# Lars Valerian Ahlfors

### Born: 18 April 1907 in Helsingfors, Finland, Russian Empire (now Helsinki, Finland)

Died: 11 October 1996 in Pittsfield, Massachusetts, USA

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**Lars Ahlfors**described his ancestors in [6]:-

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.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.

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]:-

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]:-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.

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]:-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.

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]:-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.

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]:-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.

He published his breakthrough in two papers, both appearing in 1929: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.

*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:-

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

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:-I was in for the surprise of my life when in1936, 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.

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

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

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]:-When I was able to leave Finland and go to Sweden, I was not allowed to take more than10crowns 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.

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]:-The plane was a reconditioned Frying 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, ages5and6, were quite capable of understanding the implications of danger.

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]:-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 from7to9o'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 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].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 of1947Ahlfors gave a course on the method of the extremal metric, which was then in its infancy. In the year1947-48he gave a course on the calculus of variations. In the year1948-49he 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.

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:-

The citation for the Wolf Prize gave a fine overview of his achievements:-... for seminal discoveries and the creation of powerful new methods in geometric function theory.

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.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.

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."

**Article by:** *J J O'Connor* and *E F Robertson*

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**List of References** (31 books/articles)

**Mathematicians born in the same country**

**Additional Material in MacTutor**

- Reviews of Lars Ahlfors' books
- Preface to Lars Ahlfors' Complex Analysis
- Extracts from Lars Ahlfors'
*Complex analysis* - Obituary: New York Times

**Honours awarded to Lars Ahlfors**

(Click the link below for those honoured in this way)

1. | Speaker at International Congress | 1936 | |

2. | Fields Medal | 1936 | |

3. | SIAM John von Neumann Lecture | 1960 | |

4. | Speaker at International Congress | 1962 | |

5. | Speaker at International Congress | 1978 | |

6. | Wolf Prize | 1981 | |

7. | AMS Steele Prize | 1982 |

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