Sofia Kovalevskaya Claire Larkin Kovalevskaya’s life

Sofia Kovalevskaya was born in , 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.

Figure 5: Karl Weierstrass, friend and colleague of Sofia.

4 Figure 6: Augustin Louis Cauchy, the mathematician who initially thought of the problem that was solved by Kovalevskaya, which is now known as the Kovalevskaya Thereom.

Historical events that marked Kovalevskaya’s life.

Many things were going on throughout Kovalevskaya’s life in Europe. When she was just three years old the Crimean War began, where Russia fought against the alliance of France, Britain, the Ottoman Empire, and Sardinia. The war was over the rights of the of Roman Catholics and Eastern Orthodox Church. Although she was very young when this all occurred, it still had an impact on the rest of the world forever. It was the first time modern technologies were used in war, marking the beginning of advanced warfare.

The mid-19th century was a time of feminist reform in Russia. The feminist reform was led by women of the privileged classes. These women fought for rights such as equal education and equal opportunity to work. One step forward for education was that St. Petersburg University allowed women to audit classes, or to take them without credit. However, this rule was removed after four years.

Some of the first feminist reform came from a group called the ”Tchaikovsky” circle, a socialist group of students of both sexes. While this was not a feminist group, reform was made through this group. They involved women in political discussion and activity on the same level as men, a significant step forward for that time period. Then, beginning in 1870, there women’s labor movement began, consisting of strikes towards equal rights for women workers. Many of the women who fought together worked in the textile industry, where they got low wages and had no rights. The harsh conditions in the factory forced the women to oppose their bosses, leading to many strikes from 1870 to1890. One outcome of this movement was the law that banned children and women from working the night shift. While this was a significant step forward for women in work, the later years would continue to bring women more equality in other areas as well.

In 1877, Russian declared war on the Ottoman Empire. For almost two years Russia enclosed on the Ottoman empire and attempted to win the battles. However, the were slightly outnumbered. They did have the advantage of being on the offense. On the other hand, the Ottoman Empire had more soldiers but always

5 played the defensive role. Russia’s goal was to end the Ottoman force’s influence in the Balkans. After one of the bloodiest wars of that time, Russia signed the Treaty San Stefano, where the Ottomans gave up part of Armenia and Bulgaria to the .

Figure 7: Russian stamp showing unity between man and woman after the advancements made during the feminist revolution in the 19th century.

Figure 8: In 1877 Russia declared war on the Ottoman Empire.

Significant historical events around the world during Kovalevskaya’s life

In 1848 the Women’s Rights Movement in the United States was just beginning. In 1848 the first gathering devoted to women’s rights was held in Seneca Falls, New York, known as the Seneca Falls Convention. Here they created a ”Declarations of Sentiments, Grievances, and Resolutions”, which echoed the Declaration of Independence and brought light to the issues that these women wanted to reform. The Women’s Rights Movement came in many waves. It initially started with addressing social and institutional barriers, such as

6 family responsibilities, labor equality, and educational opportunities.

Throughout the 1850’s they fought for equal economic freedoms. Susan B. Anthony and Elizabeth Cady Staton were two of the leaders of the movement. They lobbied Congress to include women in the 14th and 15th amendements, but were unsuccessful. It was not until the start of the Civil War that the reformers started focusing their fights solely on women’s suffrage. Staton and Anthony created the National Woman Suffrage Association which focused on changing federal law to include women in the right to vote. Another leader, Lucy Stone, formed the American Woman Suffrage Association. While both of these groups fought continuously for the right to vote, not much came of it until the middle class began to fight along their side. The two groups united into one, the National American Woman Suffrage Association. For the next 20 years they would lobby in states and for the vote. 1917 was the first turning point for their group. Arkansas granted partial voting rights and New York granted full rights. In 1919, the 19th amendment was created, providing full voting rights to women across the country.

Figure 9: United States in the late 19th century; women marching to fight for equality.

In 1860 president Lincoln was elected President in the United States. This marks a turning point for the whole world because the United States then entered the Civil war in 1861. It was the first time the strengths of the democracy were being tested. The Union victory also brought passionate feelings towards the Republic and equality to Western Europe. Most of Europe was made up of monarchies and dictatorships and if the Union would have fallen, the beliefs that democracies would not work would have been confirmed. However, the Union victory confirmed that democracies could work and then set up the model for all democracies that followed. The victory also freed the slaves, a major step forward for the entire world.

Throughout the 1800’s there were important technological advancements in Europe and the United States. Between 1840 and 1870 the Second Industrial Revolution was taking place. In Europe, some of the most influential advancements that came were expanded steel production, heavy industrial operations, and new processes of iron smelting. These were extremely important to mass production and increasing a country’s economic export. The industrial revolution came to the United States slightly after it spread in Europe. In the United States, important technological advancements were the cotton loom, steamships, and refrigeration. The US also grew exponentially in agricultural output, sending crops over seas to the rest of the world.

