拉尔斯·阿尔福斯Lars Ahlfors

拉尔斯·阿尔福斯在[6]中描述了他的祖先：-

我的父亲出生在奥兰群岛，该群岛是芬兰的一部分，但居民是瑞典人，并且完全讲瑞典语。我的祖先从 mainland 上的小城市比约讷堡来到奥兰群岛的玛丽港。我想他们中有些人是船长，拥有帆船，在世界各地进行贸易。我的祖父开了一家杂货店，他一定做得相当不错，因为他能够送孩子到 mainland 上学。这很好，因为奥兰群岛上几乎没有什么学校。这就是我父亲能够先成为工程师，然后成为赫尔辛基理工学院工程学教授的原因。

据他讲述，他的父亲卡尔·阿克塞尔Mauritz Ahlfors（1874-1961），使用阿克塞尔这个名字，是赫尔辛福斯理工学院机械工程教授。不幸的是，他的母亲西耶沃·玛蒂尔达·赫兰德（1881-1907）在他出生时死于难产。阿尔福斯有两个姐姐，奥内（1902-1921）和伊萨（1905-1990）。在他生命的最初三年，阿尔福斯由奥兰群岛上的两个姑妈照顾。他三岁时回到赫尔辛福斯与父亲同住。虽然阿尔福斯被父亲照顾得很好，但他后来形容父亲是一个非常严厉的人。这个家庭讲瑞典语，阿尔福斯后来才学会芬兰语。我们在此可以提到，他的家人用昵称“拉塞”称呼他。同样值得注意的是，他的父亲再婚娶了海莱娜·爱丽丝·帕尔姆罗斯（1884-1973），她使用爱丽丝这个名字。他们的女儿温加，阿尔福斯的同父异母妹妹，出生于1917年。

在阿尔福斯成长的时代，芬兰虽然是俄罗斯帝国的一部分，但受到合理对待，他的父亲阿克塞尔相当富裕，雇有一个女仆、一个厨师和一个家庭教师。然而这种情况没有持续下去，因为1914年第一次世界大战爆发，随后1917年俄国革命，芬兰获得独立。然而，革命的红军控制了社会民主党，并在1918年接管了赫尔辛福斯和芬兰南部的其他城镇。内战随之而来，食物匮乏，Axel Ahlfors被红军俘虏。然而，在德国对白军的支持下，红军很快被击败，Axel Ahlfors获释。阿尔福斯在一所私立瑞典语高中，新瑞典语学校接受教育。他在[3]中描述了他的早年：-

作为一个孩子，我对数学着迷，却不理解它是什么，但我绝不是神童。事实上，除了在最高年级，我无法接触到数学文献。见过许多神童被雄心勃勃的父母毁掉，我只能感谢我父亲的克制。高中课程不包括任何微积分，但我最终设法自学了一些，这要归功于偷偷访问我父亲的工程图书馆。

这些年里，阿尔福斯对学业之外的事情兴趣寥寥。他讨厌体育，只有上学日才开心，讨厌星期天和学校假期。他最大的快乐是做数学作业，这让他与典型的学校学生颇为不同！在他喜欢的学校科目中，我们应当提到语言，而在他厌恶的科目中，我们应当提到历史。他说[6]：-

为什么要记住与各种事件和人物相关的年份？人们大可以同样去记电话号码。这对我来说毫无意义，而且显得相当愚蠢。我的历史老师不太喜欢我。

起初他父亲想让他成为工程师，但意识到儿子对数学充满热情，且无法掌握任何机械方面的东西，当阿尔福斯大约15岁时，他父亲告诉他，他会成为数学教授。他于1924年17岁时进入赫尔辛福斯大学，在那里由恩斯特·列奥纳德·林德勒夫和罗尔夫·内万林纳授课[6]：-

