Died: 20 November 1986 in Princeton, New Jersey, USA

Arne Beurling studied at Uppsala and undertook research with Anders Wiman as his thesis advisor. He was also strongly influenced by another of his teachers Erik Holmgren. He obtained a doctorate there in 1933 with his thesis Études sur un problème de majoration. However, parts of the thesis were written in 1929, in particular his the proof of the Denjoy conjecture concerning asymptotic values of an entire function. However, he was not the first to publish a proof of this conjecture since he took a vacation with his father (they went crocodile hunting!) and Lars Ahlfors published his proof of the conjecture in 1929. However, Beurling's thesis was published in 1933 and is described by Ahlfors and Carleson as follows [2]:-

It was not a mere collection of interesting and important results, but also a whole programme for research in function theory in the broadest sense. As such it has been one of the most influential mathematical publications in recent times. ... Beurling's leading idea was to find new estimates for the harmonic measure by introducing concepts, and problems, which are inherently invariant under conformal mapping. The novelty in his approach was to apply the majorization to entities, mostly of a geometric character, which are not by themselves invariant, but whose extreme values, in one sense or another, possess this property. The method may have been used before, but not in this systematic manner.

He taught at Uppsala from 1932 to 1954 becoming professor of mathematics there in 1937. During the session 1948-49 Beurling was a visiting professor at Harvard in the United States. Then, in 1954 he emigrated to the United States and became a professor at the Institute for Advanced Study at Princeton. He retired in 1973 and was named professor emeritus.

During World War II, Beurling worked on cracking the German codes. Many other top mathematicians did similar work but details are still hard to obtain. Kjellberg writes in [4]:-

Beurling was one of the most charming persons you could meet. He had a very strong feeling for justice and fair play. During world war II he decoded in two weeks (in summer 1940) the German G-Schreiber message code, so all German troop movements were known to the Swedish command.

Ulfving writes [6]:-

It is now known that Professor Arne Beurling was the man behind the breaking of the German Geheimschreiber. David Kahn writes in "The Codebreakers":

Quite possibly the finest feat of cryptoanalysis performed during the Second World War was Arne Beurling's solution of the secret of the Geheimschreiber.

Arne Beurling's greatness is given by the fact he had at his disposal only the teleprinter tapes with the cipher text. He had no access to any machine. Everything had to be reconstructed, something which was done in a remarkably short time. It is known that he based his analysis on only 24 hours of traffic intercepted on 25 may 1940. A quick analysis showed that the first assumptions probably were correct. A check was made with the traffic intercepted on 27 May. Two weeks later the construction principles for the cipher machine were solved. On the other hand it is not known how he set about it. That secret Arne Beurling took with him to the grave.

Beurling worked on the theory of generalized functions, differential equations, harmonic analysis, Dirichlet series and potential theory. The concepts of energy and the Dirichlet integral took Beurling to a global axiomatic theory called the theory of Dirichlet spaces for complex functions. Among his papers let us mention Exceptional sets (1935), Sur les intégrales de Fourier absolument convergentes et leur application à une transformation fonctionelle (1938), Sur les spectres des fonctions (1949), (with Ahlfors) On the boundary correspondance under quasi-conformal map (1956), and (with Paul Malliavin) On the closure of characters and the zeros of entire functions (1967). Describing their work on this last mentioned paper, Malliavin writes [3]:-

We devoted half of the academic year 1960-1961 to this problem; very often I stayed at Beurling's house for a full night of common work. I was quite welcome there by Mrs Beurling, a former distinguished PhD student at Uppsala University, where she has been president of the association of graduate students. Mrs Beurling worked in a Chemistry lab at Princeton University. Although very occupied by her scientific work, she was kind enough to prepare a supper for our half night break. We obtained in June 1961 all our results. I presented mimeographed notes of their proofs at the summer school in Harmonic Analysis, organized by Peter Lax at Stanford University in August 1961. Nevertheless Beurling was not "esthetically" satisfied with these proofs. It took us the fall quarter 1966 at the Institute to write the final version which appeared in Acta 1967.

