Lars Valerian Ahlfors (1907–1996) Raoul Bott; Clifford Earle; Dennis Hejhal; James Jenkins; Troels Jorgensen; Steven G. Krantz, Editor; Albert Marden; Robert Osserman Lars Valerian Ahlfors was born in Helsingfors, the first two Fields Medals ever Finland, on April 18, 1907. He studied mathemat- awarded. The prestige was perhaps not ics with Ernst Lindelöf at Helsingfors University yet the same as it is now, but in any case and earned the doctorate in 1928. Ahlfors accepted I felt singled out and greatly honored. a position at Harvard University and moved to The citation by Carathéodory mentions Cambridge in 1935. He spent most of his profes- explicitly my paper “Zur Theorie der sional life at Harvard, with a period in Europe (pri- Überlagerungsflächen,” which threw marily at the University of Helsinki) from 1938 to some new light on Nevanlinna’s theory 1946. of meromorphic functions. The award Lars Ahlfors retired from his position at Harvard contributed in great measure to the in 1977. He continued to study mathematics until confidence I felt in my work. the end of his life on October 11, 1996. During World War II Ahlfors was forced to pawn Lars Valerian Ahlfors was arguably the preemi- his Fields Medal in order to obtain the money to nent complex function theorist of the twentieth secure safe passage from Finland to Zurich at the century. With a career spanning more than sixty end of the war. Ahlfors’s remarks on the incident years, Ahlfors made decisive contributions to areas were: ranging from meromorphic curves to value distri- I can give one very definite benefit [of bution theory, Riemann surfaces, conformal geom- winning a Fields Medal]. When I was etry, extremal length, quasiconformal mappings, able to leave Finland to go to Sweden, and Kleinian groups ([7] serves as a map of I was not allowed to take more than 10 Ahlfors’s contributions to the subject). Ahlfors crowns with me, and I wanted to take was both role model and mentor to his graduate a train to where my wife was waiting for students and to the many mathematicians around me. So what did I do? I smuggled out the world who learned from his example. He is re- my Fields Medal, and I pawned it in the membered warmly, both as a mathematician and pawn shop and got enough money. I as a man. had no other way, no other way at all. Ahlfors’s life contained many firsts. In 1936 he And I’m sure that it’s the only Fields received, along with Jesse Douglas, one of the first Medal that has been in a pawn shop… two Fields Medals. Ahlfors’s remarks concerning As soon as I got a little money, some that occasion, taken from his Collected Papers [4], people in Switzerland helped me re- typify his honesty and humility: trieve it. I was in for the surprise of my life when Another remarkable achievement of Lars Ahlfors in 1936, at the International Congress is that he was invited to be a plenary speaker at in Oslo, I was told only hours before the the International Congress of Mathematicians on ceremony that I was to receive one of three different occasions. No other mathematician in the past seventy years has achieved such a dis- Steven G. Krantz is professor of mathematics at Wash- tinction. His plenary addresses were about three ington University in St. Louis. His e-mail address is completely distinct areas in which he had made [email protected]. seminal contributions. 248 NOTICES OF THE AMS VOLUME 45, NUMBER 2 Ahlfors’s text Complex Analysis [1], which was first published in 1953 and has gone through three editions, has been the definitive text in complex function theory for the past forty-five years; there are few other texts in modern mathematics that have played such a dominant role. At a conference held in Storrs, Connecticut, to commemorate Ahlfors’s seventy-fifth birthday, Ahlfors remarked that “Retirement is wonderful. I can’t perish anymore, so I don’t have to publish.” The upshot of his remarks was that he could de- vote himself full time to understanding the new de- velopments in his field without the pressure of writ- ing them up. He seemed to think of this as a mathematician’s paradise. The NSF program offi- cer in complex analysis told me about fifteen years ago that Lars Ahlfors’s entire NSF proposal con- sisted of a single sentence: “I will continue to study the work of Thurston.” Ahlfors was seventy-five years old at the time, and he was devoting himself to learning the newest ideas in the subject! In what follows, seven mathematicians relate Early photograph of Lars Ahlfors, who is reputed to have worn their personal reminiscences of Lars Ahlfors. a beret until the day he died. —Steven G. Krantz, Editor reactive rather than presenting