我大一就修了高等微积分，这本不该是我修的。开学第一天，我去罗尔夫·内万林纳那里请求允许修他的课时，他问我：“你懂微积分吗？”“哦，懂，”我说，“我学过微积分。”我没有告诉他这全是我自学的。恩斯特·列奥纳德·林德勒夫不喜欢我跳级，他让我参加所有我没有上过的先修课程的全部考试。我喜欢高等微积分；我对我的学习非常感兴趣。

他于1928年从赫尔辛福斯大学毕业，获得候选人学位。1928/29学年，罗尔夫·内万林纳在赫尔曼·外尔赴美国进行学术访问期间，接替了赫尔曼·外尔在苏黎世的位置。恩斯特·列奥纳德·林德勒夫告诉阿尔福斯，他应该和罗尔夫·内万林纳一起去苏黎世，并去找阿尔福斯的父亲，说服他儿子将从这次经历中受益匪浅。在苏黎世，罗尔夫·内万林纳的讲课水平与他在芬兰时不同。阿尔福斯说[6]：-

苏黎世是我第一次接触到本应如此教授的数学。那是当代数学，而不是我们课程通常那样的数学史。在苏黎世，罗尔夫·内万林纳能够在研究层面上讲授，这给我留下了完全不同的印象。我开始理解数学是什么，以及我应当做数学，而不仅仅是学习数学。这一点在此之前对我来说并不清楚。

罗尔夫·内万林纳谈到了阿尔诺·当茹瓦在21年前提出的关于一个整函数的渐近值数目的猜想。阿尔福斯解决了这个猜想，从而突然在国际上闻名。他在[3]中谦虚地写道：-

我极其幸运地想到了一种基于共形映射的新方法，在罗尔夫·内万林纳和乔治·波利亚的大力帮助下，这导致了对整个猜想的证明。

他在两篇论文中发表了他的突破，两篇都发表于1929年：Sur le nombre des valeurs asymptotique d'une fonction entire d'ordre finiⓉ（论有限阶整函数的渐近值数目）和Über die asymptotischen Werte der ganzen Funktionen endlicher OrdnungⓉ（论有限阶整函数的渐近值）。在苏黎世之后，阿尔福斯与罗尔夫·内万林纳去了巴黎三个月，然后返回芬兰。在巴黎，他遇到了阿尔诺·当茹瓦，后者告诉他，21现在成了他最喜欢的数字，因为他的猜想在提出21年后被一位21岁的数学家解决了。回到芬兰后，他被任命为图尔库的瑞典语奥博学术大学的数学讲师。他在1930年提交了他的学位论文，其中给出了他对阿尔诺·当茹瓦猜想的证明。该猜想指出，如果一个整函数有$p$个不同的渐近值，那么它的阶≥½$p$。阿尔诺·当茹瓦证明了该猜想的一个特例，而Torsten Carleman在1921年证明了该猜想的一个较弱版本。Mikhail Sodin在[30]中概述了阿尔诺·当茹瓦、Carleman和阿尔福斯的工作的证明，他还提到了后来的发展。

在1930-32年间，阿尔福斯在洛克菲勒基金会奖学金的支持下多次访问巴黎及其他欧洲中心。1933年，他担任了赫尔辛基大学的副教授职位。同样在1933年，他与Erna Martha Liesbeth Lehnert结婚，她出生于维也纳，但随父母先移居瑞典，后移居芬兰。阿尔福斯和Erna 阿尔福斯育有三个女儿：Cynthia Mary、Vanessa Elisabeth和Caroline Gertrud。他们的儿子Christopher在婴儿时期便去世了。

1935年，康斯坦丁·卡拉西奥多里——阿尔福斯在旅行期间在慕尼黑见过他——推荐他去美国哈佛大学任职。阿尔福斯同意了三年的试用期。这里似乎适合从[13]中简要引用关于他早期数学贡献的内容：-

函数论在芬兰已有悠久的传统，阿尔福斯在他的老师罗尔夫·内万林纳所发展的理论的第一波浪潮刚过顶峰后便开始了他的职业生涯。……阿尔福斯工作的主要方向是回到该学科的基础，并从更几何（以及微分几何）的角度来看待它。这为几乎所有后来的推广提供了基础。我们在这里也看到了后来在拟共形映射、共形度量和极值长度方面工作的起源。

