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After several days of no writing, I will make up by publishing some edited notes that I used during one guest lecture this week. This is a short bio of a brilliant female mathematicians who lived about 150 years ago. Her story is an example of personal determination and brilliant mind, but also can make us reflect upon discrimination against women happening in the past as well as today. In a world where 2/3 of illiterate people are women and where such lack of education results in violence, rape, child and forced marriage, and low quality of life, we cannot stop questioning how we can support girls and women to reach a state of complete equality.

##### Sofia's family

Sofia was born under the family name of Korvin-Krukovskaya in Moscow in 1850. Her family background surely had a strong influence on the development of her personality and aspirations. Her mother was a woman from aristocratic family, 20 years younger than her husband. He used to treat her as a child, being probably rights, since she did not care much for her daughters. She was an aspiring actress who could never realise her dream. This made her a weak figure in Sofia’s early life and maybe also her failure in life was a warning. On the other hand, her father was an educated men, with high military rank, and in early childhood Sofia grew naturally close to him. Sofia was the second born, when her parents were hoping for a son. Her older sister Anna was the incarnation of the ideal girl, while Sofia had a more intense and rebel personality, darker hair and carnation. This created a natural bonding between Anna and her mother, and as a consequence Sofia grew closer to her father.

Despite such strong differences, her sister Anna, “Aniuta”, was never an enemy. When she grew older she became more independent and rebel, and opposed with strength their father. She became an important role model to Sofia. Anna was lively and active in the local cultural literary scene and quickly abandoned all frivolousness she learned from her mother: she was so strong and fascinating, that she even got a marriage proposal from Dostoevsky (which she rejected)! Sofia was looking up to Anna, but wanted to find her own way and perhaps this was one of the reasons that led her to science and mathematics. Also, two of her uncles were amateur scientists and she could listen to their conversations when they were visiting. In her memoirs, Sofia tells about one unused room in the family estate of Palibino (in the current Belarus) that had no wallpaper, and was temporarily covered with some lectures notes in calculus, which she spent hours looking at, when very young.

When teenagers, Sofia and Anna were craving for freedom. As women, they were not allowed to travel to Europe to study without permission of their father or their husband. At the time, it was frequent for girls to persuade boys into fictitious marriages to gain freedom. Therefore around 1868 (Sofia being about 18), she therefore proposed to Vladimir Kovalevsky, a scientific writer. Her father opposed to the marriage, but Sofia found a trick: she sent around some letters to his friends declining some regular invitations, telling her fiancé and her were too busy preparing the wedding and could not attend. Her father was faced with the choice of admitting she was rebelling to his authority in front of his close friends, or playing along, and chose the latter.

##### Studying in Europe

The fake marriage was the plan for the two sisters to freely travel to Europe. Anna told her parents the couple would have chaperoned her but secretly traveled to Paris alone. Sofia and her husband started living in Heidelberg (Germany). Here, after a long struggle with the administration, she was allowed to follow university courses when the teachers would grant permission. She studied under Bunsen, Kirchhoff, Helmholtz, P. du Bois-Reymond (integral equations and Fourier series) and Koenigsberger (elliptic functions, differential equations).

The newlyweds soon started having serious money problems, that would lead Vladimir to commit suicide eventually. In 1870 Sofia travelled to Berlin, hoping that Weierstrass, one of the best mathematicians at the time, would grant her admission to the university. Because of university rules, he could not admit her to his lectures, but he was so impressed by her talent that despite having little time, he started giving her private lessons.

In 1871, Sofia lived a life-changing experience. At the time France was losing in the war against Prussia declared by Napoleon. People in Paris got agitated and tried to establish an independent government of the capital (*La Commune*). Anna and her lover were fighting as supporters of La Commune and Paris was surrounded by the Prussian army. In the spring of that year, Vladimir and Sofia rushed there, masking it as a study trip of Vladimir, to save them. Anna was captured but managed to escape, while her lover was saved by her parents who travelled there and intervened by using their good name and bribes; in exchange they required Anna and her lover to legally marry (in front of them!). All the family was miraculously safe at the end.

When Sofia came back, a turning point in the relationship with Weierstrass happened. She revealed him the secret behind her marriage. Weierstrass grew humanly closer to Sofia’s struggling, how she was prevented from attending courses and graduating, despite of her talent. From that moment, he became Sofia's first champion.

During the time span of 18 months (1873-1874), Sofia wrote **three** dissertations, two of which were considered excellent by Weierstrass. He recommended her work to the more liberal University of Goettingen (Germany) and she was finally awarded the doctoral degree, summa cum laude. She was the first woman in modern Europe to receive such a title in mathematics.

##### The Cauchy-Kovalevskaya theorem and the majorant method

One of her dissertations contained important material, not only for the final result, but also for the method used to obtain it. Let us give a simple version of the Cauchy-Kovalevskaya theorem:

Let be a function analytic near . Then there exist a neighbourhood of in and a *unique* solution to the initial value problem:

(1)

that is analytic in .

There are more general versions of this theorem where you have higher order derivatives, several analytic given functions instead of the only and several variables. The result is valid also for complex analyticity.

I will here give a simple explanation of the majorant method that Kovalevskaya used to prove it, following the example of Cauchy before her. Given the system, by using the chain rule of differentiation:

...

In general, the -derivative of can be written as a polynomial with **non-negative** integer coefficients depending on . If you imagine the following computations, there is no way a minus sign can show up. In symbols, we write that:

When evaluating those derivatives at , we can eliminate the dependence on the previous derivatives by noticing that:

( depends only on ) ...

