Sofya Kovalevskaya
Part 2 - Berlin and Beyond
Berlin
After an initial visit with Karl Weierstrass August where he agreed to teach Sofya privately, the couple made the move to Berlin arriving October of 1870.
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Karl Weierstrass was a renowned German mathematician who was known for his contributions to the development of mathematical analysis. Perhaps best known for his work on the theory of functions, which helped to lay the foundation for modern calculus, his dedication to mathematical rigor and precision had a profound impact on the field of mathematics, and his work continues to influence research in mathematics and related disciplines today.
Though he shared societies prejudice against higher education for women, after a few months of working with Sofya he quickly regarded her as the most brilliant and promising of all his students, sharing his ideas and unpublished work with her.
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Sofya’s studies were interrupted for large portions of 1871 because of her concern for her sister still caught in the chaos of the Franco-Prussian War.
Vladimir had concerns about the destruction and theft of important fossils and museum specimens by the Germans invading Paris.
In April of 1871 the couple infiltrated German troops and walked over 25 miles on foot to gain access to the besieged city. Upon arrival Sofya worked alongside her sister in the hospitals while Vladimir visited the paleontological collections of the city’s museums.
By May troops had entered Paris and over 20,000 Parisians were executed with no respect for age or sex. Anna’s husband Victor Jaclard was arrested and was facing a death sentence. Upon hearing the plight of his eldest daughter, the 70 year old General Korvin-Krukovsky traveled to Paris to assist in the escape of Victor to Switzerland using the passport of Vladimir.
Once Anna and her husband were save in Geneva, Vladimir returned to his paleontology studies across Europe eventually getting his PhD while Sofya returned to Berlin where she found her old frield Julia Lermontova waiting for her.
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After leaving Heidelberg, Julia managed to make a private arrangement to continue studying chemistry while Sofya began intense work with Weierstrass. The University had a strict prohibition on women students, they were not allowed through the doors at all.
Weierstrass was teaching Sofia the Theory of Abelian Functions, Introduction to Analytic Functions and Calculus of Variations. These were lectures he gave to the male students, but Sofya received these lessons privately in his personal office.
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Vladimir received his doctorate from the University of Jena with his dissertation on the paleontology of ungulates with the support of Charles Darwin.
In the six years since their marriage trouble had grown within the relationship. The financial strains and long distance lead to resentment of each other’s work. Both thought they were playing second fiddle to the other.
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In 1873 the Russian Tsar Alexander II sent out a decree requiring all Russian students, studying abroad to return at once. Non-compliance would lead to denial of employment opportunities and social ostracism. The decree was primarily directed towards the women studying medicine at the University of Zurich, but Sofya and Julia must have felt the significance of the degree because both women rushed to defend their dissertations and return to Russia to avoid consiquences.
Göttingen University - Triple Dissertation
“Without ever being officially accepted as a student, Sofya earned a doctorate in mathematics entirely on the strength of her original contributions to the field”
Weierstrass did not fully understand why Sofia wanted an official degree, as he believed she was simply pursuing intellectual satisfaction. He did not realize the importance of official recognition of her scientific competence, which would have made it easier for her to secure employment in the male-dominated field of mathematics. Because women could not enroll as regular students at the University of Berlin, eventually Weierstrass campaigned his contemporaries at Göttingen University to accept her dissertations in absentia. Weierstrass specifically chose Gottingen for its liberal reputation, because Sofia's great grandfather had studied there and because the university had a long history of subtly supporting women in science. He wanted to ensure that her dissertations were strong enough to stand on their own, without the need for an oral defense that would require strong German language skills. He also feared that Sofia might be nervous when questioned by senior scholars.
Ultimately, Sofia's academic achievements and contributions to mathematics spoke for themselves, and she earned her doctor of philosophy in mathematics summa cum laude supported by Weierstrass, DuBoi-Reymond, Lazarus Fuchs and others in July 1874.
Julia Lermontova received her doctorate in chemistry from Göttingen University in the same year.
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The first paper was titled "On the Theory of Partial Differential Equations", and it dealt with partial differential equations with constant coefficients. This paper introduced a new method for solving certain types of differential equations, which would later become known as the Cauchy-Kovalevskaya theorem.
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The second paper was titled "On the Problem of the Rotation of a Solid Body About a Fixed Point", and it dealt with the dynamics of rigid bodies.
The main mathematical areas covered in the paper include partial differential equations, the theory of functions of a complex variable, and the theory of surfaces. Specifically, Kovalevskaya introduced a new method for solving partial differential equations, which was later named the Kovalevskaya-Painlevé method.
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The third paper was titled "On the Theory of the Motion of a Rigid Body", and it expanded on the second paper by considering the motion of a rigid body around a fixed point subject to various forces.