1936年，他是奥斯陆国际大会上首批两位约翰·查尔斯·菲尔兹奖章获得者之一。他写道[3]：-

1936年，在奥斯陆举行的国际数学家大会上，就在颁奖典礼前几个小时，我被告知我将获得首届两项菲尔兹奖之一，这让我经历了人生中最大的惊喜。当时这项荣誉的声望或许还不像现在这样高，但无论如何，我感到自己受到了特别的关注，深感荣幸。康斯坦丁·卡拉西奥多里的颁奖词中明确提到了我的论文《Zur Theorie der Überlagerungsflächen》，该文为罗尔夫·内万林纳的亚纯函数理论带来了一些新的启示。这个奖项在很大程度上增强了我对自己工作的信心。

正是在这篇论文中——该文是他获得约翰·查尔斯·菲尔兹奖章的重要因素——"拟共形映射"这一术语首次出现。康斯坦丁·卡拉西奥多里说，这篇论文开辟了分析的一个全新篇章，可以称之为"度量拓扑学"。阿尔福斯在[3]中对这篇论文的评论中写道：-

当时我几乎不知道拟共形映射会在我的工作中扮演如此重要的角色。

1938年，阿尔福斯获得了赫尔辛基大学的数学讲席职位，由于相当思乡，他接受了这一职位，而不是永久留在哈佛。然而，随着第二次世界大战即将开始，一段艰难的时期正在逼近。战争给芬兰带来了严重的问题，大学被迫关闭。阿尔福斯因身体条件不适合服兵役，因此正如他在[3]中所说：-

矛盾的是，我自己在战争期间反而能够做大量的工作，尽管无法利用可及的图书馆。

战争期间，阿尔福斯的家人被疏散到瑞典，与亲戚住在一起。阿尔福斯留在赫尔辛基，那里的大学因男生太少而关闭。这意味着尽管缺乏图书馆资源，他仍能专注于研究，其中大部分研究是在防空洞中进行的。冬季战争之后，他的家人回到了赫尔辛基，但在1941年德国进攻苏联后，芬兰支持了德国。起初，苏联忙于阻止德军推进，无暇过多关注芬兰，但一旦他们扭转了对德战局，便进攻了芬兰。1944年，当阿尔福斯获得苏黎世的讲席职位时，这似乎是改善他和家人所处境况的好机会。此时阿尔福斯的健康状况很差，心律不齐，因此他获准与家人一起前往瑞典，在那里Arne Beurling给了他们大量的帮助和友谊[6]：-

当我能够离开芬兰前往瑞典时，我被允许携带的克朗不超过10枚。那么我做了什么呢？我把我的约翰·查尔斯·菲尔兹奖章偷偷带了出来，然后把它典当了！我确信这是唯一一枚进过当铺的约翰·查尔斯·菲尔兹奖章。等我一拿到一点钱，瑞典的一些人就帮我把它赎了回来。

然而，战争使得前往瑞士就任苏黎世职位之行几乎不可能。在一个没有月亮的夜晚，安排了一架从斯德哥尔摩飞往苏格兰普雷斯特威克的航班，1945年3月，他们完成了这次旅行[3]：-

飞机是一架翻新的飞行堡垒，大约有十几名乘客。飞机没有增压，呼吸依靠个人氧气面罩。所有人都穿着救生衣。我们的孩子，一个5岁一个6岁，完全能够理解危险的含义。

他们从格拉斯哥乘火车到伦敦，然后艰难地穿越英吉利海峡到迪耶普，再经巴黎穿越法国到达瑞士。他写道[3]：-

我的第一个失望是得知我将负责画法几何，这门学科不知为何在瑞士高中和本科课程中保留了下来。我的第二个震惊是这门课要在早上7点到9点讲授。尽管如此，我还是慢慢适应了我的工作，其中甚至包括一些严肃但并不高深的数学。Fueter教授和他的同事保罗·芬斯勒教授年事已高，显然邀请我的原因是找不到有能力的本土继任者。我接手了一个处于成长阶段的学生班级，我很高兴地说，许多人一直是我的朋友，现在他们本身已是重要的数学家。我不能诚实地说我在苏黎世很快乐。战后时期不是一个陌生人在瑞士扎根的好时机。……我和妻子在直接同事圈子之外并不感到受欢迎。

