玛丽·卡特赖特Mary Cartwright

玛丽·卡特赖特的父亲是William Degby 卡特赖特（约1865年生于伦敦蒂尔尼街），在卡特赖特出生时，他是Aynho的牧师。她的母亲是Lucy Harriette Maud Cartwright（约1869年生于伦敦帕丁顿）。卡特赖特有四个兄弟姐妹：弗瑞兹·约翰（约1896年出生）、Nigel（约1898年出生）、Jane（约1905年出生）和William（约1907年出生）。

当她十一岁时，卡特赖特被送到学校，先就读于利明顿高中，后来就读于索尔兹伯里的戈多尔芬学校。她在学校最擅长的科目是历史，但它的缺点是需要花费大量精力学习无穷无尽的事实清单。当她在学校的最后一年受到鼓励学习数学时，卡特赖特意识到这是一个无需长时间学习事实就能取得成功的科目。它成了她想要在大学学习的科目。

1919年10月，卡特赖特进入牛津大学圣休学院学习数学。当时，她是整个大学中仅有的五位学习数学的女性之一。这是进入大学的一个困难时期，因为第一次世界大战刚刚结束，有大量男性从军队返回，他们要么重新开始战前已开始的大学学业，要么首次开始他们的学业。讲堂里挤满了人，卡特赖特常常不得不抄写她因人群拥挤而无法进入的讲座笔记。

经过两年的学习，她参加了数学 Moderations 考试，并获得二等成绩。不仅卡特赖特觉得拥挤的条件难以应付，因为那一年一等成绩的获奖者非常少。然而，这并没有阻止她因未能获得她所追求的一等成绩而产生的深深失望，卡特赖特认真考虑完全放弃数学，回到她最初热爱的历史。这是一个痛苦的决定，她为此苦恼了一段时间。然而，她非常享受数学，她仍然记得在学校学习历史时长时间学习事实的经历。她决定坚持她的数学课程，但[4]：-

她留在数学领域的决定并没有减少她对历史的兴趣。她的许多数学论文都包含历史视角，为她的工作增添了有趣的维度。她写了几篇传记回忆录，展现了她非凡的历史感。

在决定完成数学荣誉课程后，卡特赖特幸运地遇到了一位同学V C Morton。Morton是一位优秀的数学家——他后来成为阿伯里斯特威斯大学的数学教授——他给了她非常明智的建议。他告诉她阅读埃德蒙·泰勒·惠特克和乔治·奈维尔·沃森的Modern analysis，并参加戈弗雷·哈罗德·哈代的晚间课程。这些课程在周一晚餐后举行，包括一次演讲，然后进行数学讨论，直到晚上较晚的时候。在获得许可参加这些鼓舞人心的课程后（[5]或[6]）：-

……奠定了她职业生涯的方向……

卡特赖特最终在 Final Honours 中获得一等学位，并于1923年从牛津大学毕业。

卡特赖特随后在学校任教四年，之后返回牛津攻读博士学位。1923年没有继续攻读博士学位的原因，似乎是她不想给家庭财务带来更多压力。她先是在伍斯特的爱丽丝·奥特利学校任教，然后在白金汉郡的威科姆阿比学校任教。第二个职位还附带助理女教师的职位，卡特赖特很快发现，行政工作使她偏离了教学。另见[3]：-

……学校的教学和方法内容受到严格规定。没有实验空间让卡特赖特对自己的职业感到不满。

1928年回到牛津后，她的博士研究由戈弗雷·哈罗德·哈代指导。在1928-29学年，戈弗雷·哈罗德·哈代在普林斯顿，正是爱德华·查尔斯·蒂奇马什接任了导师的职责。她关于整函数零点的学位论文由利特尔伍德审查，她是在博士学位论文口试中第一次见到他作为外部考官。在那紧张的时刻，她绝不可能猜到，自己将在多年间与利特尔伍德成为重要的合作者。

