Sofia Kovalevskaya 1850-1891 A 19th century pioneer for women in mathematics June Barrow-Green The Open University Florence Nightingale Day Lancaster University 17 December 2015

Women mathematicians born before 1850 Hypatia of Alexandria (c.370-415) Wrote commentaries on Diophantus’ Arithmetica and Apollonius’ Conics. Marquise du Chatelet (1706-1749) French translation of Newton’s Principia (1759). Maria Agnesi (1718-1799) Institutzioni Analitche (1748), first text book to include calculus. Caroline Herschel (1750-1848) Worked on astronomy with her brother William; discoverer of eight comets. Sophie Germain (1776-1831) Grand Prix of the French Academy of Sciences for work on the vibration of elastic surfaces (1816). Mary Somerville (1780-1872) The Mechanism of the Heavens (1831). Ada Lovelace (1815-1852) Translated and wrote extensive notes on Menabrea’s Mémoire on Charles Babbage’s Analytical Engine (1846). Florence Nightingale (1820-1910) Made pioneering studies of mortality statistics.

Moscow b.1850

Moscow b.1850 Palabino 1858

Moscow b.1850 Palabino 1858

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)]

St Petersburg 1868 Moscow b.1850 Palabino 1858

St Petersburg 1868 ‘Fictitious’ marriage to Vladimir Kovalevski

St Petersburg 1868

St Petersburg 1868 London 1869

St Petersburg 1868 London 1869

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’

St Petersburg 1868 London 1869 Heidelberg 1869

St Petersburg 1868 London 1869 Berlin 1870 Heidelberg 1869

Berlin 1870-1874 Studies with Karl Weierstrass, one of the leading German mathematicians of the day, and acknowledged as a great teacher.

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’

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.

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 1 1− 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

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’

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’

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 − 1 2 𝑎 2 𝛾 2 (cos4𝑡+cos4 𝑡 1 ) 1+ 𝛽 2 −2cos𝑡cos 𝑡 1 −2 𝛽 2 sin𝑡sin 𝑡 1 + 1 2 1− 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

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’

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

St Petersburg & Moscow 1874-1881

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.

St Petersburg & Moscow 1874-1881 1877 Writes scientific and literary articles 1878 Recommences correspondence with Weierstrass Birth of her daughter Fufa 1880 Mittag-Leffler starts trying to get her a job 1881 Leaves Vladimir 1879 starts working on mathematics again

Berlin 1881

Berlin 1881 Paris 1881

Paris 1881-1883

Paris 1881-1883 1883 Vladimir commits suicide Sofia obtains a position in Stockholm

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.

Stockholm 1883-1891

Stockholm 1883-1888 1883 Privat Docent (5 year appointment) 1884 Appointed to editorial board of Acta Mathematica 1886 Announcement of question for Prix Bordin of 1888 Fufa comes to Stockholm 1887 The Struggle for Happiness (two plays written jointly with Anna-Carlotta Leffler) 1888 Submits entry to Prix Bordin

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

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

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.

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

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

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]

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

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”