.
While Weierstrass in general didn’t believe that married women needed university degrees or careers, Sofya convinced him to help her work towards a doctoral degree. To achieve this Sofya wrote three papers, one on partial differential equations (PDEs), one on elliptic integrals and one on the dynamics of Saturn’s rings.
The introduction to Sofya’s paper on PDEs
Her paper on PDEs came to be her most important work, as it included what is known today as the Cauchy–Kovalevskaya theorem. Cauchy had already proven that, for certain types of first order PDE, there exist unique analytical solutions. Sofya’s achievement was to generalise his result to higher order PDEs.
In her third paper on the dynamics of Saturn’s rings, she used the assumptions of the time that Saturn’s rings were made of a continuous liquid to prove that the rings were egg-shaped ovals, symmetric about a single point. She wasn’t inclined to do too much precise calculation, as she believed that new research would disprove the assumptions she was working with. Indeed, it was shown later that Saturn’s rings were made of discrete particles and not a continuous liquid.
Saturn and its rings, which are not a liquid
In 1874 and due to Weierstrass’s efforts, she was granted a doctorate from the University of Göttingen in absentia without an oral defence, thus becoming the first woman to be awarded a doctorate in mathematics.
After this, she and her husband Vladimir returned to Russia, where she wanted to teach mathematics but found she wasn’t allowed to attain the certificate she needed for this. She helped her old tutor Strannoliubsky to set up higher education courses in St Petersburg, but wasn’t allowed to teach these either. This and her rejection by most of the mathematical community in Russia, led to her taking a mathematical hiatus. A few years later, this hiatus was ended when she was asked to present a paper at the Congress of Russian Naturalists and Physicians. One of those attending her talk was Gösta Mittag-Leffler, who was one of Weierstrass’s old students. Being impressed by her talk and the high regard Weierstrass had for her, he offered to help her find a teaching position in Europe.
During their time in Russia, she and her husband had decided to act like an actual married couple and have children. However, sadly, after the birth of their daughter, they started growing apart and were having financial difficulties. They separated and later Vladimir committed suicide due to the possibility of getting arrested for financial crimes.
In 1883, Sofya was finally able to find a position at Stockholm University with the help of Gösta Mittag-Leffler, who had become the head of mathematics there. She was offered a temporary position as a sub-professor without an official affiliation to the university, and her payment coming via private arrangements with her students rather than as a salary. She hoped that by accepting this position she would become a role model for other women hoping to be part of academia. Her appointment even made front page news.
A year later, despite the prejudice against her gender, nationality and political beliefs, she was given a 5 year position of assistant professor and became an editor of the scientific journal Acta Mathematica, becoming the first woman on the board of a major scientific journal. It was during this time that she won the Prix Bordin award of the French Academy of Sciences, the second most prestigious award of the academy. Her submission for the prize was her discovery of the Kovalevskaya top. Spinning tops are rigid bodies that are fixed at one point and otherwise free to move and rotate under influence of gravity. Before Kovalevskaya, two types of top were known for which the equations that describe their motion could be solved analytically; Sofya discovered a third such type of top.
An example Kovalevskaya top, built using three spheres and a ring. If this top’s position is fixed at the orange point but is allowed to rotate, the equations that describe its motion can be solved analytically thanks to Sofya.
Her winning the prize and the high acclaim of her work meant that the university didn’t want to lose her and therefore she was offered a permanent position. This meant that in 1889, Sofya became the first woman to be appointed to a full professorship in modern times. Just two years later, aged 41 and at the height of her academic career, she passed away due to pneumonia.
Despite her recognition in Sweden and France, Sofya Kovalevskaya never won her dream professorship in Russia and her accomplishments weren’t valued there until after her death. Her influence on maths and the precedents she set for women in academia will never be forgotten.
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