1930年，卡特赖特获得Yarrow研究奖学金，前往剑桥大学格顿学院，继续研究其博士论文的主题。在听利特尔伍德的讲座时，她解决了他提出的一个未解决问题。她的定理，现在称为卡特赖特定理，给出了一个解析函数最大模的估计，该函数在单位圆盘中取同一值不超过$p$次。为了证明该定理，她使用了一种新方法，应用了拉尔斯·阿尔福斯为共形映射引入的技术。从这时起，她的研究蓬勃发展，并且（[5]或[6]）：-

……在一系列长论文中，她继续探索复（特别是整）函数理论；尤其是它们“爆炸”时的奇异行为……其中一些复杂性已被现代分形图像所捕捉，许多分形图像是通过迭代这类函数生成的。卡特赖特的工作描述了，除其他内容外，可能出现在分形边界附近的深刻而微妙的现象，并在该领域找到了新的应用。

1934年，在戈弗雷·哈罗德·哈代和利特尔伍德的推荐下，卡特赖特被任命为剑桥大学数学助理讲师，次年又被任命为兼职数学讲师。1936年，她成为格顿学院数学研究主任，1938年，她开始从事一个新项目，该项目对她研究的方向产生了重大影响。科学和工业研究部的无线电研究委员会就某些微分方程制作了一份备忘录，这些微分方程来自无线电和雷达工作的建模。他们询问伦敦数学会能否帮助找到一位能研究这些问题的数学家，卡特赖特对他们的备忘录产生了兴趣。

这些问题背后的动力学对卡特赖特来说并不熟悉，因此她向利特尔伍德寻求这方面的帮助。他们开始合作研究这些方程。利特尔伍德写道：-

为了找点事做，我们不停地钻研这个东西，完全看不到有任何“结果”的希望；突然间，解的戏剧性精细结构的整个图景展现在我们面前。

利特尔伍德在这里描述的精细结构，今天被视为“蝴蝶效应”的一个典型实例。这次合作产生了重要结果，这些结果极大地影响了现代动力系统理论所采取的方向。1947年，主要基于她在与利特尔伍德合作中的杰出贡献，她当选为皇家学会会士，尽管她不是当选该学会会士的第一位女性，但她是第一位女性数学家。

卡特赖特于1948年被任命为格顿学院院长，随后于1959年又任剑桥大学函数论讲师，担任此职直到1968年。作为格顿学院院长[5]：-

……她不强加自己的观点，但充分承担自己的责任，以明智而清醒的决定引导学院。

尽管她在格顿学院的职务所涉及的行政工作相当繁重，她仍抽出时间进行研究，特别是她在20世纪50年代关于聚值集的工作。退休后，她有了更多时间从事她如此喜爱的旅行。她在美国和欧洲的多所机构度过了较长时间；特别是她在布朗大学度过了1968-69学年，然后在克莱蒙特研究生院，1969-70年访问了威尔士大学和波兰。

她还获得了许多其他荣誉，包括1964年的皇家学会詹姆斯·约瑟夫·西尔维斯特奖章：-

……以表彰她在分析以及实变量与复变量函数理论方面的杰出贡献。

她是第一位获得詹姆斯·约瑟夫·西尔维斯特奖章的女性，也是第一位在皇家学会理事会任职的女性。卡特赖特于1961-62年担任伦敦数学会主席，成为第一位女性主席，并且至今仍是唯一一位女性主席。她还在1968年获得了该学会的奥古斯塔斯·德摩根奖章。1969年，她获得了女王授予的荣誉，成为大英帝国司令勋章获得者，即卡特赖特女爵士。

卡特赖特在许多不同方面都是一位非凡的人物，[3]中这样描述她：-

……一个具有天赋的人，能够直击事情的核心，无论是在数学还是在人事中都能看到要点。

她的幽默感在[3]、[5]和[6]中被描述为“冷面滑稽”，Caroline Series在[6]中写道：-

即便在96岁高龄，电视纪录片《我们辉煌的事业》仍捕捉到了她机智的敏锐光芒。

我[EFR]观看了这部电视纪录片，完全赞同Caroline Series的评论。看到卡特赖特眼中的光芒是一次美妙的体验。

她的其他兴趣在[1]中被提及：-

她绝非狭隘的专家，对广泛的主题都异常熟悉，其中包括绘画和音乐。Stanley Spencer为她绘制的肖像将向Girton的后代传达Dame 卡特赖特作为学者和管理者的一些印象，但他们可能会错过她朋友们所熟知的温暖幽默感和同情心。