因此，1946年哈佛大学的邀请被欣然接受，这一次，他留在那里，于1977年退休。接受哈佛任命两年后，阿尔福斯被选为数学系主任。1964年，他被命名为威廉·卡斯帕·格劳斯坦数学教授。James Jenkins在[10]中写道阿尔福斯在哈佛的早期岁月：-

阿尔福斯在会见研究生时非常尽责，但他的角色主要是被动回应，而不是提出建议或想法（至少在我的情况是这样）。1947年春季，阿尔福斯讲授了一门关于极值度量方法的课程，当时这一方法还处于起步阶段。1947-48学年，他讲授了一门变分法课程。1948-49学年，他讲授了一门关于波恩哈德·黎曼曲面的课程，主要方向是将他关于赫尔曼·阿曼杜斯·施瓦茨引理的工作推广到有限边界的波恩哈德·黎曼曲面。在我停留期间，阿尔福斯从未定期举办讨论班。有一年，他做了一系列关于奥斯瓦尔德·泰希米勒论文的报告，大概是为了给自己研究拟共形映射做准备。

关于阿尔福斯数学工作的出色综述见[17]，其中他的贡献按以下标题列出：共形几何；克莱因群；以及拟共形映射。这些部分转载于[16]。

他的著作具有持久的重要性。这些著作是Complex analysis（1953年）、Riemann surfaces（与Leo Sario合著）（1960年）、Lectures on quasi-conformal mappings（1966年）、Conformal invariants: topics in geometric function theory（1973年）和Möbius transformations in several dimensions（1981年）。

我们从THIS LINK上对这些著作的评论中摘录了稍作修改（或翻译）的简短片段。

请允许我[EFR]就阿尔福斯的Complex analysis说一句个人感想。这是Edward Copson推荐给我的教材，他教我复分析；Copson本人写过一本极好的复分析著作，却推荐阿尔福斯的书而不是自己的书，这确实是对阿尔福斯的致敬。我发现阿尔福斯的Complex analysis写得极美，是数学教材最高品质的典范，将清晰与对主题的激情融为一体。

我们让读者有机会在THIS LINK上阅读第一版序言，并在THIS LINK上阅读正文的简短摘录。

阿尔福斯因其对数学的杰出贡献获得了许多荣誉。上文提到的首届约翰·查尔斯·菲尔兹奖章的颁发，必定是最重要的，但其他重大荣誉还包括1968年芬兰国际奖以及1981年沃尔夫数学奖：-

……因在几何函数论中的开创性发现和强大新方法的创立。

沃尔夫奖的颁奖词对他的成就作了很好的概述：-

半个多世纪以来，复变量函数论一直受到（荣休）教授阿尔福斯的思想和工作的指引。他的成就包括证明阿尔诺·当茹瓦猜想、罗尔夫·内万林纳理论的几何推导、赫尔曼·阿曼杜斯·施瓦茨引理的重要推广、（与Beurling一起）发展极值长度方法，以及在波恩哈德·黎曼曲面、拟共形映射和奥斯瓦尔德·泰希米勒空间理论中的众多决定性成果。阿尔福斯关于克莱因群的著名有限性定理，以及他关于极限集的工作，重振了一个重要的研究领域。他现在正在从事高维拟共形形变的开创性工作。阿尔福斯的影响广泛而有益。他的方法将深刻的几何洞察与精妙的分析技巧结合起来；他表述这些方法时极为清晰简洁。他一次又一次地攻克并解决某一学科的核心问题。其他数学家一次又一次地受到他多年前所做工作的启发。今天每一位复分析学者在某种意义上都是他的学生。

