1. Sofia Kovalevskaya
A 19th century pioneer for women in mathematics
June Barrow-Green
The Open University
Florence Nightingale Day
Lancaster University
17 December 2015

2. Women mathematicians born before 1850
Hypatia of Alexandria (c ) Wrote commentaries on Diophantus’ Arithmetica and Apollonius’ Conics.
Marquise du Chatelet ( ) French translation of Newton’s Principia (1759).
Maria Agnesi ( ) Institutzioni Analitche (1748), first text book to include calculus.
Caroline Herschel ( ) Worked on astronomy with her brother William; discoverer of eight comets.
Sophie Germain ( ) Grand Prix of the French Academy of Sciences for work on the vibration of elastic surfaces (1816).
Mary Somerville ( ) The Mechanism of the Heavens (1831).
Ada Lovelace ( ) Translated and wrote extensive notes on Menabrea’s Mémoire on Charles Babbage’s Analytical Engine (1846).
Florence Nightingale ( ) Made pioneering studies of mortality statistics.

3. Moscow
b.1850

4. Moscow
b.1850
Palabino
1858

5. Moscow
b.1850
Palabino
1858

6. University education for women in the 19th century
Russia 1869
University Courses for women are opened, which opens the profession of teacher, law assistant and similar lower academic professions for women
UK 1878
London University first university in UK to award degrees to women
[Oxford (1920), Cambridge (1947)]

7. St Petersburg
1868
Moscow
b.1850
Palabino
1858

8. St Petersburg 1868
‘Fictitious’ marriage to Vladimir Kovalevski

9. St Petersburg
1868

10. St Petersburg
1868
London
1869

11. St Petersburg
1868
London
1869

12. Kovalevskaya meets Herbert Spencer in London, 1869
George Eliot at once turned to [Spencer]. “I’m so glad you have come today” she said, “I can introduce you to the living refutation of your theory – a woman mathematician. Allow me to present my friend,” she continued, turning to me still without mentioning his name, “only I have to warn you that he denies the very existence of a woman mathematician. … Try to make him change his mind!”
Sofia Kovalevskaya ‘My recollections of George Eliot’

13. St Petersburg
1868
London
1869
Heidelberg
1869

14. St Petersburg
1868
London
1869
Berlin
1870
Heidelberg
1869

15. Berlin
Studies with Karl Weierstrass, one of the leading German mathematicians of the day, and acknowledged as a great teacher.

16. PhD 1874
Awarded PhD summa cum laude in absentia from University of Göttingen
Partial Differential Equations
Abelian Integrals
Shape of Saturn’s Rings
[describe a wide variety of phenomena such as sound, heat, fluid flow etc.] Sofia had examined equations for heat conduction and discovered that certain PDEs have no (what are called) analytic solutions even when they have formal series solutions. This was her first important work, and its significance was acknowledged by Weierstrass as well as by several other leading mathematicians, including the great French mathematician Henri Poincaré.
Elliptic integrals are part of higher calculus – they originally arose in connection with the problem of giving the arc length of an ellipse’

17. Partial Differential Equations
PDEs contain the partial derivatives of a function of more than one variable.
E.g. the wave equation [in modern notation]:
𝜕 2 𝑢 𝜕 𝑡 2 = 𝑐 2 𝜕 2 𝑢 𝜕 𝑥 2
first formulated by Jean-Baptiste le Rond d’Alembert in
1747 in connection with his solution of the problem of
the vibrating string.
[describe a wide variety of phenomena such as sound, heat, fluid flow etc.] Sofia had examined equations for heat conduction and discovered that certain PDEs have no (what are called) analytic solutions even when they have formal series solutions. This was her first important work, and its significance was acknowledged by Weierstrass as well as by several other leading mathematicians, including the great French mathematician Henri Poincaré.
Elliptic integrals are part of higher calculus – they originally arose in connection with the problem of giving the arc length of an ellipse’
Kovalevskaya’s work included the definitive formulation of what is now known as the Cauchy-Kovalevskaya theorem.