Mary Cartwright's father was William Degby Cartwright (born in Tilney Street, London about 1865), and at the time of Mary's birth he was the Vicar at Aynho. Her mother was Lucy Harriette Maud Cartwright (born in Paddington, London about 1869). Mary had four siblings: John (born about 1896), Nigel (born about 1898), Jane (born about 1905) and William (born about 1907).

When she was eleven years old, Mary Cartwright was sent away to school, first attending Leamington High School, then later attending the Godolphin School in Salisbury. Her best subject at school was history but it had the disadvantage of requiring much effort in learning endless lists of facts. When she was encouraged in her studies of mathematics in her final year at school, Mary realised that it was a topic where one could succeed without the long hours of learning facts. It became the topic that she wanted to study at university.

In October 1919 Cartwright entered St Hugh's College in Oxford to study mathematics. At that time she was one of only five women in the whole university who were studying mathematics. This was a difficult time to enter university since, World War I having just ended, there were large numbers of men returning from the army who were either restarting the university studies they had begun before the war or were taking up their studies for the first time. The lecture halls were crowded and often Cartwright had to copy up notes of lectures which she could not get into because of the crowds.

After two years of study she took her Mathematical Moderations examinations and was awarded second class. It was not only Cartwright who had found the crowded conditions hard, for there were very few first class awards that year. This did nothing, however, to stop a deep sense of disappointment at failing to get the first class that she had aimed for, and Cartwright seriously considered giving up mathematics altogether and returning to her first love of history. It was a painful decision, over which she agonised for some time. However, she was enjoying mathematics so much and she still remembered the long hours of learning facts when studying history at school. She decided to stick with her mathematics course but [4]:-

Her decision to remain in mathematics did not diminish her interest in history. Many of her mathematical papers include historical perspectives that add an interesting dimension to her work. She wrote several biographical memoirs that portray her exceptional sense of history.

Having made her decision to complete the mathematics honours course, Cartwright was fortunate to meet a fellow student V C Morton. Morton was a fine mathematician - he later became Professor of Mathematics at Aberystwyth - and he gave her very sensible advice. He told her to read Whittaker and Watson's Modern analysis and to attend Hardy's evening sessions. These sessions took place after dinner on Monday and consisted of a talk and then mathematical discussions until later in the evenings. Having received permission to attend these inspiring sessions which ([5] or [6]):-

... set the course of her career ...

Cartwright went on to be awarded a first class degree in Final Honours, and graduated from Oxford in 1923.

Cartwright then taught in schools for four years before returning to Oxford to read for her D.Phil. The reason for not continuing to her doctorate in 1923 seems to have been a wish not to impose further strains on her family's finances. She taught first at Alice Ottley School in Worcester, and then at Wycombe Abbey School in Buckinghamshire. This second post also carried with it a position of assistant mistress and Cartwright soon found that she was being diverted from teaching by the administration. Also [3]:-

... teaching and method content were strictly dictated at the school. Having no room to experiment led Cartwright to feel discontent with her career.

Back to Oxford in 1928, she was supervised by Hardy in her doctoral studies. During the academic year 1928-29 Hardy was at Princeton, and it was Titchmarsh who took over the duties as a supervisor. Her thesis on zeros of integral functions was examined by J E Littlewood whom she met for the first time as an external examiner in her oral examination for the D.Phil. She could never have guessed on that stressful occasion that she would become a major collaborator with Littlewood over many years.