This paper builds on her earlier work and focuses on the mathematical analysis of the motion of rigid bodies in the presence of external forces. The paper utilizes advanced mathematical techniques such as the Hamiltonian formalism, the theory of partial differential equations, and the theory of integrable systems. Kovalevskaya derived a number of important results in this paper, including the Kovalevskaya exponents and the Kovalevskaya matrix. These results are still used in modern physics and mechanics to study the motion of rigid bodies.
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In September of 1870 while Sofya was traveling across western Europe, she asks Charles Darwin for help locating a specific book: Fundamenta Nova Theoriae Functionum Ellipticarum was a seminal work by Carl Gustav Jakob Jacobi published in 1829. The title translates from Latin as "New Foundations of the Theory of Elliptic Functions."
This work is one of Jacobi's most significant contributions to mathematics. In it, he laid the groundwork for the modern theory of elliptic functions, a branch of mathematics that deals with functions derived from the functions used to describe the shape of an ellipse.
Elliptic functions are important in many areas of mathematics and physics, including number theory, algebraic geometry, and string theory. They are doubly periodic functions of a complex variable with two independent periods, and they are characterized by the ability to be expressed in terms of the inverse of elliptic integrals.
Jacobi's work on elliptic functions contributed significantly to their understanding and application. It was well-regarded in his time and continues to be a fundamental reference in the study of elliptic functions.
Return to Russia
Even with a doctoral degree Sofya was unable to find work in her home country. The stigma against learned women was too pervasive across society at the time, but she did return as a sort of celebrity, someone to be stared at and discussed, but not taken seriously. Frustrated, she packed away her mathematics work and let it gather dust for the next six years going so far as ignoring the letters from Weierstrass who by this time regarded her as a daughter, appalled by her wasting her gifts.
During this period in Russia, Sofya and Vladimir attempted to live as a more traditional husband and wife. Sofya along with Vladimir made their way though the extensive literary and intellectual circles in St. Petersburg. The couple was in need of financial support, Vladimir was unsuccessful finding a university professorship and the best job Sofya could find was teaching arithmetic to elementary girls. Using the inheritance that came to Sofya from her father’s death, Vladimir entered into the business of buying and selling houses. Unfortunately his plans vastly exceeded his capital quickly leading to mounting debts and angry creditors.
In 1878 she helped organize the “Higher Courses for Women” to help establish a women’s university within Russia. Vladimir helped with administration and fundraising for the school along with donating some of his publishing catalog to the school. But, after offering her services to teach without a salary, she was passed over as a member of the faculty. It was a blow to her ego.
Either due to the responsibility she felt towards her parents or an obligation to honor Vladimir, the couple ended up having their first child in October named Sofya Vladimirovna, nicknamed Fufa.
Meanwhile, Vladimir’s business opportunities were drastically worsening and Sofya didn’t handle the situation well. In a notebook she satirized them both, her husband for his comments on the incapacity of women to do creative work and herself as the erudite wife.
In January 1880, the Sixth Congress of Natural Scientist was held in St. Petersburg. The mathematician P.L. Chebyshev urged Sofya to present a paper on some of her work. Reluctantly, she returned to her unpublished dissertation on Abelian integrals. In a single night she translates the paper into Russian and the following morning read it to the audience with huge success. Despite six years elapsing, no other work had been done in this field. In one day she reestablished herself as a world class mathematician. In attendance was Gosta Mittag-Leffler, a former student of Weierstrass and a professor at the University of Helsingfors. Weierstrass had sent him to the congress with instructions to bring Sofya back into the fold of mathematics.
Paris
“you write that no woman has created anything important, but it is just because of this that it is essential for me, while I still have energy and tolerable circumstances, to position myself so that I may show whether I can achieve anything or whether I lack brains.”
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The couple had a final rendezvous in Paris in the apartment of Anna and her husband. The couple agreed that a separation was best. Sofya would live abroad for a while splitting her time between Paris & Berlin where she worked on a new research project concerning the refraction of light in crystals along with Charles Hermite and Henri Poincare while waiting for a teaching job promised by Mittag-Leffler. Their daughter Fufa was frequently left in Russia, under the care of Julia or Vladimir’s brother Alexander.
By the spring of 1883, Vladimir was desperate. He faced legal problems due to the actions of the petroleum company he worked with and was increasingly frustrated by the lack of scientific opportunities. Vladimir took his own life by inhaling a bottle of chloroform. Sofya was in Paris when she received the news. She shut herself into her room for five days refusing food eventually leading to her collapse and loss of consciousness. When she finally came to, she immersed her self in mathematical computations crying and cussing the event, blaming herself for abandoning Vladimir in his greatest time of need.