18. Abelian Integrals Abelian integrals are named after the Norwegian
mathematician Niels Henrik Abel (1802–1829).
Most integrals are impossible to solve using elementary functions. The simplest type of non-elementary integral are elliptic integrals.
E.g.
𝐹 𝜑,𝑐 = 0 𝜑 − 𝑐 2 sin 2 𝜃 𝑑𝜃
Elliptic integrals originally arose in connection with the problem
of finding the arc length of an ellipse.
[describe a wide variety of phenomena such as sound, heat, fluid flow etc.] Sofia had examined equations for heat conduction and discovered that certain PDEs have no (what are called) analytic solutions even when they have formal series solutions. This was her first important work, and its significance was acknowledged by Weierstrass as well as by several other leading mathematicians, including the great French mathematician Henri Poincaré.
Elliptic integrals are part of higher calculus – they originally arose in connection with the problem of giving the arc length of an ellipse’
Kovalevskaya investigated a certain class of Abelian Integrals which reduce to elliptic integrals

19. Shape of Saturn’s rings
1675 Rings first seen by Domenico Cassini
1787 Pierre-Simon Laplace suggested that the rings are
formed of solid ringlets; assumed their cross-section
to be elliptical
1856 James Clerk Maxwell showed that solid rings are
unstable and would break apart
[describe a wide variety of phenomena such as sound, heat, fluid flow etc.] Sofia had examined equations for heat conduction and discovered that certain PDEs have no (what are called) analytic solutions even when they have formal series solutions. This was her first important work, and its significance was acknowledged by Weierstrass as well as by several other leading mathematicians, including the great French mathematician Henri Poincaré.
Elliptic integrals are part of higher calculus – they originally arose in connection with the problem of giving the arc length of an ellipse’

20. Shape of Saturn’s rings
Published in 1885
Kovalevskaya used Fourier series to represent the
cross-section which gave an oval (rather than an ellipse)
This led her, via some impressive juggling which involved elliptic integrals, to a system of infinitely many equations in infinitely many unknowns!
[describe a wide variety of phenomena such as sound, heat, fluid flow etc.] Sofia had examined equations for heat conduction and discovered that certain PDEs have no (what are called) analytic solutions even when they have formal series solutions. This was her first important work, and its significance was acknowledged by Weierstrass as well as by several other leading mathematicians, including the great French mathematician Henri Poincaré.
Elliptic integrals are part of higher calculus – they originally arose in connection with the problem of giving the arc length of an ellipse’

21. Shape of Saturn’s rings
Published in 1885
Kovalevskaya used Fourier series to represent the
cross-section which gave an oval (rather than an ellipse)
This led her, via some impressive juggling which involved elliptic integrals, to a system of infinitely many equations in infinitely many unknowns!
In the simplest case she was led to expressions of the form
𝑎 2 𝛾 2 +𝛼𝛽𝛾 cos𝑡+cos 𝑡 1 −2𝛼𝛽𝛾 sin𝑡sin2 𝑡 1 +sin2𝑡sin 𝑡 1
× −2 𝑎 2 𝛾 2 sin2𝑡sin2 𝑡 1 −𝛼𝛽𝛾 cos3𝑡+cos3 𝑡 1 − 𝑎 2 𝛾 2 (cos4𝑡+cos4 𝑡 1 ) 1+ 𝛽 2 −2cos𝑡cos 𝑡 1 −2 𝛽 2 sin𝑡sin 𝑡 − 𝛽 2 (cos2𝑡+cos𝑡2 𝑡 1 )
which she didn’t compute!
[describe a wide variety of phenomena such as sound, heat, fluid flow etc.] Sofia had examined equations for heat conduction and discovered that certain PDEs have no (what are called) analytic solutions even when they have formal series solutions. This was her first important work, and its significance was acknowledged by Weierstrass as well as by several other leading mathematicians, including the great French mathematician Henri Poincaré.
Elliptic integrals are part of higher calculus – they originally arose in connection with the problem of giving the arc length of an ellipse’
Kovalevskaya’s techniques were found to be useful in other contexts

22. Adrien Marie Legendre (1752–1797)
Important work on elliptic integrals
3 volume book published 1825–1830
[describe a wide variety of phenomena such as sound, heat, fluid flow etc.] Sofia had examined equations for heat conduction and discovered that certain PDEs have no (what are called) analytic solutions even when they have formal series solutions. This was her first important work, and its significance was acknowledged by Weierstrass as well as by several other leading mathematicians, including the great French mathematician Henri Poincaré.
Elliptic integrals are part of higher calculus – they originally arose in connection with the problem of giving the arc length of an ellipse’

23. Adrien Marie Legendre in 1820
Important work on elliptic integrals
3 volume book published 1825–1830
Caricature discovered in 2005
[describe a wide variety of phenomena such as sound, heat, fluid flow etc.] Sofia had examined equations for heat conduction and discovered that certain PDEs have no (what are called) analytic solutions even when they have formal series solutions. This was her first important work, and its significance was acknowledged by Weierstrass as well as by several other leading mathematicians, including the great French mathematician Henri Poincaré.
Elliptic integrals are part of higher calculus – they originally arose in connection with the problem of giving the arc length of an ellipse’
Adrien Marie Legendre in 1820
Louis Legendre (1752–1797)
French politician

