Sofia Kovalevskaya Claire Larkin Kovalevskaya's life Sofia Kovalevskaya was born in Moscow, Russia on January 15th, 1850. She was born into a wealthy, well- educated family with an older sister and a younger brother. Sofia was often seen as the neglected middle child because both of her siblings were well-mannered and well-admired; she was always the odd one out. For these reasons, Sofia was not a happy child. She also had a governess who tried to make her into a young lady. However, Sofia always wanted to do math and science, subjects that were not acceptable for young women to do during that time period. One reason for her acquiring her interest in mathematics was her uncle, Pyotr Vasilievich Krukovsky, who spoke about the subject often in front of her. When she was 11, she grew a deep interest for the subject. She would study it on the wall of her nursery and then began studying it with a tutor. However, her father did not like the idea of his daughter doing math, so he put a stop to all of it. This forced her to learn math in the shadows of her family. Sofia was also extremely lucky that she lived next to Professor Tyrtov, a well known professor of mathematics and physics as well as author of a physics book Sofia read while she was young. Professor Tyrtov was beyond dumbfounded when he found out that Sofia had read and understood his book. He even claims that she understood and explained it in the same fashion that the founders of trigonometry did. Professor Trytov impacted Sofia’s life in one of the greatest ways possible: allowing her to continue her studies in math. After he found out about her great talents in math, he convinced her father to allow her to take private lessons in math. After she finished her secondary schooling, she wanted to continue at the university level. However, this was quite difficult for Sofia because no universities were open to women in Russia. The closest university that did accept women was in Switzerland, but women were also not permitted to travel alone. She then decided to marry a man named Vladimir Kovalevsky, a friend of her sister, Anya. Vladimir was originally supposed to marry Anya, but when he met Sofia he was automatically in love. The main reason the Sofia agreed to marry him was because her father would not allow her to travel alone. In order to study at the university level, she had to travel to a different country, and Vladimir was the key to her leaving the country. They got married and then began their travels throughout Europe. Sofia and Vladimir stayed in St. Petersberg right after they were married and then moved to Heidelberg. She studied at the University of Heidelberg and where people were shocked by her brilliance. However, she continually struggled with others patronizing her because she was a woman. After studying in Heidelberg for two years, the couple moved to Berlin so Sofia could study with Karl Weierstrass. However, she was not able to study at the University of Berlin because they did not allow women to take classes. This ended up working in her favor because she was personally tutored by Weierstrass for four years, during which she developed three papers. In 1878 Sofia gave birth to her daughter, Sofia. This brought her husband Vladimir and her closer than they had been in a few years, but after one year of raising her daughter she left her daughter under the care of her sister. Vladimir and her then split up, and in 1883 he committed suicide. The shock pushed her further into her mathematical work in hopes of distracting her from her emotions. After years of opposition and struggle, Kovalevskaya was able to secure a position of privat docent at Stockholm University in Sweden. This meant that she was not being officially paid by the university, but still received some compensation for her lectures. She obtained this position because she knew Mittag-Leffler, who she knew through her work with Weierstrass. Shortly after she began lecturing there, she was appointed to a five year extraordinary professorship. This was a ranking below a real professor, but was still a significant improvement in ranking, especially due to her being a female. After about five years, she was appointed as Professor Ordinarius, meaning she was a chair holder. She was the first women to hold a chair position in a European University. This was a huge step for women in all of mathematics and science. During her years at the University, Kovalevskaya taught courses on analysis and was the editor of the journal Acta Mathematica. She also participated in international conferences, all of which began to bring her fame and attention from Europe and all over the world. The French Academy of Sciences hosted a Prix Bordin every year, which was a contest with a monetary prize. Kovalevskaya entered her work, now known as the Kovalevskaya Top, and won the contest and was awarded a prize for her outstanding work. Sadly, Kovalevskaya died at the young age of 41 due to complications with influenza. Figure 1: Sofia Kovalevskaya, 1873 Figure 2: Vladimir Kovalevsky, husband of Sofia Kovalevskaya Kovalevskaya's mathematical works After working with Weierstrass for four years, Kovalevskaya had produced three mathematical papers in hopes of being awarded a degree. The three papers are on Partial Differential Equations, Abelian Integrals, and 2 Saturn's Rings. Her paper, "On the Theory of Partial Differential Equations", was a major contribution to math and was published in Crelle's Journal in 1875. It is known as the Cauchy-Kovalevskaya Theorem. Weierstrass had previously worked on theorems involving ordinary differential equations and wanted to expand to partial differential equations. Cauchy created the first "general existence theorem in partial differential equations", which showed that differential and partial differential equations have analytic solutions. Going off of this, Sofia was supposed to simplify this result for her dissertation. This theory proved "Under analyticity assumptions on all coefficients, and the non-characteristic condition, there is a unique local analytic solution u to the equations". In simpler terms, she proved that there are solutions to a system of differential equations when the coefficients are analytic functions. Figure 3: One of the greatest of Kovalevskaya's mathematical works, the Kovalevskaya Theorem. Kovalevskaya's second paper was about the Abelian Integral, which was said to have little importance at the time, but still demonstrated very high levels of mathematical competence. Abelian Integrals are elliptical integrals with a polynomial degree greater than four. Kovalevskaya investigated how to simplify abelian integrals to elliptic integrals. With the assistance of Weierstrass, she was able to produce a paper of the highest ranks. However, no one at the time was interested in this topic, therefore it was not able to be replicated or published until six years later. Kovalevskaya's third paper produced under the mentorship of Weierstrass was about Saturn's Rings. Her work proved that Saturn's rings are egg-shaped ovals and and symmetric around a single point. Although this work was later disproved, it was still important at the time because of how she manipulated the math and the methods that came from her work. For example, the method which Kovalevskaya used for dealing with infinite sets of unknowns was one of the most important takings from her work. Another significant mathematical contribution Kovalevskaya made was on the rotation of an unsymmetrical solid body around a fixed point, well known as the Kovalevskaya Top. There were three major cases of rotational body motion Euler, Lagrange, and Kovalevskaya. The Kovalevskaya top is one of the cases where a top under the influence of gravity is integrable. This is one of her most well known pieces of mathematics. 3 Figure 4: The Kovalevskaya Top is a rotational body that is integrable under the influence of gravity. Collaboration with other scholars Sofia worked with many famous scholars throughout her lifetime. Not only did she, but her husband did as well. Vladimir was a paleontologist who at some points worked with the famous Charles Darwin. Sofia studied under Weierstrass, who is often given the name "father of modern analysis", for four years before receiving her PhD. Sofia said that she would not have become the person she was if it had not been for Weierstrass. Together they tackled some of the most challenging mathematical problems of their time. The greatest achievement was the Cauchy-Kovalevskaya Theorem, generally referred to as the Kovalevskaya Theorem. Although this was Kovalevskaya's work, Weierstrass gave her the initial problem and then mentored her throughout her work. After graduating from University of Berlin, Sofia found out she could not teacher because she was female. This forced her to make the decision to step back from mathematics and focus on her family. However, after six years of being away from her passion she decided to become even more dedicated to the subject. She once again entered the world of mathematics and began to collaborate with other mathematicians and connect with other people who would help her gain ranks in mathematics. In an evaluation of her own mathematical career, Kovalevskaya comments that it was the work of other scholars that influenced her own research. The French physicist Lame inspired her work on the refraction of light in crystals as well many other her other pieces of work. She also said that the work of other scientists and mathematicians greatly influenced the direction that she took with all of her own research.