Lennart Axel Edvard Carleson (born 18 March 1928) is a

Swedish Swedish or ' may refer to: Anything from or related to Sweden, a country in Northern Europe. Or, specifically: * Swedish language, a North Germanic language spoken primarily in Sweden and Finland ** Swedish alphabet, the official alphabet used by ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

, known as a leader in the field of

harmonic analysis Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an ex ...

. One of his most noted accomplishments is his proof of

Lusin's conjecture Carleson's theorem is a fundamental result in mathematical analysis establishing the pointwise (Lebesgue) almost everywhere convergence of Fourier series of functions, proved by . The name is also often used to refer to the extension of the re ...

. He was awarded the

Abel Prize The Abel Prize ( ; no, Abelprisen ) is awarded annually by the King of Norway to one or more outstanding mathematicians. It is named after the Norwegian mathematician Niels Henrik Abel (1802–1829) and directly modeled after the Nobel Pri ...

in 2006 for "his profound and seminal contributions to harmonic analysis and the theory of smooth dynamical systems."

Life

He was a student of

Arne Beurling Arne Carl-August Beurling (3 February 1905 – 20 November 1986) was a Swedish mathematician and professor of mathematics at Uppsala University (1937–1954) and later at the Institute for Advanced Study in Princeton, New Jersey. Beurling worked ...

and received his Ph.D. from

Uppsala University Uppsala University ( sv, Uppsala universitet) is a public research university in Uppsala, Sweden. Founded in 1477, it is the oldest university in Sweden and the Nordic countries still in operation. The university rose to significance during ...

in 1950. He did his post-doctoral work at

Harvard University Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of highe ...

where he met and discussed Fourier series and their convergence with

Antoni Zygmund Antoni Zygmund (December 25, 1900 – May 30, 1992) was a Polish mathematician. He worked mostly in the area of mathematical analysis, including especially harmonic analysis, and he is considered one of the greatest analysts of the 20th century. ...

and

Raphaël Salem Raphaël Salem (Greek: Ραφαέλ Σαλέμ; November 7, 1898 in Salonika, Ottoman Empire (now Thessaloniki, Greece) – June 20, 1963 in Paris, France) was a Greek mathematician after whom are named the Salem numbers and Salem–Spencer sets ...

who were there in 1950 and 1951. He is a

professor emeritus ''Emeritus'' (; female: ''emerita'') is an adjective used to designate a retired chair, professor, pastor, bishop, pope, director, president, prime minister, rabbi, emperor, or other person who has been "permitted to retain as an honorary title ...

at Uppsala University, the

Royal Institute of Technology The KTH Royal Institute of Technology ( sv, Kungliga Tekniska högskolan, lit=Royal Institute of Technology), abbreviated KTH, is a public research university in Stockholm, Sweden. KTH conducts research and education in engineering and technolog ...

Stockholm Stockholm () is the capital and largest city of Sweden as well as the largest urban area in Scandinavia. Approximately 980,000 people live in the municipality, with 1.6 million in the urban area, and 2.4 million in the metropo ...

, and the

University of California, Los Angeles The University of California, Los Angeles (UCLA) is a public land-grant research university in Los Angeles, California. UCLA's academic roots were established in 1881 as a teachers college then known as the southern branch of the Californ ...

, and has served as director of the

Mittag-Leffler Institute The Mittag-Leffler Institute is a mathematical research institute located in Djursholm, a suburb of Stockholm. It invites scholars to participate in half-year programs in specialized mathematical subjects. The Institute is run by the Royal Sw ...

Djursholm Djursholm () is one of four suburban districts in, and the seat of Danderyd Municipality, Stockholm County, Sweden. Djursholm is included in the multi-municipal Stockholm urban area. Djursholm is divided into a number of different areas: Djursholm ...

outside Stockholm 1968–1984. Between 1978 and 1982 he served as president of the

International Mathematical Union The International Mathematical Union (IMU) is an international non-governmental organization devoted to international cooperation in the field of mathematics across the world. It is a member of the International Science Council (ISC) and supports ...