24. St Petersburg & Moscow 1874-1881

25. 1876 Meets Gösta Mittag-Leffler
St Petersburg
1876 Meets Gösta Mittag-Leffler
“More than anything else in St. Petersburg what I found most interesting was getting to know Kovalevskaya ... As a woman, she is fascinating. She is beautiful and when she speaks, her face lights up with such an expression of feminine kindness and highest intelligence, that it is simply dazzling. As a scholar she is characterised by her unusual clarity and precision of expression ... I understand fully why Weierstrass considers her the most gifted of his students.

26. St Petersburg & Moscow 1874-1881
Writes scientific and literary articles
Recommences correspondence with Weierstrass
Birth of her daughter Fufa
Mittag-Leffler starts trying to get her a job
Leaves Vladimir
starts working on mathematics again

27. Berlin
1881

28. Berlin
1881
Paris
1881

29. Paris

30. Paris
1883 Vladimir commits suicide
Sofia obtains a position in Stockholm

32. This lady, a native of Russia, is a celebrated mathematician, who lectured last winter at the University of Stockholm, and who has just been appointed Professor of Mathematics at that University. We believe that this is the first time, since the Middle Ages (in Italy), that a woman has been appointed to an academical chair at any University in Europe. Sweden is a country where much interest has been felt in the claims of the fair sex to a full opportunity of acquiring and exercising intellectual accomplishments. The position now conceded to Madame Kowalevski is worthy of notice as a sign of the times, and will be observed with gratification by many English friends and advocates of higher education for women.

33. Stockholm

34. Stockholm 1883-1888 1883 Privat Docent (5 year appointment)
1884 Appointed to editorial board of Acta Mathematica
Announcement of question for Prix Bordin of Fufa comes to Stockholm
1887 The Struggle for Happiness (two plays written jointly with Anna-Carlotta Leffler)
Submits entry to Prix Bordin

35. Prix Bordin 1888
“Improve, in some important point, the theory of the movement of a rigid body”

36. Prix Bordin 1888
“Improve, in some important point, the theory of the movement of a rigid body”
1 June Sofia sends in half-finished memoir
Late summer Sofia sends in revised (but still incomplete) memoir
“On the Problem of the Rotation of a Solid Body about a Fixed Point”
15 entries received for the competition

37. Prix Bordin 1888
“Improve, in some important point, the theory of the movement of a rigid body”
1 June Sofia sends in half-finished memoir
Late summer Sofia sends in revised (but still incomplete) memoir
“On the Problem of the Rotation of a Solid Body about a Fixed Point”
December Sofia learns she has won the competition
Prize money raised from 3,000 to 5,000 francs
15 entries received for the competition
Sofia impressed the committee by applying the recently developed and highly abstract theory of Abelian functions to solve a problem in physics.

38. Paris 1888-1889 24 December 1888 Sofia receives the Prix Bordin
Spring/Summer 1889
Travels in Europe
Looks for a position in Paris
Mittag-Leffler urgently trying to get her a permanent position in Stockholm

39. Stockholm 1889-1891 1889 Professor in Stockholm
Publication of Prix Bordin memoir

40. Stockholm 1889-1891 1889 Professor in Stockholm
Publication of Prix Bordin memoir
1890 Wrote Nihilist Girl (also called
Vera Vorontsova)
[printed in Switzerland in 1892;
in Russia in 1906, then banned;
a Czech translation allowed in Russia in 1908]

41. Stockholm 1889-1891 1889 Professor in Stockholm
Publication of Prix Bordin memoir
1890 Wrote Nihilist Girl (also called
Vera Vorontsova)
[printed in Switzerland in 1892;
in Russia in 1906, then banned;
a Czech translation allowed in Russia in 1908]
1891 Dies of pneumonia

42. Sofia Kovalevskaya 1850–1891
Her competence and poise as a member of the faculty in Stockholm established that society can, without falling apart, allow women to become the colleagues of men on university faculties. Although the realisation of all her ambitions for women in mathematics is not yet complete, it continues apace. The impetus she gave to this movement a century ago and the permanent advances in knowledge which her works gave to the world form the enduring foundation of her fame.
Roger Cooke
The Mathematics of Sonya Kovalevskay (1984)
“It is impossible to be a mathematician without being a poet in soul”