7 Figure 10: During the mid-19th century there were many technological advancements taking place due to the Second Industrial Revolution. This is one example of the growth during that time- the new process of iron smelting. Significant mathematical progress during Kovalevskaya’s life

Mathematical progress came in many forms between 1850-1900. Progress came through new discoveries, changes in education, and the formation of mathematical societies. the progress that was made paved the way for the rest of the years to come and for math to advance at an extremely fast rate.

During the 1850’s, there was progress made on the math classes students have to take at the high school and college level. While it was originally only required for college students to go through algebra and geometry, it was then becoming more often mandatory for students to study calculus as well, unless their area of study did not need it.

In addition, there were many mathematical societies created during this time period. One of the first math- ematical societies to be created was the Moscow Mathematical Society. This was a group of Russian mathe- maticians aimed at developing mathematics. The American Mathematical Society was also created during this time period in 1888. Along with those, the London Mathematical Society, French Mathematical Society, and the Edinburgh Mathematical society were created. The creation of these societies gave rise to the importance of math and encouraged collaboration among scholars. They also assisted in publishing scholars work and served as a debate platform for many scholars to discuss among.

During the beginning years of Kovalevskaya’s life, Bernhard Riemann was alive. Riemann is known as one of the greatest mathematicians of all time, and throughout his life he made significant mathematical progress. There are multiple theorems named after Riemann in topics ranging from topology, complex analysis, and algebraic geometry. He is also well-known for his contributions to prime numbers, for which he found a way to determine approximately how many prime numbers there are between two given numbers. This was one of the greatest findings of the time because prime numbers had been a fascination for such a long time.

Another great achievement during the time that Kovalevskaya lived was the Set Theory, by Georg Cantor. Cantor realized that it was possible to add and subtract infinities, and that the size of the infinity was infinite. He also coined a new term, transfinite, which was used to distinguish between the levels of infinity and absolute infinity. His Set Theory began to appear after these discoveries. Although sets had been used previously, it was Cantor who clearly defined a set and showed that a set could be infinite, compared to the finite sets being used before. This theory was not well accepted by the other mathematicians. He faced a lot of difficulty from contemporary mathematicians, and there were very few colleagues who he could talk about his work with.

8 Figure 11: Georg Cantor created the Set Theory, a highly disputed theory created during the time Kovalevskaya lived. Connections between history and the development of mathematics

One of the most significant connections between what was going on in the world during this time and the development of mathematics was that it was becoming more accepted for women to enter the subject of study. In Europe and the United States there were major women’s rights movements beginning, which assisted in Kovalevskaya being able to do her work. In addition, the work that Kovalevskaya did was with the help of many male mathematicians. Without their acceptance of women in the field, Kovalevskaya would have never risen to the point that she did. They helped her defy the odds of being a woman in math and taught her the rigorous mathematics needed to succeed, something many women did not have the opportunity of learning.

During this time, there was also a lot of governmental conflict in Europe. At the time she was entering the world of mathematics, Russia was trying to gain as much take over the Ottoman Empire, which they successfully did. There were also many other conflicts between the European countries along with conflict occurring in the United States. This sparked competition among the European countries, which then competed to be the best they could. This affected mathematics because the mathematicians not only wanted to prove something for the sake of mathematics, but for the pride of their country.

Much of Europe was experiencing the industrial revolution at this time, which impacted the way that they used mathematics. For example, France was focused on practical usefulness of mathematics, while Russia was focused on pure mathematics. The Industrial Revolution influenced France because they saw how much mathematics could do in advancing technology. Russia and much of the rest of Eastern Europe focused on the advancement of pure mathematics, which was also influenced by the Industrial Revolution because the people wanted to focus on even greater things that could one day make a difference in the world.

Remarks

Overall, Sofia Kovalevskaya made major advancements in the mathematical world while also making strides for women in mathematics. She overcame gender inequality by pursuing her passions and never giving up on herself. Her advancements as a women were helped by the women’s rights movements occurring all throughout the world at that time. The protests moved women’s rights up as an issue of upmost importance that must be addressed. This also brought the issue to everyones’ minds, whether they supported the movement or not. The men who did agree with it were able to help Kovalevskaya succeed in her career. Although she did have help from many male mathematicians, the whole reason they wanted to help her was because of her intelligence.

9 She was a phenomenal mathematician with a great mind. She may not be the most well-known mathematician, but made significant contributions to mathematics during her time.

References https://www.britannica.com/topic/history-of-Europe/Realism-in-the-arts-and-philosophy#ref311206 http://www.storyofmathematics.com/19th.html https://www.agnesscott.edu/lriddle/women/kova.htm http://russiapedia.rt.com/prominent-russians/science-and-technology/sofia-kovalevskaya/ http://www-gap.dcs.st-and.ac.uk/history/Biographies/Kovalevskaya.html http://www.marxist.com/emancipation-women-russia.htm http://history.house.gov/Exhibitions-and-Publications/WIC/Historical-Essays/No-Lady/Womens-Rights/ http://www.nwhp.org/resources/womens-rights-movement/history-of-the-womens-rights-movement/ http://www.pballew.net/mathbooks.html http://fabpedigree.com/james/grmatm5.htm http://www-groups.dcs.st-andrews.ac.uk/~history/Miscellaneous/Kovalevskaya/biog.html

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