. Carleson married Butte Jonsson in 1953, and they had two children: Caspar (born 1955) and Beatrice (born 1958). He has supervised 29 PhD students. They include Svante Janson,

Kurt Johansson Kurt Ivar Björn Johansson (25 February 1914 – 8 August 2011) was a Swedish shooter who competed at the 1948, 1960 and 1968 Olympics. In 1948 in London he placed fourth in the free rifle, three positions, 300 m event. In 1960 he finished 19t ...

Warwick Tucker Warwick Tucker is an Australian mathematician at Monash University (previously deputy Chair and Chair at the Department of Mathematics at Uppsala University 2009–2020) who works on dynamical systems, chaos theory and computational mathematics. H ...

, Bengt Rosén, Per Sjölin, Hans Wallin and Ingemar Wik.

Work

His work has included the solution of some outstanding problems, using techniques from

combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many a ...

and

probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...

(especially

stopping time In probability theory, in particular in the study of stochastic processes, a stopping time (also Markov time, Markov moment, optional stopping time or optional time ) is a specific type of “random time”: a random variable whose value is inter ...

s). In the theory of

Hardy space In complex analysis, the Hardy spaces (or Hardy classes) ''Hp'' are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper . I ...

s, Carleson's contributions include the

corona theorem In mathematics, the corona theorem is a result about the spectrum of the bounded holomorphic functions on the open unit disc, conjectured by and proved by . The commutative Banach algebra and Hardy space ''H''∞ consists of the bounded ...

(1962), and establishing the

almost everywhere In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...

convergence of Fourier series for

square-integrable function In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value ...

s (now known as Carleson's theorem). It was a famous old problem by

Joseph Fourier Jean-Baptiste Joseph Fourier (; ; 21 March 1768 – 16 May 1830) was a French mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series, which eventually developed into Fourier analysis and ha ...

when he invented

Fourier analysis In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph ...

in 1807 and formalised by Nikolai Luzin in 1913 as the

Kolmogorov Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...

proved a famous negative result of the conjecture for ''L''¹ function in 1928 and stated that the conjecture must be false. It was so until 38 years later when Carleson gave his proof at the

International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the Nevanlinna Prize (to be rena ...

at Moscow in 1966. But his proofs were very hard and only understood in the late 80s and early 90s when a general theory of operators arrived and brought mathematicians closer to using his striking ideas with ease. He is also known for the theory of

Carleson measure In mathematics, a Carleson measure is a type of measure on subsets of ''n''- dimensional Euclidean space R''n''. Roughly speaking, a Carleson measure on a domain Ω is a measure that does not vanish at the boundary of Ω when compared to the su ...

s. His tools and methods have been of fundamental importance to analysis as well as many areas of mathematics. The theorem for Fourier multipliers developed by Carleson and Per Sjölin has been standard in the study of the Kakeya problem. In 1974 he solved the extension problem for

quasiconformal mapping In mathematical complex analysis, a quasiconformal mapping, introduced by and named by , is a homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded eccentricity. Intuitively, let ''f'' : ''D' ...

s, and gave important new results in the Bochner–Riesz mean in dimension two. In the theory of

dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...

s, Carleson has worked in

complex dynamics Complex dynamics is the study of dynamical systems defined by iteration of functions on complex number spaces. Complex analytic dynamics is the study of the dynamics of specifically analytic functions. Techniques *General **Montel's theorem ** P ...

. His proof with Michael Benedicks, of the existence of

strange attractors In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain ...

in the Hénon map in 1991 led to a new field in dynamical systems. In addition to publishing some landmark papers, Carleson has also published two books: First, an influential book on

potential theory In mathematics and mathematical physics, potential theory is the study of harmonic functions. The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gra ...