他获得的其他荣誉包括波士顿大学（1953年）、奥博学术大学（1970年）、苏黎世大学（1977年）和伦敦大学（1978年）的荣誉学位。他当选为芬兰科学学会、Finnish Academy of Science、National Academy of Sciences、Royal Swedish Academy of Sciences和丹麦皇家科学院的成员。

让我们引用Troels Jorgensen [10]的话来结束这篇传记：-

我有幸认识阿尔福斯四分之一世纪。他是一位鼓舞人心的数学家，也是一个非常特别的人。他对数学的直觉和热爱，加上强大的工作能力，使他在跨越六十多年的职业生涯中成为一个和谐的工匠。他享受所获得的认可和奖项，但对自己的成就非常谦虚。在庆祝他七十岁生日的宴会上，阿尔福斯说：“我喜欢去有鱼的地方钓鱼，而不是一味地追求大鱼。”

Lars Ahlfors described his ancestors in [6]:-

My father was born in the Aland Islands, which is a part of Finland but has a Swedish population and is completely Swedish-speaking. My ancestors came to Mariehamm in the Aland Islands from Bjorneborg, a smaller city on the mainland. I think that some of them were sea captains, with sailing ships that traded all over the world. My grandfather had a grocery store, and he must have done fairly well because he was able to send his children to school on the mainland. This was good since there were practically no schools on the Aland Islands. That's how my father was able to become first an engineer and then a professor of engineering at the Polytechnical Institute in Helsinki.

As he related, his father, Karl Axel Mauritz Ahlfors (1874-1961), who used the name Axel, was professor of mechanical engineering at the Polytechnic Institute in Helsingfors. Tragically his mother, Sievä Matilda Helander (1881-1907), died in childbirth when he was born. Lars had two older sisters, Aune (1902-1921) and Isa (1905-1990). For the first three years of his life, Lars was looked after by two of his aunts on the Aland Islands. He returned to Helsingfors to live with his father when he was three years old. Although Lars was looked after well by his father, he later described him as a very stern man. The family were Swedish speaking and Lars only learnt Finnish later in life. We could mention at this point that his family called him by the nickname 'Lasse'. Also worth noting is that his father remarried Héléne Alice Palmroth (1884-1973), who used the name Alice. Their daughter Unga, Lars' half-sister, was born in 1917.

At the time that Ahlfors was growing up, Finland, although part of the Russian Empire, was treated reasonably and his father Axel was quite well off employing a maid, a cook, and a governess. However this situation did not last, since in 1914 World War I broke out and, following the Russian Revolution of 1917, Finland gained independence. However, the revolutionary Reds gained control of the Social Democratic Party and, in 1918, they took over Helsingfors and other towns in the south of Finland. Civil war followed, food was scarce and Axel Ahlfors was taken prisoner by the Reds. However, with German support for the White army, the Reds were quickly defeated and Axel Ahlfors was released. Lars Ahlfors was educated at a private Swedish language high school, the Nya Svenska Samskolan. He describes his early years in [3]:-

As a child I was fascinated by mathematics without understanding what it was about, but I was by no means a child prodigy. As a matter of fact I had no access to mathematical literature except in the highest grades. Having seen many prodigies spoilt by ambitious parents, I can only be thankful to my father for his restraint. The high school curriculum did not include any calculus, but I finally managed to learn some on my own, thanks to clandestine visits to my father's engineering library.

Ahlfors had little interest outside his school work during these years. He hated sports and was only happy on school days, hating Sundays and school vacations. His greatest joy was doing his mathematics homework making him rather different from the typical school student! Among his favourite school subjects we should mention languages and among those he detested we should mention history. He said [6]:-

Why memorize the years associated with various events and people? One might just as well memorize telephone numbers. It didn't make any sense to me, and it seemed rather silly. My history teacher was not very fond of me.

