**Lennart Carleson**completed his secondary schooling in Karlstad, Sweden, graduating in 1945. He then entered Uppsala University, obtaining his first degree (Fil. kand.) in 1947, and his Master's Degree (Fil. lic.) in 1949. Carleson's research thesis was supervised by Arne Beurling and he was awarded his doctorate in 1950 for

*On a Class of Meromorphic Functions and Its Exceptional Sets*. He spoke of his supervisor in 1984 [1]:-

Following the award of his doctorate he was appointed as a lecturer in mathematics at Uppsala University.It was my great fortune to have been introduced to mathematics by Arne Beurling; the tradition he, T Carleman and Marcel Riesz initiated is very obviously responsible for the good standard of mathematics in our country. Personally I am very happy for this opportunity to express to Arne Beurling my gratitude for having guided me into a fruitful area of mathematics and for having given an example that only hard problems count.

Carleson spent session 1950-51 in the United States, undertaking post-doctoral work at Harvard University. There he was greatly influenced by A Zygmund and R Salem who were both at Harvard that year and, as we explain below, it was Zygmund's influence which set him on the path to proving his most famous result. Carleson returned to Sweden, taking up his lectureship at Uppsala University at the beginning of session 1951-52. In 1954 he was appointed to a professorship at the University of Stockholm but he returned to Uppsala in the following year, holding a chair of mathematics there until 1993. During this time he made a number of research visits to the United States, being a visiting research scientist at MIT during the autumn of 1957, spending session 1961-62 at the Institute for Advanced Studies, Princeton, being a guest professor at Stanford University in session 1965-66, and holding a similar position at MIT during 1974-75.

Among many important roles which Carleson has occupied, we should mention three in particular. First his very significant role as Director of the Mittag-Leffler Institute, Stockholm, from 1968 to 1984, during which time he built the Institute from a small base into one of the leading mathematical research institutes in the world. His other highly significant roles were that of editor of *Acta Mathematica* from 1956 to 1979, and as President of the International Mathematical Union from 1978 to 1982. In this last mentioned position, he worked tirelessly to have the People's Republic of China represented on the Union and was the main driving force behind the creation of the Nevanlinna Prize honoring the contributions of computer science to mathematics by rewarding young theoretical computer scientists.

Carleson's mathematical contributions have been far too many, and much too deep, to be described in any detail in a biography of this type. However we will try to give some idea of the importance of his contributions. We begin with a quote from Marcus du Sautoy who writes:-

A major problem solved by Carleson in 1962 was the famous 'corona problem' in the paperThe mark of a great mathematician is someone who not only cracks a big open problem that has defeated previous generations of mathematicians but who then goes on to create tools for future generations. During his career as a mathematician Carleson has been influential in several major areas of analysis and dynamical systems over nearly half a century of mathematical activity. Carleson's mathematics is characterized by a deep geometric insight combined with an amazing control of the branching complexities of the proofs. His contributions have provided future generations with tools to carry out a systematic study of analysis and dynamical systems.

*Interpolations by bounded analytic functions and the corona problem*. As so often in his work, not only did he solve the problem but in doing so he introduced what are today called 'Carleson measures' which went on to become a fundamental tool in complex analysis and harmonic analysis. In 1967 Hörmander introduced some ideas to simplify Carleson's proof and Carleson lectured on

*The corona theorem*to the Fifteenth Scandinavian Congress in Oslo in 1968. The conference Proceedings contains a complete proof by Carleson:-

In 1966 Carleson solved one of the outstanding problems of mathematics in his paperIn the paper ... a complete proof is given incorporating Hörmander's ideas. Moreover, the presentation is quite clear, so that the proof, while remaining non trivial, is now reasonably easy to read.

*On convergence and growth of partial sums of Fourier series*. Fourier, in 1807, had claimed that every function equals the sum of its Fourier series. Of course Fourier was thinking about 'well-behaved' functions so his initial claim has to be modified somewhat. A major research area throughout the 19

^{th}century concerned the convergence of Fourier series, and continuous functions whose Fourier series diverges at a dense set of points were constructed by du Bois-Reymond. In 1913 Luzin conjectured that if a function

*f*is square Lebesgue integrable then the Fourier series of

*f*converges pointwise to

*f*almost everywhere. Kolmogorov proved results in 1928 which seemed to suggest that Luzin's conjecture must be false but Carleson amazed the world of mathematics when he proved Luzin's long-standing conjecture in 1966. He explained in [4] how he was led to prove the theorem:-