In 1930 Cartwright was awarded a Yarrow Research Fellowship and she went to Girton College, Cambridge, to continue working on the topic of her doctoral thesis. Attending Littlewood's lectures, she solved one of the open problems which he posed. Her theorem, now known as Cartwright's Theorem, gave an estimate for the maximum modulus of an analytic function which takes the same value no more than $p$ times in the unit disc. To prove the theorem she used a new approach, applying a technique introduced by Ahlfors for conformal mappings. From this time her research flourished and ([5] or [6]):-

... in a long series of papers she continued to explore the theory of complex (especially entire) functions; particularly their strange behaviour where they "blow up" ... Some of the complexity has been captured in modern pictures of fractals, many of which are created by iterating functions of this kind. Cartwright's work describes, among other things, deep and delicate phenomena which can appear near fractal boundaries, and has found new applications in this field.

Cartwright was appointed, on the recommendation of both Hardy and Littlewood, to an assistant lectureship in mathematics in Cambridge in 1934, and she was appointed a part-time lecturer in mathematics the following year. In 1936 she became director of studies in mathematics at Girton College, and in 1938 she began work on a new project which had a major impact on the direction of her research. The Radio Research Board of the Department of Scientific and Industrial Research produced a memorandum regarding certain differential equations which came out of modelling radio and radar work. They asked the London Mathematical Society if they could help find a mathematician who could work on these problems and Cartwright became interested in their memorandum.

The dynamics which lay behind the problems was unfamiliar to Cartwright and so she approached Littlewood for help with this aspect. They began to collaborate studying the equations. Littlewood wrote:-

For something to do we went on and on at the thing with no earthly prospect of "results"; suddenly the whole vista of the dramatic fine structure of solutions stared us in the face.

The fine structure which Littlewood describes here is today seen to be a typical instance of the "butterfly effect". The collaboration led to important results, and these have greatly influenced the direction that the modern theory of dynamical systems has taken. In 1947, largely on the basis of her remarkable contributions in the collaboration with Littlewood, she was elected a Fellow of the Royal Society and, although she was not the first woman to be elected to that Society, she was the first woman mathematician.

Cartwright was appointed Mistress of Girton in 1948 then, in addition, a Reader in the Theory of Functions in Cambridge in 1959, holding this appointment until 1968. As Mistress of Girton [5]:-

... without imposing her views, but carrying her responsibilities to the full, she guided the College with wise and clear headed decisions.

Although the administration involved in her role at Girton was substantial she found time for research, particularly her work on cluster sets in the 1950s. After retiring she found more time for the travelling which she so enjoyed. She spent extended periods at various institutions in the United States and Europe; in particular she spent the academic year 1968-69 at Brown University then Claremont Graduate College, with the University of Wales and Poland being visited in 1969-70.

She received many other honours including the Sylvester Medal of the Royal Society in 1964:-

... in recognition of her distinguished contributions to analysis and the theory of functions of a real and complex variable.

She was the first woman to receive the Sylvester Medal and, the first woman to serve on the Council of the Royal Society. Cartwright was President of the London Mathematical Society in 1961-62 becoming the first woman president, and until now the only woman president. She also received the De Morgan Medal of the Society in 1968. In 1969 she received the distinction of being honoured by the Queen, becoming Dame Mary Cartwright, Commander of the Order of the British Empire.

A remarkable person in many different ways, Cartwright is described in [3] as:-

... a person who had a gift for going to the heart of a matter and for seeing the important point, both in mathematics and in human affairs.

Her sense of humour is described in [3], [5], and [6] as "wry" and Caroline Series writes in [6]:-

Even at the age of 96, the TV documentary "Our brilliant careers" captured the sharp sparkle of her wit.

I [EFR] watched the TV documentary and fully agree with Caroline Series' comment. It was a wonderful experience to see the sparkle in Cartwright's eyes.

Her other interests are mentioned in [1]:-

She was no narrow specialist, being exceptionally well informed on a wide variety of subjects, among them painting and music. Stanley Spencer's portrait of her will convey to future generations at Girton some idea of Dame Mary as scholar and administrator, but they may miss the warm sense of humour and sympathy that her friends knew.