If you iterate and eliminate the dependence on step by step, you can conclude that:

(*)

This shows that if such analytic solution exists, it must be unique, because its Maclaurin coefficients depend only on the given function . Now, let's recall we assumed that was analytic near . This means that its Taylor series near converges:

for near

We can surely find some numbers such that:

The term is the -th Mclaurin coefficient of the analytic function .

Consider the twin initial value problem:

(2)

Such problem has the following explicit solution (it's an exercise to verify): , that converges in an open neighbourhood of .

By previous calculations, that are valid also now, we already know that . Since the polynomials have non-negative coefficients, we can majorize as follows:

(3)

We now stand at this point. We know that (1) has an analytic solution that we can build it *uniquely* by (*). The convergence in an open disc near is now guaranteed by formula (3) and the local analyticity of . There is one thing left to prove, that is being a solution to (1).

Since are analytic, their composition is and has the following expansion:

(4)

but by construction, . If you write the Taylor series of (exercise) you can conclude it coincides with (4) and this concludes the proof.

##### Back to Russia and afterwards

After graduating, Sofia was exhausted and the family moved back to Russia, hoping to find jobs in the academia (Vladimir graduated as well during their stay in Europe). In 1875, after 7 years of platonic relationship, her marriage turned into a real one. In 1878 their first and only daughter Sofia “Fufa” was born. Shortly afterwards, Sofia’s mother died and Sofia and her husband lost an important investment that was supposed to grant them security, while finding a job was so hard. Despite their titles, they were fighting against unemployment. Sofia was discriminated as a woman and told she could teach to elementary schoolgirls at most. She was also struggling with her new family responsibilities and with the feeling of the waste of effort to gain a high education. Anyway, following Weierstrass' advice, she managed to send a paper to the 6^{th} Congress of Mathematicians and Physicians. This was a lucky move, because one of the readers was the Swedish mathematician Mittag-Leffler. Mittag-Leffler at the time was a professor at University of Helsinki (1877-1881). He became Sofia’s second champion and tried very hard to get a position for her in Helsinki. Ironically, she did not have a chance, not because she was a woman, but because she was Russian. At the time, Finland was struggling to affirm her own cultural independence after centuries of domination by Sweden and Russia. In 1881 Mittag-Leffler was appointed professor at the University of Stockholm, a more modern university than Uppsala and Lund, and some hope for Sofia to get a position got back.

In the meantime, Sofia’s marriage was facing a deep crisis: Vladimir was depressed for the previous financial loss and was acting irrationally, even wasting job opportunities. Sofia started planning life for herself and her child, and moved to Berlin. In 1883 Vladimir commited suicide and Sofia reacted by letting herself starve to coma. Luckily her friends took good care of her and despite her family tragedy, shortly afterwards that she finally was appointed in Stockholm, even thought not with a paid nor permanent position. In 1884 she gave her first lecture. She was not paid by university but from her students, but since she was a very good lecturer, this was not a problem. In 1886 Sofia went to Russia to assist her severely sick Anna, who died at the end of 1887.

One year later, despite struggling with her personal life tragic events, she won a prestigious prize for her work on the rotation dynamics of a top with a fixed point.

Before her result, only the case where the center of gravity and the fixed point were the same, and the case were such 2 points were on the symmetry axis were known. The jury even doubled the prize because of originality and important of her results. This achievement granted her to become full professor in Stockholm in 1889. Unfortunately she could not enjoy the position for long: in 1891, after returning from a brief holiday, she ended up under a pouring rain. This resulted in a flu that degenerated to pneumonia and led her quickly to death.

##### After Sofia

Sofia left us a great inheritance, not only in terms of mathematical results, but also being a pioneer of women in science and mathematics. Even though much progress in terms of gender equality has occurred, we are still far from good.

A report of the Department of Mathematics and Statistics of University of Helsinki (2008), shows the horrible situation of the local gender gap with a single graph:

To date, we still have no women math professors here. As far as I know, there are only two female full-professors in mathematics in all Finland. Now, this is not a question of aiming to a perfect 50-50 situation, neither to push too hard women to take part, but I think this figures are worrying and should at least make us wonder.

Sofia and many women before her (Maria Agnesi, Mary Sommerfield, Hypatia, ...) managed to exploit their talent, pursue their dreams, and succeed because they had a male champion. Mentoring still stands as one of the acclaimed possible solutions to encourage women to follow their aspirations in science. With the ultimate goal being getting free from such constraint and having women completely free from social pressure and able to pursue a scientific career, I anyway encourage all my colleagues, women and men, to dedicate some of their time and effort to mentor girls, female students, and female colleagues, to share the privilege and build all together a more fair and even more productive work environment.

**Bibliography**

http://www.epigenesys.eu/en/science-and-you/women-in-science/739-sofia-kovalevskaya

http://en.wikipedia.org/wiki/Cauchy–Kovalevskaya_theorem

http://www.encyclopediaofmath.org/index.php/Cauchy–Kovalevskaya_theorem

http://en.wikipedia.org/wiki/Kovalevskaya_Top

http://scienceworld.wolfram.com/physics/KovalevskayaTop.html

Cooke, *The Mathematics of Sonya Kovalevskaya*, Springer-Verlag, 1984.

Sofya Kovalevskaya, *A Russian Childhood*, Springer-Verlag, 1978 edition.

Gantumur, *The Cauchy-Kovalevskaya theorem*, Math 580 lecture notes.