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Lars Ahlfors Entered the University of Helsinki, Where His Teachers Were Two Internationally Known Math- Ematicians, Ernst Lindelöf and Rolf Nevanlinna

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Lars Ahlfors Entered the University of Helsinki, Where His Teachers Were Two Internationally Known Math- Ematicians, Ernst Lindelöf and Rolf Nevanlinna NATIONAL ACADEMY OF SCIENCES LARS VALERIAN AHLFORS 1907–1996 A Biographical Memoir by FREDERICK GEHRING Any opinions expressed in this memoir are those of the author and do not necessarily reflect the views of the National Academy of Sciences. Biographical Memoirs, VOLUME 87 PUBLISHED 2005 BY THE NATIONAL ACADEMIES PRESS WASHINGTON, D.C. LARS VALERIAN AHLFORS April 18, 1907–October 11, 1996 BY FREDERICK GEHRING PERSONAL AND PROFESSIONAL HISTORY ARS AHLFORS WAS BORN in Helsinki, Finland, on April 18, L 1907. His father, Axel Ahlfors, was a professor of mechanical engineering at the Institute of Technology in Helsinki. His mother, Sievä Helander, died at Lars’s birth. As a newborn Lars was sent to the Åland Islands to be taken care of by two aunts. He returned to his father’s home in Helsinki by the age of three. At the time of Lars’s early childhood, Finland was under Russian sovereignty but with a certain degree of autonomy, and civil servants, including professors, were able to enjoy a fairly high standard of living. Unfortunately, all this changed radically during World War I, the Russian Revolution, and the Finnish civil war that followed. There was very little food in 1918, and Lars’s father was briefly imprisoned by the Red Guard. For historical reasons the inhabitants of Finland are divided into those who have Finnish or Swedish as their mother tongue. The Ahlfors family was Swedish speaking, so Lars attended a private school where all classes were taught in Swedish. He commented that the teaching of math- ematics was mediocre, but credited the school with helping 3 4 BIOGRAPHICAL MEMOIRS him become almost fluent in Finnish, German, and English, and less so in French. A sister of Lars was two years ahead of him in school. Hence Lars was able to learn high school mathematics by doing her homework and by means of clandestine visits to his father’s engineering library. Indeed, the teachers eventually relied on Lars to provide problems for the class. Lars spent a summer vacation as a young student with a Finnish-speaking family to improve his knowledge of Finnish. He had hoped to pay for his keep that summer by teaching mathematics, but having found no takers, he earned his board by giving lessons to the children of the family in the German language and playing the cello. In 1924 Lars Ahlfors entered the University of Helsinki, where his teachers were two internationally known math- ematicians, Ernst Lindelöf and Rolf Nevanlinna. At that time the university was still run on the system of one professor for each subject. Lindelöf was the professor for mathematics. There were no graduate courses then, and all advanced reading was done under the supervision of Lindelöf. Lars remembered well the Saturday mornings when he had to visit Lindelöf in his home at 8 a.m. to be praised or scolded— as the case may have been. In the spring of 1928 Lars completed all examinations for the master’s degree, and in the fall of the same year he accompanied Nevanlinna to Zürich. Nevanlinna had been invited to the Eidgenösische Technische Hochschule in Zürich for a year to fill Hermann Weyl’s chair while Weyl was on leave of absence. Lindelöf persuaded Lars’s father to let his son accompany Nevanlinna to Zürich. Later Lars emphasized the importance of this visit to Zürich: “I found myself suddenly transported from the periphery to the center of Europe.” Nevanlinna’s course of lectures at Zürich was Lars’s first LARS VALERIAN AHLFORS. 5 exposure to contemporary function theory, and he became “addicted” to complex analysis. Among other things, Nevanlinna introduced the class to a 21-year-old conjecture made by the French mathematician Denjoy on the asymptotic values of an entire function, namely that an entire function of order k can have at most 2k finite asymptotic values. Lars created a sensation in the mathematical world when he found a different way to approach this problem and obtained a beautiful proof of this conjecture by means of conformal mapping (1929). His name became known to all those work- ing in this area and, as he remarked later, “my future was made.” Lars returned to Finland and undertook his first teaching assignment as lecturer at Åbo Akademi, the Swedish- language university in Turku. At the same time he began work on his thesis, which he defended in the spring of 1930. During 1930-1932 Lars made several trips to continental Europe, including a longer stay in Paris supported by a fellowship from the Rockefeller Foundation. In 1933 he returned to Helsinki as an adjunct professor at the University of