suggestions or ideas (at least in my case). James Jenkins In the spring of 1947 Ahlfors gave a course on Lars Ahlfors returned to Harvard in the fall of the method of the extremal metric, which was then 1946, the same time that I arrived to begin my grad- in its infancy. In the year 1947–48 he gave a course uate work there. David Widder was at that time the on the calculus of variations. In the year 1948–49 chairman of the mathematics department, and he he gave a course on Riemann surfaces, directed was recruiting people to work with Ahlfors. Since largely to a generalization of his work on Schwarz’s I had some background in function theory from my lemma to finite-bordered Riemann surfaces. work at the University of Toronto, he approached Ahlfors did not conduct a regular seminar at any me in this context (not that I was reluctant). time during my stay. One year he gave a series of In those days there was no mathematics build- talks on Teichmüller’s papers, probably with a ing at Harvard; indeed, the faculty members did view to preparing himself for his work on quasi- not have regular offices. Some of the senior peo- conformal mappings. ple had faculty studies in Widener Library, and Ahlfors shared one with Julian Coolidge, profes- Ahlfors had a remarkable command of English sor emeritus and a geometer of the old school. I for one for whom it was not the native language. had my first few meetings with Ahlfors there. On However, English is tricky. One semester, when he one occasion he had the duty of testing my profi- was teaching a course in elementary calculus, he ciency in foreign languages with Coolidge present. asked students on the examination to find the In a passage in French making a mildly pejorative three first derivatives of an explicit function of one statement about “géomètres”, I translated the word real variable (creating considerable havoc). Also, as “geometers”, and Ahlfors was quick to point out under stress one tends to revert: at a tense moment that it was used in the more general sense of in a lecture “j ” came out as “yot ”. “mathematicians”. There were occasionally people who came to Every faculty member at Harvard had an affili- work with Ahlfors, for example, a Swiss by the ation with one of the residential houses (which had name of Haefeli. In the year 1948–49 Arne Beurl- a certain analogy with colleges at the English uni- ing came to Harvard as visiting professor. He gave versities). Ahlfors’s affiliation was with Dunster two rather well-known one-semester courses on his House, and he had a sort of office there. We met brands of functional analysis and harmonic analy- there a few times, but afterwards always at his sis. He and Ahlfors were working on a book on ex- home. Ahlfors was very conscientious in meeting tremal length and in fact had a rough draft of with his graduate students, but his role was largely some of the material. No book ever appeared, but James Jenkins is professor of mathematics emeritus at some of the contents showed up in Ahlfors’s book Washington University in St. Louis. Conformal Invariants [2]. FEBRUARY 1998 NOTICES OF THE AMS 249 tendance would be ideally suited to work on. After Robert Osserman we returned to Harvard Ahlfors suggested that I When I reached the point in my graduate studies work on one of the two problems, and he had where it was time to select a Ph.D. advisor, I nat- some ideas about how to proceed. It turned out that urally consulted fellow students who were more the particular approach he suggested did not work, advanced. One of those was a young student from but I was able to solve the problem by a different Finland named Vidar Wolontis. He may not have method, and it became part of my Ph.D. thesis. been totally objective in his recommendation that A high point of the Princeton conference was a I study with his countryman Ahlfors, but he did cameo appearance by Albert Einstein, who ex- mention one reason that I still remember. He said, pressed his own indebtedness to Riemann and the “Look at Ahlfors’s bibliography. It is really short. importance to him in particular of Riemann’s in- Only a relatively small number of papers, but every troduction of the idea of a Riemann surface. What one significant.” I later learned of Gauss’s famed Einstein was referring to was of course the idea of motto: “Pauca sed matura” or “Few, but ripe.” Per- a Riemannian manifold, and it was a legitimate con- haps Ahlfors’s unstated motto was “Pauca sed fusion to think that a Riemann surface might be egregia”—“Few, but outstanding.” That was an idea just a two-(real-like)-dimensional Riemannian man- that appealed to me greatly, and I did become ifold.