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Sofya began the study of the refraction of light in crystals, she started by studying the work of French physicist Lamé
Gabriel Lamé was a French mathematician and physicist known for his contributions to the study of elasticity, surfaces and curves in three-dimensional space, and number theory. His work on elasticity theory laid the foundation for the modern theory of elasticity. In addition, he developed mathematical techniques for describing the shapes of surfaces and curves, and made important contributions to number theory. Lamé's work on elasticity theory contributed to the study of refraction of light in crystals, which arises due to the interaction of light with the crystal lattice of a material. His work on the behavior of solid materials under stress and strain provided a mathematical framework for describing the behavior of light in a crystalline medium, and his contributions continue to have an impact on modern research in this area.
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In 1882, Sofya Kovalevskaya became the first woman to be granted full membership in the Mathematical Society of Paris. This was a significant achievement for her as the society was one of the most prestigious mathematical societies in Europe. As a member, she was able to attend meetings, present papers, and interact with other leading mathematicians of the time, including Charles Hermite, Emile Picard, Henri Poincaré, and Joseph Bertrand.
Kovalevskaya's membership in the society helped to raise her profile and establish her reputation as a mathematician in France. She presented several papers at the society's meetings, including her work on partial differential equations, and was respected for her contributions to the field. Her involvement in the society also gave her the opportunity to collaborate with other mathematicians and to continue her research on a variety of mathematical topics.
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Vlad tragically took his own life by inhaling chloroform and putting a bag over his head. In a letter to his brother, he expressed regret for the pain he caused Sofia and how he had ruined her life. Upon hearing the news, Sofia was devastated and isolated herself in her room for five days, eventually losing consciousness and requiring medical intervention. However, a few days later, she awoke and immediately turned to mathematics, starting work on a problem.
Weierstrass had not felt comfortable actively seeking university positions for Sofia until after Vlad's death, and after further negations by Mittag-Leffler the University of Stockholm offered her an assistant professor position with no salary.
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In 1888, Sofya Kovalevskaya was awarded the Bordin Prize of the Paris Academy of Sciences for her work on the rotation of solid bodies, specifically on the motion of Saturn’s rings.
The Bordin Prize was a prestigious award given every five years to a mathematician under the age of 40 who had made important contributions to mathematical analysis. The prize was named after the French mathematician Jean-Charles de Borda and was first awarded in 1791.
Kovalevskaya was the first woman to receive the Bordin Prize, which was a significant achievement considering the discrimination against women in the field of mathematics at the time.
“My work had something of an impact in the mathematical world, I had viewed it from a different, entirely new standpoint.”
Stockholm
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Sofya moved to Stockholm, Sweden in 1883, taking the unpaid position of assistant professor. Her welcome to the new city included a scathing article attacking Sofya writing:
“a female professor of mathematics is a pernicious and unpleasant phenomenon - even, one might say, a monstrosity; and her invitation to a country where there are so many male mathematicians far superior in in learning to her can be explained only by the gallantry of the Swedes toward the female sex.”
By 1884, her natural skills as a teach prevailed and she was promoted to Professor of Mathematics for a five year period. She became editor of the influential journal Acta Mathematica.
The following year in 1885, she was given the additional position of Chair of Mechanics. This was a significant achievement, as it made her the first woman in Europe to hold a professorship in mathematics.
Her sister Anna succumbed to chronic illness in the autumn of 1887, she was the last thread connecting Sofya to her family, childhood and past life in Russia.
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The Prix Bordin competition is sponsored by the French Academy of Sciences.
The award was for substantive contributions to the solution of a long-standing problem in mechanical physics. The problem being the rotation of a solid body around a fixed point, the solution was so elusive it had earned the title of “the mathematical water nymph,” had, for years, eluded world class mathematicians including Euler, Lagrange and Poisson. Strict mathematical solutions had only been established in isolated cases and the prize had only been awarded ten times in the fifty years since the problem was announced.
Mittag-Leffer decided his protege must enter the contest to help further his career and possibly secure a position in France.
Fifteen entries were submitted anonymously to the Academy of Sciences, but one stood well above the others. The submission made such a significant contribution to the general solution the prize money was increased from 3000 francs to 5000. When the sealed envelop was opened, Sofya was revealed as the winner. In December she returned to Paris to receive the award along with her new close friend Maxim Maximovich. Unfortunately he wasn’t one to play second fiddle and left casting a sad shadow on the festivities.
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Sofya wrote down her recollections of growing up in Russia during the turbulent mid 19th Century. The thinly veiled autobiography was initially published under the title From Russian Life: the Rajevski Sisters. It was an immediate success, not necessarily because she was a distinguished mathematician, but because her life as a “mathematical lady” was a bizarre and fascinating phenomenon.
In this period she also wrote A Nihilists Girl and two plays.
Sofya found parallels in literature and mathematics, she wrote “It seems to me that the poet must see what others do not see, must see more deeply than other people. And the mathematician must do the same.”