Carleson lectured on his spectacular result at the International Congress of Mathematicians at Moscow in 1966 when he gave the address... the problem of course presents itself already when you are a student and I was thinking about the problem on and off, but the situation was more interesting than that. The great authority in those days was Zygmund and he was completely convinced that what one should produce was not a proof but a counter-example. When I was a young student in the United States, I met Zygmund and I had an idea how to produce some very complicated functions for a counter-example and Zygmund encouraged me very much to do so. I was thinking about it for about15years on and off, on how to make these counter-examples work and the interesting thing that happened was that I realised why there should be a counter-example and how you should produce it. I thought I really understood what was the background and then to my amazement I could prove that this "correct" counter-example couldn't exist and I suddenly realised that what you should try to do was the opposite, you should try to prove what was not fashionable, namely to prove convergence. The most important aspect in solving a mathematical problem is the conviction of what is the true result. Then it took2or3years using the techniques that had been developed during the past20years or so.

*Convergence and summability of Fourier series*. He began his address with the words:-

In 1967 he published the bookI do not intend to give in this lecture any survey of the very large field covered by the title. I rather want to present my personal interests which are concentrated on the almost everywhere behaviour of the partial sums. Also the subject of summability will only be touched upon.

*Selected problems on exceptional sets*which Ahlfors describes as follows:-

Carleson received the Wolf Prize 1992 together with John G Thompson. The authors of [6] write:-The author announces that he had originally prepared a survey of the theory of small sets in1959. At that time several books covering parts of the subject were published, and he found that a survey was less desirable. In1961he collected those parts that seemed to contain new or less known aspects, methods of proof or results. Some later results were added, and the author states that the selection reflects his own personal tastes. Readers will agree that he has successfully eliminated the dull parts. A substantial portion of the results are original and once again bear witness to the author's extraordinary technical skill.

In 2006 Carleson received his greatest honour when he received the Abel Prize:-The citation emphasizes not only Carleson's fundamental scientific contributions, the best known of which perhaps are the proof of Luzin's conjecture on the convergence of Fourier series, the solutions of the corona problem and the interpolation problem for bounded analytic functions, the solution of the extension problem for quasiconformal mappings in higher dimensions, and the proof of the existence of 'strange attractors' in the Hénon family of planar maps, but also his outstanding role as scientific leader and advisor.

The citation by the Abel Committee states:-... for his profound and seminal contributions to harmonic analysis and the theory of smooth dynamical systems.

Carleson is always far ahead of the crowd. He concentrates on only the most difficult and deep problems. Once these are solved, he lets others invade the kingdom he has discovered, and he moves on to even wilder and more remote domains of Science. ...

*Carleson's work has forever altered our view of analysis. Not only did he prove extremely hard theorems, but the methods he introduced to prove them have turned out to be as important as the theorems themselves. His unique style is characterized by geometric insight combined with amazing control of the branching complexities of the proofs*.

On 23 May 2006 he received the Prize from Queen Sonja. He said in reply:-

Peter W Jones in [5] gives this summary:-Carl Friedrich Gauss once described mathematics at the queen of science, and for a servant of this queen like me to stand here in these beautiful surroundings and receive the grand Abel Prize from a real queen is really an overwhelming event in my life.

Carleson has received a host of honours for his truly outstanding contributions. Many learned societies around the world have been eager to elect him to membership. These are the Royal Swedish Academy of Sciences, the American Academy of Arts and Sciences, the Russian Academy of Sciences, the Royal Society, London, the French Academy of Sciences, the Royal Danish Academy of Sciences and Letters, the Norwegian Academy of Science and Letters, the Royal Norwegian Society of Sciences and Letters, the Finnish Academy of Science and Letters, and the Hungarian Academy of Sciences. He had also won numerous prizes, some of which we have mentioned above. These are the Leroy Steel Prize from the American Mathematical Society (1984), the Wolf Prize (1992), the Lomonosov Gold Medal from the Russian Academy of Sciences (2002), the Sylvester Medal from the Royal Society, London (2003), and the Abel Prize (2006). He has been awarded honorary doctorates by the University of Helsinki (1982), the University of Paris (1988), and the Royal Institute of Technology, Stockholm (1989).Carleson's influence extends far beyond his research, a fact well known to the broad mathematical community. Besides his papers Carleson has published an influential book on potential theory 'Selected problems in the theory of exceptional sets' and helped make accessible the unpublished work of Arne Beurling(i.e., as co-editor with P Malliavin, J Neuberger, and J Wermer of 'The collected works of Arne Beurling'2Vols1989)... But Carleson's influence extends far beyond his publications. He has trained many PhD students, and many more mathematicians who came from around the world to learn from him. As director of the Mittag-Leffler Institute, he not only developed a world-class research centre, but moulded an entire generation of analysts. His research in analysis is a series of towering and fundamental discoveries. His friends know well his generosity, encouragement and selfless giving of himself.

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

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