At first his father wanted him to become an engineer but, realising that his son was passionate about mathematics and unable to master anything mechanical, when Ahlfors was about 15 years old his father told him that he would become a professor of mathematics. He entered Helsingfors University at the age of 17 in 1924, and there he was taught by Ernst Lindelöf and Rolf Nevanlinna [6]:-

I took advanced calculus as a freshman, which I was not supposed to do. On the first day of school, when I went to Nevanlinna to ask permission to take his course, he asked me, "Do you know any calculus?" "Oh, yes," I said, "I have studied calculus." I didn't tell him it was all on my own. Lindelöf did not like my jumping ahead, and he made me take all of the tests from the prerequisite courses that I had not attended. I liked advanced calculus; I was very interested in my work.

He graduated from Helsingfors with a Candidate's degree in 1928. Nevanlinna replaced Hermann Weyl in Zürich for the session 1928/29 while Weyl was on a research visit to the United States. Lindelöf told Ahlfors that he should go to Zürich with Nevanlinna and he approached Ahlfor's father persuading him that his son would benefit greatly from the experience. In Zürich Nevanlinna lectured on a different level to what he had done in Finland. Ahlfors said [6]:-

Zürich was my first encounter with mathematics as it should be taught. It was contemporary mathematics, not the history of mathematics as our courses usually were. In Zürich, Nevanlinna could talk on the research level, which made a completely different impression on me. I came to understand what mathematics was about, and that I was supposed to do mathematics, not just learn it. This had not been clear to me before.

Nevanlinna talked about Denjoy's conjecture, made 21 years earlier, on the number of asymptotic values of an entire function. Ahlfors suddenly became known internationally when he solved this conjecture. He modestly writes in [3]:-

I had the incredible luck of hitting upon a new approach, based on conformal mappings, which, with very considerable help from Nevanlinna and Pólya, led to a proof of the full conjecture.

He published his breakthrough in two papers, both appearing in 1929: Sur le nombre des valeurs asymptotique d'une fonction entire d'ordre fini Ⓣ and Über die asymptotischen Werte der ganzen Funktionen endlicher Ordnung Ⓣ. After Zürich, Ahlfors went to Paris with Nevanlinna for three months before returning to Finland. In Paris he met Arnaud Denjoy who told him that 21 had now become his favourite number since his conjecture was solved by a 21 year old mathematician 21 years after he made it. Back in Finland he was appointed lecturer in mathematics at the Swedish language Abo Akademi in Turku. He presented his doctoral thesis in 1930, in which he gave his proof of Denjoy's conjecture. This states that if an entire function has $p$ distinct asymptotic values, then its order is ≥ ½$p$. Denjoy had proved a special case of the conjecture and Torsten Carleman had proved a weaker version of the conjecture in 1921. Mikhail Sodin sketches proofs of the work of Denjoy, Carleman, and Ahlfors in [30] where he also mentions later developments.

In the year 1930-32 Ahlfors made a number of visits to Paris, supported by a fellowship from the Rockefeller Foundation, and to other European centres. In 1933 he took up the position of adjunct professor at the University of Helsinki. Also in 1933 he married Erna Martha Liesbeth Lehnert who had been born in Vienna but had moved with her parents first to Sweden and then to Finland. Lars and Erna Ahlfors had three daughters: Cynthia Mary, Vanessa Elisabeth, and Caroline Gertrud. Their son Christopher, died while still an infant.

In 1935, Constantin Carathéodory, whom Ahlfors had met in Munich during his travels, recommended him for a post at Harvard in the United States. Ahlfors agreed to a three year trial period. This seems a good place to quote briefly from [13] about his early mathematical contributions:-

Function theory already had a long tradition in Finland and Ahlfors began his career just after the first waves from the theory developed by his teacher, Rolf Nevanlinna, had crested. ... The main direction of Ahlfors's work was to return to the foundations of the subject, and view it from a more geometric (and differential geometric) point of view. This has provided the basis of almost all subsequent generalizations. We also find here the genesis of later work in quasiconformal mappings, conformal metrics and extremal length.

