Outstanding Woman in Science (SOFIA KOVALEVSKAYA (1850–1891)

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Women’s rights are human rights.
—Feminist slogan famously quoted by Hillary Clinton during the 1995 United Nations World Conference on Women



SOFIA KOVALEVSKAYA (1850–1891)
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Sofia Vasilyevna Kovalevskaya was a Russian mathematician who authored significant original contributions to analysis, partial differential equations, and mechanics. She was also the first woman appointed to a full professorship in northern Europe and one of the first women to work for a scientific journal as an editor (Figure 1, right). Along with her sister, the socialist and feminist Anya Jaclard, Sofia (sometimes called Sonya) advocated for women’s rights in the 19th century.

In Recollections of Childhood, Sofia vividly describes her early years: her education by a strict governess of English extraction; life on a country estate and then a move to Saint Petersburg, where her family’s social circle included the novelist Fyodor Dostoevsky (1821–1881); and her parents’ dismay with the new ideas espoused by the two sisters.

It was her struggle that started to open universities’ doors to women. In addition, her groundbreaking work in mathematics made her male counterparts reconsider their archaic notions of women’s inferiority to men. Sofia was raised in posh surroundings, although she was not precisely a happy child, as she felt neglected in a family that included a widely admired, firstborn daughter, Anya, and a younger male heir, Fedya—all of which led her to the nervous and withdrawn personality she manifested throughout her lifetime.

Mathematics attracted Sofia while she was very young, and she often spent time studying her father’s old calculus notes. She credited her Uncle Peter, too, for first sparking her curiosity in mathematics. At 14 years old, she taught herself trigonometry to understand the optics section of a physics book whose author, a certain Prof. Tyrtov, was incidentally her neighbor: Tyrtov was extremely impressed and convinced her father to allow her to attend school in Saint Petersburg. After concluding her secondary education, Sofia wanted to continue at the university level; however, the closest universities open to women were in Switzerland, and young, unmarried women were not permitted to travel alone—unbelievable customs of those days, indeed!
To resolve the problem, Sofia, then just 18, entered into a marriage of convenience with Vladimir Kovalevsky in 1868. The newlyweds remained in St. Petersburg for a few months and, thereafter, traveled to Vienna and Heidelberg, where Sofia gained some notice. She was allowed to study physics there but could not find a mathematics professor to work with. Sofia took courses on a variety of other subjects, while her husband studied geology and paleontology. During this time, the couple traveled extensively and had social contacts with the leading intellectuals of that time, including Charles Darwin (1809–1882), Herbert Spencer (1820–1903), and Thomas Huxley (1825–1895) [1], [2]. While in Heidelberg, in 1869, Sofia took courses with the outstanding German physiologist Emil du Bois-Reymond (1818–1896), as well as Hermann von Helmholtz (1821–1894) and Gustav R. Kirchhoff (1824–1887)—an impressive group of high-level scientists. Soon thereafter, in 1870, Sofia pursued studies under Karl Weierstrass (1815–1897) at the University of Berlin. Weierstrass was considered among the most renowned mathematicians of his time (he formalized the continuity of a function, proved the intermediate value theorem, and studied the properties of continuous functions on closed-bounded intervals). Meanwhile, Vladimir moved to Jena to obtain his doctorate.
After some time, Weierstrass realized Sofia’s caliber and immediately set to work privately tutoring her because the university still would not permit women to attend. Sofia continued her studies under Weierstrass for four years. She is quoted as saying, “These studies had the deepest influence on my entire career in mathematics. They determined finally and irrevocably the direction I was to follow in my later scientific work: All my work has been done precisely in the spirit of Weierstrass.” At the end of her four years, she had produced three papers. The first of these, “On the Theory of Partial Differential Equations,” was published in Crelle’s Journal, the common name of (in English) the Journal for Pure and Applied Mathematics founded by August Leopold Crelle in 1826—a very high honor for an unknown mathematician.

Finally, in July 1874, Sofia Kovalevskaya was granted a Ph.D. degree from the University of Göttingen. Yet, even with such a prestigious degree and the help of Weierstrass, who had grown quite fond of her, she was not able to find employment. She and Vladimir decided to return to her home. Shortly after, her father died unexpectedly. It was during this period of sorrow that Sofia and Vladimir bore a daughter (theirs had been an essentially arranged marriage, but during this time it blossomed into a more deeply emotional attachment). While at home, Sofia neglected her work in mathematics and instead developed her literary skills in fiction, theater reviews, and scientific articles for a newspaper.


Soon, however, in 1880, Sofia returned to mathematics with a new fervor. She presented a paper on Abelian integrals at a scientific conference. [Originally, Niels Henrik Abel (1802–1829), the famous Norwegian mathematician, had solved for those difficult integrals.] Once again, though, she was faced with the dilemma of finding employment, and she decided to return to Berlin, working again with Weierstrass. Sadly, not long after, news reached her of Vladimir’s death; he had committed suicide when all of his business ventures collapsed. A grieving Sofia threw herself into her work more passionately than ever.