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Sofya’s five year position with the University of Stockholm was coming to a conclusion and she repeated her efforts to secure a position in Russia. She had hoped admittance to the Russian Academy of Sciences would raise her profile and position within the country but unfortunately she as disappointed.
“Inasmuch as faculty positions in our universities are completely closed to women, no matter how great their ability and learning, there is no place in our country for Madame Kovalevskaya which would be as distinguished and remunerative as the one she now occupies in Stockholm.”
The doors were closed to Sofya in France as well despite her high connections and efforts put forth on her behalf.
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Only six months after the excitement of her autobiography spread across Europe, Sofya caught a bad case of pneumonia and passed away at the age of 41. Over the next few years her autobiography was translated into several languages including Czech, Dutch, Danish, German French, Polish, Japanese and two English translations
“Why, this is the first handsome mathematical lady I have ever seen!”
The mathematics underlying Kovalevskaya's work on the motion of Saturn's rings involves a number of different concepts and techniques. Here are some of the key ideas:
Rotating fluids: The rings of Saturn can be thought of as a system of small particles orbiting around the planet. However, because the rings are not a solid object, they behave more like a fluid. Kovalevskaya's analysis therefore focused on the motion of small particles in a rotating fluid, which she believed could provide insights into the behavior of the rings.
Partial differential equations: To study the motion of the particles in the rotating fluid, Kovalevskaya used partial differential equations, which describe how the fluid velocity and pressure vary with time and position. These equations are notoriously difficult to solve, but Kovalevskaya was able to make progress using a technique called Fourier analysis.
Fourier analysis: Fourier analysis involves breaking down complex functions into simpler components, called Fourier series, which are based on trigonometric functions. This technique can be used to decompose the complex motion of the particles in the rotating fluid into simpler components, which can then be analyzed more easily.
Kovalevskaya's constant: One of the key contributions of Kovalevskaya's work was the introduction of a new concept, now known as Kovalevskaya's constant. This constant describes the maximum density of particles that can be supported in a rotating fluid without collapsing into a single mass. By calculating this constant for the rings of Saturn, Kovalevskaya was able to determine under what conditions the rings could be stable.
Mathematical modeling: Finally, Kovalevskaya's work is a classic example of the use of mathematical modeling in science. By using mathematics to describe the behavior of a complex physical system, she was able to make predictions about the behavior of Saturn's rings that could be tested experimentally. Her work helped to establish the field of celestial mechanics as a legitimate area of research, and it demonstrated the power of mathematics to explain and predict phenomena in the natural world.
Sofya Kovalevskaya's work on the motion of Saturn's rings was based on the principles of fluid dynamics, partial differential equations, and Fourier analysis.
To begin with, Kovalevskaya assumed that the motion of small particles in a rotating fluid could be used as a model for the rings of Saturn. She then developed a set of partial differential equations to describe the motion of these particles, taking into account the forces of gravity, centrifugal force, and the pressure of the fluid.
Partial differential equations are a type of mathematical equation that describe how a quantity changes over time and space, taking into account multiple factors. They are used in many areas of science and engineering to model complex phenomena, such as fluid flow, heat transfer, and electromagnetic waves.
Once Kovalevskaya had formulated the equations for the motion of the particles, she needed to solve them to obtain a mathematical description of the behavior of the fluid. This is where Fourier analysis came in.
Fourier analysis is a mathematical technique that involves breaking down a complex motion into simpler components. Specifically, it involves expressing a function as a sum of sinusoidal functions, which describe the way in which the function varies with time and space.
Kovalevskaya used Fourier analysis to express the motion of the particles in the fluid as a sum of sinusoidal functions, which allowed her to separate out the different components of the motion. By doing this, she was able to simplify the equations for the motion of the particles and obtain a mathematical expression for the density of particles in the fluid.
Kovalevskaya then introduced the concept of Kovalevskaya's constant, which describes the maximum density of particles that can be supported in a rotating fluid without collapsing into a single mass. This constant was based on her mathematical analysis of the fluid dynamics of the particles, and it provided a critical threshold for determining the stability of Saturn's rings.
Kovalevskaya's analysis showed that if the density of particles in Saturn's rings was less than the critical value given by Kovalevskaya's constant, then the rings could be stable over long periods of time. This was a major breakthrough in the study of celestial mechanics, as it helped to explain why the rings of Saturn had not collapsed into a single mass over the course of millions of years.
Overall, Sofya Kovalevskaya's work on the motion of Saturn's rings was a remarkable achievement, as it involved the use of advanced mathematical techniques to solve a complex problem in celestial mechanics. Her ideas and methods continue to influence the study of fluid dynamics, partial differential equations, and other areas of mathematics today.
deeper dive into the math
Vera Sergeevna Goncharova
Vera was the original model for Sofya’s Nihilist Girl…find out more about her. She was one of the first group of four women who were permitted to study at the Faculty of Medicine in Paris.