In 1936 he was one of the first two recipients of a Fields Medal at the International Congress in Oslo. He wrote [3]:-

I was in for the surprise of my life when in 1936, at the International Congress in Oslo, I was told only hours before the ceremony that I was to receive one of the first two Fields medals ever awarded. The prestige was perhaps not yet the same as it is now, but in any case I felt singled out and greatly honoured. The citation by Carathéodory mentions explicitly my paper "Zur Theorie der Überlagerungsflächen," which threw some new light on Nevanlinna's theory of meromorphic functions. The award contributed in great measure to the confidence I felt in my work.

It was in this paper, which was a major factor in his receiving the Fields Medal, that the term 'quasiconformal mapping' appears for the first time. Carathéodory said that this paper opened a completely new chapter in analysis, one that could be called "metric topology." Ahlfors, in a commentary on this paper in [3] wrote:-

Little did I know at the time what an important role quasiconformal mappings would come to play in my own work.

In 1938 Ahlfors was offered a chair in mathematics at the University of Helsinki and, being rather homesick, he accepted this rather than remain permanently at Harvard. However a difficult time was approaching with World War II about to begin. The war led to severe problems in Finland and the universities were closed. Ahlfors was unfit for military service so, as he states in [3]:-

Paradoxically I was myself able to do a lot of work during the war, although without the benefit of accessible libraries.

Ahlfors' family was evacuated to Sweden during the war where they lived with relatives. Ahlfors remained in Helsinki where the university was closed because there were too few male students. This meant that despite the lack of library resources, he was able to concentrate on research, much of which was carried out in air raid shelters. After the winter war, his family returned to Helsinki but after Germany attacked the Soviet Union in 1941 Finland supported Germany. At first the Soviet Union was too occupied trying to stop the German advance to pay much attention to Finland but once they had turned the tide against the Germans they attacked Finland. When Ahlfors was offered a chair in Zürich in 1944 it seemed a good chance to improve the difficult position that he and his family were in. At this stage Ahlfors' health was poor and he had an irregular heartbeat so he was allowed to go with his family in Sweden, where Arne Beurling gave them a great deal of help and friendship [6]:-

When I was able to leave Finland and go to Sweden, I was not allowed to take more than 10 crowns with me. So what did I do? I smuggled out my Fields Medal, and I pawned it! I'm sure it is the only Fields Medal that has been in a pawn shop. As soon as I got a little money some people in Sweden helped me retrieve it.

However the war made the trip to Switzerland to take up his position in Zürich close to impossible. A flight from Stockholm to Prestwick in Scotland was arranged on a moonless night and, in March 1945, they made the trip [3]:-

The plane was a reconditioned Flying Fortress, with perhaps a dozen passengers. It was not pressurised, and breathing was accomplished by individual oxygen masks. Life jackets were worn by all. Our children, ages 5 and 6, were quite capable of understanding the implications of danger.

From Glasgow they travelled by train to London, then they made the difficult journey across the Channel to Dieppe, across France via Paris to Switzerland. He writes [3]:-

My first disappointment came when I learned that I would be responsible for Descriptive Geometry, a subject that for some reason had survived in the Swiss high school and undergraduate curriculum. My second shock was that it was to be taught from 7 to 9 o'clock in the morning. Nevertheless, I slowly adjusted to my work, which even included some serious, although not very advanced, mathematics. Professor Fueter and his colleague Professor Finsler were getting on in years, and it became clear that the reason for inviting me was that no competent native successor was in sight. I took over a class of students in their formative years, and I am happy to say that many have remained my friends and are now important mathematicians in their own right. I cannot honestly say that I was happy in Zürich. The post-war era was not a good time for a stranger to take root in Switzerland. ... My wife and I did not feel welcome outside the circle of our immediate colleagues.