Helsinki. That same year he married Erna Lehnert, an Austrian who with her parents had first settled in Sweden and then in Finland. Half a century later, in the preface to his Collected Papers, Lars wrote, “This was the happiest and most important event in my life.” In 1935 Lars was first offered and then accepted a three- year appointment at Harvard University as a visiting lecturer. One year later, at the quadriennal International Congress of Mathematicians in Oslo, he was awarded a Fields Medal, the equivalent for mathematicians of a Nobel Prize. During the 1924 International Congress of Mathematicians in Toronto, the president of the congress, Professor J. C. Fields of Canada, had proposed that two gold medals be awarded at each International Congress of Mathematicians 6 BIOGRAPHICAL MEMOIRS “for outstanding achievements in mathematics.” The winners were to be selected by an international jury. The 1932 Congress in Zürich had approved Fields’s proposal and named the medals after Fields, who had died just before the 1932 Congress. The first two Fields Medals were then awarded to Lars Ahlfors and Jesse Douglas of the United States in 1936. That he was to be awarded a Fields Medal came as a com- plete surprise for Lars. Indeed, he was told officially that he was to receive this honor only hours before the formal award ceremony. Lars Ahlfors’s appointment at Harvard was for three years. He and his wife had found life in Cambridge very reward- ing and they had to decide in 1938 whether to stay at Harvard or return to Finland, where he had been offered a professor- ship at the University of Helsinki. In the end, patriotic feelings and loyalty to his teachers drew them back to Helsinki, where they had a happy year. Unfortunately, World War II broke out in 1939, and Lars’s wife and two children were evacuated and found refuge with relatives in Sweden. Helsinki was bombarded, the university was closed for lack of male students, but other- wise life went on. Lars was never called to military duty because of an earlier physical condition, and it is quite ironic that one of his best papers was written while he was in an air raid shelter. Soon after the winter war the Ahlfors family was able to return to Helsinki and resume a seemingly normal life. How- ever, politics in Finland took an unfortunate turn, and, when Hitler attacked the Soviet Union in 1941, Finland was his ally. When the Russians were able to repulse the Germans, they intensified the war in Finland with foreseeable results. The Finnish-Russian war ended with a separate armistice in September 1944, and Finland was forced to expel the German troops that had been stationed there. LARS VALERIAN AHLFORS. 7 The war continued off and on until 1944, and Lars felt he could not remain in Finland without sacrificing his research. He therefore accepted an invitation to the University of Zürich and, after a very difficult journey, joined their faculty in the spring of 1945. He and his wife found the postwar era a difficult time for strangers to take root in Switzerland. Hence in 1946 Lars was delighted to accept an invitation to return to Harvard, where he remained for the rest of his career. We consider next excerpted descriptions of three aspects of Ahlfors’s research by Robert Osserman, Irwin Kra, and the present author, respectively, which appear in Volume 45 of the Notices of the American Mathematical Society. CONFORMAL GEOMETRY BY ROBERT OSSERMAN There are two directions in which one can pursue the relations between Riemann surfaces and Riemannian mani- folds. First, if a two-dimensional Riemannian manifold is given, then not only lengths but also angles are well defined, so that it inherits a conformal structure. Furthermore, there always exist local isothermal coordinates, which are local conformal maps from the plane into the surface. The set of all such local maps forms a complex structure for the mani- folds, which can then be thought of as a Riemann surface. One then has all of complex function theory to bring to bear in studying the geometry of the surface. The most notable successes of this approach have been in the study of minimal surfaces, as exemplified in the contributions to that subject made by some of the leading function theorists of the nineteenth century: Riemann, Weierstrass, and Schwarz. In the other direction, given a Riemann surface one can consider those metrics on the surface that induce the given 8 BIOGRAPHICAL MEMOIRS conformal structure. By the Koebe uniformization theorem, such metrics always exist. In fact, for “classical Riemann surfaces” of the sort originally considered by Riemann, which are branched covering surfaces of the plane, there is the natural Euclidean metric obtained by pulling back the standard metric on the plane under the projection map. One can also consider the Riemann surface to lie over the Riemann sphere and to lift the spherical metric to the surface.
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