In 1883, she received an invitation from an acquaintance and former student of Weierstrass, Gosta Mittag-Leffler (1846–1927), a Swedish mathematician, to lecture at the University of Stockholm. Then came a series of great accomplishments: a tenured position at the university, her appointment as editor of a mathematics journal, publication of her first paper on crystals, and, in 1885, the coveted appointment as chair of mechanics. At the same time, she cowrote a play, The Struggle for Happiness, with a friend, Anna Leffler, perhaps to vent her inner troubles and turmoil.


In 1887, Sofia again received devastating news: the death of her sister, Anya. This was particularly hard on Sofia because the two had always been very close. Not long afterward, as a kind of compensation, Sofia achieved her greatest personal triumph: in 1888, her paper “On the Rotation of a Solid Body About a Fixed Point” won the Prix Bordin awarded by the French Academy of Sciences. In this paper, Sofia developed the theory for a nonsymmetrical body where the center of its mass is not on an axis in the body. The paper was so highly regarded that the prize money was increased from 3,000 to 5,000 francs.


At this time, a new man entered her life: Maxim Kovalevsky (Vladimir’s cousin), who had come to Stockholm for a series of lectures. The two had a scandalous affair—although both were too passionate about their work to give it up for the other. Maxim’s work took him to France, and he wanted Sofia to give up her hardearned positions to simply be his wife. She flatly rejected such an idea but still could not bear the loss of him. She remained with him in France for the summer and fell into another one of her frequent depressions. Again, she turned to her writing, returning in the fall of 1889 to Stockholm. She was still miserable at the loss of Maxim, even though she frequently traveled to France to visit him. She eventually became ill with pneumonia, exacerbated by her depression. On 10 February 1891, Sofia Kovalevskaya died, at only 41, and the scientific world mourned her loss. During her career, she had published, in addition to her literary works, ten papers in mathematics and mathematical physics [3]–[4].



Let us stress her accomplishments. During Sofia’s years at Stockholm, she carried out her most important research and taught courses (in the spirit of Weierstrass) on the newest and most advanced topics in analysis. She completed research already begun on the subject of the propagation of light in a crystalline medium. In mathematics, her name is mentioned most frequently in connection with the Cauchy–Kovalevsky theorem, which is basic in the theory of partial differential equations. Augustin Louis Cauchy (1789–1857), a French mathematician considered a pioneer of analysis, had examined a fundamental issue in connection with the existence of solutions, but Sofia pointed to cases that neither he nor anyone else had considered. Thus, she was able to give his results a more polished and general form. In short, Cauchy and, later, Kovalevsky, sought necessary and sufficient conditions for the solution of a partial differential equation to exist and be unique.
In the case of an ordinary differential equation, the general solution contains arbitrary constants and therefore yields an infinity of formulas (representing curves). In the general solution of a partial differential equation, arbitrary functions occur, and the plethora of formulas is even greater than in the ordinary case. Hence, additional data in the form of initial or boundary conditions are needed if a unique, particular solution is required. Abel (mentioned previously) died within a year of the research he started in that area, leaving to Weierstrass and his pupils the strenous and exciting task of developing the theory of general Abelian functions and the corresponding Abelian integrals.


Sofia’s doctoral research contributed to that theory by showing how to express a certain species of Abelian integrals in terms of the relatively simpler elliptic integrals. Complex analysis and nonelementary integrals were also a feature of the paper that led her to the Bordin Prize. In her paper, she referred to works by Swiss mathematician Leonhard Euler (1707– 1783), Siméon Denis Poisson (1781–1840), and Joseph L. Lagrange (1736–1813). They had considered two elementary cases concerning the rotation of a rigid body about a fixed point. Her predecessors had treated two symmetric forms of the top or the gyroscope, whereas she solved the problem for an asymmetric body. This case is an exceedingly difficult one, and she was able to solve the differential equations of motion by the use of hyperelliptic integrals. Her solution was so general that no new case of rotatory motion about a fixed point has been researched to date.


In a different field, she studied the form of Saturn’s rings, once more with a great predecessor: in this case, Pierre- Simon Laplace (1749–1827), whose work she generalized. Whereas, for example, he thought certain cross sections to be elliptical, she proved that they were merely egg-shaped ovals symmetric with respect to a single axis. Although it was at the time claimed that Saturn’s rings could not possibly be continuous bodies, either solid or molten, and, hence must be composed of myriad discrete particles, Sofia considered the general problem of the stability of motion of liquid ring-shaped bodies—that is, the question of whether such bodies tend to revert to their primary motion after disturbance by external forces or whether deviation from that motion increases with time. These were groundbreaking theories and the impetus for future discoveries.

Sofia was remembered by her daughter who, at the age of 72, was guest speaker when the centenary of her mother’s birth was celebrated in the Soviet Union. There is no question that Sofia Krukovsky Kovalevskaya was an incredible scientist and human being, an example to remember and respect.
 
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