Dame Mary Lucy Cartwright was born on 17 December 1900 at Aynho in Northamptonshire, the daughter of William Digby Cartwright, rector of the parish, and Lucy Harriette Maud Cartwright, née Bury. She grew up in a clerical household as one of five children and was educated in a period when advanced mathematical study for women was still exceptional. In 1919 she entered St Hugh’s College, Oxford, to read mathematics. The lecture rooms were crowded with men returning from the First World War, and in her first year she often worked from copied notes. She nearly left the subject after disappointing early results, but she persisted and in 1923 became the first woman to obtain a first-class degree in the final honours examinations in mathematics.
After Oxford she taught for several years at the Alice Ottley School in Worcester and at Wycombe Abbey School in Buckinghamshire. School teaching did not end her interest in research. In 1928 she returned to Oxford to begin a doctorate under G. H. Hardy. Her thesis, on the zeros of integral functions of special types, was examined by J. E. Littlewood and completed in 1930. That examination began a professional relationship that would shape much of her later work. In the same year she was awarded a Yarrow Research Fellowship and moved to Girton College, Cambridge, where she continued the function-theoretic investigations of her thesis.
At Cambridge she attended Littlewood’s lectures and solved an open problem he had posed. The result, now known as Cartwright’s theorem, gives a sharp estimate for the maximum modulus of an analytic function that takes any given value at most a fixed number of times inside the unit disc. The theorem belongs to classical complex analysis, yet it has continued to be used in contexts far from pure mathematics, including aspects of signal processing. She published steadily through the 1930s on integral functions, conformal mapping, and related questions in the theory of functions. In 1935 she proved an inequality in conformal mapping that had resisted earlier attempts. Girton appointed her fellow and director of studies in mathematics in 1934; the university made her a lecturer in 1935. She remained resident at Girton for nearly four decades.
Her research changed direction in 1938, when the Radio Research Board of the Department of Scientific and Industrial Research circulated a memorandum asking mathematicians to examine certain nonlinear differential equations that arose in radio engineering. The equations were related to the work of the Dutch physicist Balthasar van der Pol on relaxation oscillations in electronic circuits. Cartwright took up the problem and persuaded Littlewood to collaborate with her. Their joint study of the forced van der Pol equation, especially in the regime of large nonlinearity, occupied them through the war years and beyond.
What they found was unexpected. For ranges of the parameters the equation possesses infinitely many unstable periodic solutions, and the long-term behaviour of solutions can depend in an extremely sensitive way on the initial conditions. Littlewood later described the phenomenon in language that anticipates the popular image of the butterfly effect: a minute change in the starting state can lead, after sufficient time, to an entirely different outcome. Cartwright and Littlewood did not use the later vocabulary of chaos, strange attractors, or horseshoes. Those terms belong to the 1960s and after, particularly to the work of Stephen Smale and others. Their papers of the mid-1940s, however, gave a rigorous topological and analytic account of complicated recurrent behaviour in a concrete dissipative system, and they are now widely regarded as one of the earliest mathematical investigations of what came to be called chaos. The practical side of the same work improved the understanding of radio amplifiers and oscillators used during the Second World War. Alongside this research Cartwright served as a volunteer with the British Red Cross from 1940 to 1944.
In 1947, largely on the strength of the collaboration with Littlewood and her earlier work in analysis, she was elected a Fellow of the Royal Society. She was the first woman mathematician to receive that distinction, and later the first woman to serve on the Society’s council. In 1949 she became Mistress of Girton College, a post she held until 1968, the longest tenure in the college’s history. The administrative burden was heavy, yet she continued to publish. Her Cambridge tract Integral Functions appeared in 1956, distilling a larger manuscript on the subject that the war had interrupted. In the 1950s she also returned to questions about cluster sets. Over her career she wrote more than ninety papers, ranging from classical function theory to the topological dynamics of nonlinear oscillations.
Honours accumulated in the later part of her career. She was president of the Mathematical Association in 1951 and president of the London Mathematical Society from 1961 to 1963, the first woman in each office. The Royal Society awarded her the Sylvester Medal in 1964, again the first woman to receive it, citing her contributions to analysis and the theory of functions of a real and complex variable. The London Mathematical Society gave her its De Morgan Medal in 1968. In 1969 she was appointed Dame Commander of the Order of the British Empire. She also received honorary degrees, including one from Oxford in 1966, and an honorary fellowship of the Royal Society of Edinburgh.
She retired from her readership in the theory of functions and from the Mistressship of Girton in 1968, and moved to a flat nearby in Cambridge. She remained active in mathematical life for some years and lived to see the modern theory of dynamical systems absorb and extend the phenomena she and Littlewood had described. She died in Cambridge on 3 April 1998, at the age of ninety-seven.
Cartwright’s reputation rests on two related achievements. The first is technical: precise theorems in complex analysis, and a body of work on nonlinear oscillations that exposed the intricate structure of solutions of the forced van der Pol equation. The second is institutional and historical. She entered a profession that had only recently opened its higher examinations and its learned societies to women, and she occupied, for the first time, a succession of offices that had been held only by men. She did so without abandoning research. The wartime problems that drew her from pure function theory into differential equations also show how a question posed by engineers can force the discovery of phenomena that pure mathematics is then obliged to explain. Later writers on chaos have sometimes compressed that story into a single anecdote about sensitivity to initial conditions. The fuller record is a sequence of papers, joint with Littlewood and later written alone, that used the methods of topology and classical analysis to prove the existence of highly complicated invariant sets in a specific equation. That combination of exactness and unexpected dynamical behaviour is the centre of her mathematical legacy.
CONCLUSION
A female achiever in the field of mathematics.