An offer from Harvard in 1946 was therefore gladly accepted and, on this occasion, he remained there, retiring in 1977. Two years after taking up the appointment at Harvard, Ahlfors was elected Chairman of the Mathematics Department. In 1964 he was named William Caspar Graustein Professor of Mathematics. James Jenkins writes about Ahlfors' early years at Harvard in [10]:-

Ahlfors was very conscientious in meeting with his graduate students, but his role was largely reactive rather than presenting suggestions or ideas (at least in my case). In the spring of 1947 Ahlfors gave a course on the method of the extremal metric, which was then in its infancy. In the year 1947-48 he gave a course on the calculus of variations. In the year 1948-49 he gave a course on Riemann surfaces, directed largely to a generalization of his work on Schwarz's lemma to finite-bordered Riemann surfaces. Ahlfors did not conduct a regular seminar at any time during my stay. One year he gave a series of talks on Teichmüller's papers, probably with a view to preparing himself for his work on quasiconformal mappings.

An excellent overview of Ahlfors' mathematics is given in [17] where his contributions are given under the headings: Conformal geometry; Kleinian groups; and Quasiconformal mappings. These sections are reproduced in [16].

His books are of lasting importance. These are Complex analysis (1953), Riemann surfaces (with Leo Sario) (1960), Lectures on quasi-conformal mappings (1966), Conformal invariants: topics in geometric function theory (1973) and Möbius transformations in several dimensions (1981).

We give slightly modified (or translated) short extracts from reviews of these books at THIS LINK.

Allow me [EFR] a personal note on Ahlfors' Complex analysis. This was the text recommended to me by Edward Copson who taught me complex analysis and it is indeed a tribute to Ahlfors that Copson, who had himself written a superb book on complex analysis, should recommend Ahlfors' book rather than his own. I found Ahlfors' Complex analysis beautifully written, an example of the very highest quality in mathematical texts, combining clarity with an excitement for the topic.

We give the reader the chance to read the Preface to the first edition at THIS LINK and brief extracts from the text at THIS LINK.

Ahlfors received many honours for his outstanding contributions to mathematics. The award of the first Fields medal, mentioned above, must rank as the most important but other great honours were the award of Finland's International Prize in 1968 and the Wolf Prize in Mathematics in 1981:-

... for seminal discoveries and the creation of powerful new methods in geometric function theory.

The citation for the Wolf Prize gave a fine overview of his achievements:-

For over half a century the theory of functions of a complex variable was guided by the thought and work of Professor (Emeritus) Lars Ahlfors. His achievements include the proof of the Denjoy conjecture, the geometric derivation of the Nevanlinna theory, an important generalization of the Schwarz lemma, the development (with Beurling) of the method of extremal length, and numerous decisive results in the theories of Riemann surfaces, quasi-conformal mappings and Teichmuller spaces. Ahlfors celebrated finiteness theorem for Kleinian groups, and his work on the limit set, revitalized an important area of research. He is now doing pioneering work on quasiconformal deformations in higher dimensions. Ahlfors' influence was pervasive and beneficial. His methods combine deep geometric insight with subtle analytic skill; he presents them with utter clarity and simplicity. Time and again he attacked and solved the central problem in a discipline. Time and again other mathematicians were inspired by work he did many years earlier. Every complex analyst working today is, in some sense, his pupil.

Other honours he received included honorary degrees from Boston University (1953), the Abo Adademi (1970), the University of Zürich (1977), and the University of London (1978). He was elected to the Societas Scientiarum Fennica, the Finnish Academy of Science, the National Academy of Sciences, the Royal Swedish Academy of Sciences, and the Royal Danish Academy of Sciences.

Let us end this biography by quoting Troels Jorgensen [10]:-

It was my good fortune to know Lars Ahlfors for a quarter of a century. He was an inspiring mathematician and a very special person. His intuition for and joy in mathematics, combined with a great capacity for work, made him a harmonious craftsman through a career that spanned over sixty years. He savored the recognition and awards he received yet was very modest about his accomplishments. At a banquet celebrating his seventieth birthday Ahlfors said, "I liked to go fishing where the fish are, rather than trying exclusively for the big one."