S.V. Kovalevskaya was the most gifted woman mathematician of the 19th century. She produced original research that was greatly respected by her colleges. During her career she published nine research papers dealing with topics such as the propagation of light in a crystal, the shape of the rings of Saturn, the laws describing rotations, and works pertaining to theoretical issues in partial differential equations.
Her research was in the mathematical subspecialty of analysis working primarily in partial differential equations and applying the methods and mathematical approach of her mentor Karl Weierstrass. Weierstrass was the leading mathematical analysist of the century and to a great extent changed how mathematicians thought about the concepts in the calculus.
Her graduate dissertaion was a three part paper, any part of which might have qualified her for a doctorate. The most outstanding section of the paper is cited today in texts on partial differential equations as the Cauchy-Kovalevskaia Theorem. This work resides in what can be called =pure= mathematics, that is there are no immediately obvious physical applications. The work proves that under certain conditions, members of a particular group of partial differential equations have unique solutions. She received her PhD in mathematics from the Gottingen U. in 1874 summa cum laude. Modern day mathematicians have said that this work of SK marks the beginning of the general theory of partial differential equations. (See Koblitz's quote of Oleinik, CONVERGENCE OF LIVES, p 242 as taken from a Russian math journal of 1975.) The clearness of SKs exposition in this paper, as in all her works, is frequently noted and admired by mathematicians to this day.
Her greatest achievement in mathematical research is her paper on the rotation of a solid body about a fixed point -- a top is one example of the type of object being studied, as are gyroscopes and pendulums, and planets, suns, moons. By putting restrictions on the characteristics of rotating objects (shape, center of gravity), mathematicians came up with three classes of objects for which this type of motion was defined by means of partial differential equations. The first two classes had been analyzed and solved prior to SK. (The general case, no restrictions on the rotating body, has still not been solved.) A method of solution for the last of the three defined classes had stumped mathematicians for over a hundred years and came to be known as the Mathematical Mermaid because of its fascinating and elusive nature. Kovalevskaya worked on the last of the classes whose motion is described by a particularly messy set of six differential equations in many variables. SKs breakthrough use of time as a complex variable (variables having a real and imaginary component) in these equations (as Poincare had done earlier in the n-body problem) lead to her solution of the problem. The excellence of this paper won her the Prix Bordin in December of 1888. The Prix Bordin was a mathematical contest sponsored by the Paris Academy of Science. The Paris Academy offered a prize and monetary award for the best solution of a particularly difficult mathematical problem vexing the science community. The contest rules stated that the contributor's name wasn't to be known by the judges until after the winning paper was selected. SK chose to identify her paper with the maxim: Say what you know, do what you must, and whatever will be, will be. The shape of the rotating body which SK considered in her prize winning paper has come to be called the Kovalevsky Top. (see Kochina, LOVE AND MATHEMATICS, Mir Publishers Moscow, 1985, 291-309 )
SK was the first woman in modern times to hold a tenured position on a university faculty of mathematics. She lectured in mathematics with emphasis on partial differential equations following the methods of Weierstrass and Poincare. When a colleague became ill, she also took on lectures in theoretical mechanics. She was on the faculty of the University of Stockholm with her lecture schedule beginning in Aug 1884 and ending with the fall classes of 1890. SK died suddenly in Feb 1891.
SK was the first woman to hold an editorial position on a mathematical journal. The journal, ACTA, is still published and held in esteem. Kovalevskaya joined the ACTA staff shortly after the journal's founding by Gosta Mittag-Leffler. Because of her professional skill, diplomatic personality (when she chose!) and great personal charm, she was very instrumental in assuring the publication's success on an international level at a time of strong nationalistic feelings in the science community.
Through her efforts and the power of her personality in the international mathematical community, she was instrumental especially in introducing the mathematical methods of the best Europeans to Russians, and those of Russians to Europeans. (She performed the same service for the literary community -- assisting in introducing Russians to the best of Western literature of the time and Westerners to outstanding Russian literature.)
For a complete and precise description of SKs mathematical works consult the reference THE MATHEMATICS OF SONYA KOVALEVSKAYA by Roger Cooke.
A LISTING OF SKs MATHEMATICAL WORKS as it appeared in a1995 NEH Grant Proposal by Joan Spicci Saberhagen and Natasha Kolchevska.
(Kovalevskaya published 10 works in mathematics. Two are identical except for the language, here numbered 5, 6.Two are considered to be of outstanding significance. Her work later known as Cauchy-Kovalevskaia Theorem and her paper for the Prix Bordin Competition on the rotation of a solid body about a fixed point. These items are marked with a "*". The following list of Kovalevskaya's mathematical works is from Stillman, A RUSSIAN CHILDHOOD, p249 and was compiled by P. Kochina.)
* 1. S. Kowalevsky, "Zur Theorie der partiellen Differentialgleichungen." Journal fur die reine und angewandte Mathematik, 80 (1875), 1-32.
2. _______. "Uber die Reduction einer bestimmten Klasse von Abel'scher Integrale 3-en Ranges auf elliptische Integrale." Acta Mathematica 4 (1884), 393-414.
3._________, "Zusatze und Bermerkungen zu Laplace's Untersuchung uber die Gestalt des Saturnringes." Astronomische Nachrichten 111 (1885), 37-48.
4._________, "Uber die Brechung des Lichtes in crystallinischen Mitteln." Acta Mathematica 6 (1883), 249-304.
5 __________, "Sur la propagation de la lumiere dans un milieu cristallise." Comptes rendus Acad. Sc. 98 (1884), 356-357.
6.__________, "Om ljusets fortplanting uti ett kristalliniskt medium." Of versigt af Kongl. Vetenskaps-Akademiens Forhandlinger 41 (1884), 119-121.
7.___________, "Sur le probleme de la rotation d'un corps solide autour d'un point fixe." Acta Mathematica 12 (1889), H.2, 177-232.
8.___________, "Sur une propriete du systeme d'equations differentielles qui definit la rotation d'un corps solide autour d'uin point fixe." Acta Mathematica 14 (1890), 81-93.
* 9. ___________, "Memoire sur un cas particulier du probleme de la rotation d'un corps pesant autour d'un point fixe, ou l'integration s'effectue a l'aide de fonctions ultraelliptiques du temps. Memoires presentes par divers savants a l'Academie des Sciences de l'Institut National de France, Paris, 31 (1890), 1-62.
10. _____________, "Sur un theoreme de M. Bruns." Acta Mathematica 15 (1891), 45-52.