Lennart Carleson was a research student of Beurling and completed his doctoral thesis in 1950. He explained the way that Beurling worked [3]:-

We learned much on mathematical research through Beurling's seminars. They took place every second Tuesday, 6-8 pm, when Beurling invariably would talk about his own work (he did not read much). The department was at Trädgardsgatan 18 and he would usually work at home in number 12 and at night. One should not believe that it all came by divine inspiration. His neighbours would tell how he walked back and forth (the worst being that he sometimes stopped!).

Beurling received many honours for his outstanding contributions. He was elected to membership of the Royal Swedish Academy of Sciences, the Finnish Academy of Sciences, the Royal Physiographical Society in Lund, Sweden, the Danish Academy of Sciences, and the American Academy of Arts and Sciences. The Swedish Mathematical Society also recognised his achievements by electing him to honorary membership. Among the prizes he was awarded we mention in particular the Swedish Academy of Sciences Prize in 1937 and again in 1946, the Celsius Gold Medal in 1961 (he was the first recipient), and the University of Yeshiva Science Award in 1963. In 1976-77 the Mittag-Leffler Institute in Stockholm held a "Beurling Year". It is interesting to note that Beurling had been offered the directorship of the Mittag-Leffler Institute before emigrating to the United States but had turned the offer down.

The authors of [2] write:-

Arne Beurling was a highly creative mathematician whose legacy will influence future mathematicians for many years to come, maybe even for generations. Anybody who was close to him was influenced by his strong personality and by his intense commitment to mathematics. He published very selectively and only when all details were resolved, and a sizable part of his work has never appeared in print. There are plans to publish his collected works in the near future, and they will include much that has not been previously available to the mathematical public. Beurling's personal friends and students will never forget his unquestioning loyalty and boundless generosity. His readiness to share his ideas was unselfish in the extreme.

Paul Malliavin was strongly influenced by Beurling and his work. He gave a lecture Arne Beurling - a visionary mathematician (in [3]) which he ended as follows:-

In my youth Beurling appeared to me more as a mathematician working on hard concrete problems than an abstract theory builder. Sixty years later Beurling appears now to me more as the key initiator of important theories than a problem solver. How can we explain this paradox? After having got sharp results, Beurling waited for their publication until he reached a proof which quoting his own words must be "elementary and transparent". This, sometimes strenuous, search for beauty in the proofs explains why, starting from concrete problems, Beurling reached basic general principles of universal applicability. The far reaching consequences of this Beurling's quest for Beauty illustrate magnificently the Unity of Mathematics and, by consequence, its transcendental Truth.

It seems appropriate to end this short biography with words written by Ahlfors about Beurling in The collected works of Arne Beurling (Birkhäuser, Boston, 1989):-

Articles:

1. L Ahlfors, The story of a friendship: Recollections of Arne Beurling, The Mathematical Intelligencer 15 (3) (1993), 25-27.
2. L Ahlfors and L Carleson, Arne Beurling in memoriam, Acta Math. 161 (1-2) (1988), 1-9.
3. Jubileumsskrift Arne Beurling 100 ar, U.U.D.M. Report 2007:34, Department of Mathematics Uppsala University
   http://www.math.uu.se/research/pub/Beurling100.pd
4. B Kjellberg, Memories of Arne Beurling, February 3, 1905 - November 20, 1986, The Mathematical Intelligencer 15 (3) (1993), 28-31.
5. J W Neuberger, Beurling's analyticity theorem, The Mathematical Intelligencer 15 (3) (1993), 34-38.
6. L Ulfving, The Geheimschreiber Secret : Arne Beurling and the success of Swedish signals intelligence.
   http://citeseer.ist.psu.edu/100764.html
7. J Wermer, Recollections of Arne Beurling, The Mathematical Intelligencer 15 (3) (1993), 32-33.