, ''Selected Problems on Exceptional Sets'' (Van Nostrand, 1967), and second a book on the

iteration Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. ...

analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...

s, ''Complex Dynamics'' (Springer, 1993, in collaboration with T. W. Gamelin). He was the co-editor along with Paul Malliavin, J. Neuberger and J. Wermer who collected and published the unpublished works of his mentor Arne Beurling in 1989.

Awards

He was awarded the

Wolf Prize in Mathematics The Wolf Prize in Mathematics is awarded almost annually by the Wolf Foundation in Israel. It is one of the six Wolf Prizes established by the Foundation and awarded since 1978; the others are in Agriculture, Chemistry, Medicine, Physics and Arts ...

in 1992, the

Lomonosov Gold Medal The Lomonosov Gold Medal (russian: Большая золотая медаль имени М. В. Ломоносова ''Bol'shaya zolotaya medal' imeni M. V. Lomonosova''), named after Russian scientist and polymath Mikhail Lomonosov, is awarded ...

in 2002, the

Sylvester Medal The Sylvester Medal is a bronze medal awarded by the Royal Society (London) for the encouragement of mathematical research, and accompanied by a £1,000 prize. It was named in honour of James Joseph Sylvester, the Savilian Professor of Geometry a ...

in 2003, and the

in 2006 for his profound and seminal contributions to harmonic analysis and the theory of smooth

dynamical systems In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a ...

. He is a member of the

Norwegian Academy of Science and Letters The Norwegian Academy of Science and Letters ( no, Det Norske Videnskaps-Akademi, DNVA) is a learned society based in Oslo, Norway. Its purpose is to support the advancement of science and scholarship in Norway. History The Royal Frederick Unive ...

. In 2012 he became a fellow of the

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meeting ...

.List of Fellows of the American Mathematical Society
retrieved 2012-11-10. He became a Foreign Fellow of the

Royal Society The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...

in 1993, Honorary member of the

London Mathematical Society The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical ...

in 1981, the

American Academy of Arts and Sciences The American Academy of Arts and Sciences (abbreviation: AAA&S) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, a ...

French Academy of Sciences The French Academy of Sciences (French: ''Académie des sciences'') is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research. It was at ...

Royal Swedish Academy of Sciences The Royal Swedish Academy of Sciences ( sv, Kungliga Vetenskapsakademien) is one of the royal academies of Sweden. Founded on 2 June 1739, it is an independent, non-governmental scientific organization that takes special responsibility for prom ...

Finnish Society of Sciences and Letters The Finnish Society of Sciences and Letters is a Finnish academy for natural sciences, social sciences and humanities. It is known in Latin as Societas Scientiarum Fennica, in Swedish as Finska Vetenskaps-Societeten, and in Finnish as Suomen Ti ...

Finnish Academy of Science and Letters The Finnish Academy of Science and Letters (Finnish ''Suomalainen Tiedeakatemia''; Latin ''Academia Scientiarum Fennica'') is a Finnish learned society. It was founded in 1908 and is thus the second oldest academy in Finland. The oldest is the Fi ...

, Royal Danish Academy of Sciences and Letters and the Hungarian Academy of Sciences. He has honorary doctorates from many universities such as Helsinki, Paris and KTH Royal Institute of Technology, Stockholm.

Publications

* ''Selected Problems on Exceptional Sets'', Van Nostrand, 1967 * ''Matematik för vår tid'' (Mathematics for our time), Prisma 1968 * with T. W. Gamelin
''Complex Dynamics''
Springer, 1993

References

External links

* {{DEFAULTSORT:Carleson, Lennart Axel Edvard Living people 1928 births 20th-century Swedish mathematicians 21st-century Swedish mathematicians Abel Prize laureates Foreign Members of the Royal Society Institute for Advanced Study visiting scholars Uppsala University alumni KTH Royal Institute of Technology faculty Wolf Prize in Mathematics laureates Members of the French Academy of Sciences Members of the Norwegian Academy of Science and Letters Foreign Members of the USSR Academy of Sciences Foreign Members of the Russian Academy of Sciences Directors of the Mittag-Leffler Institute Foreign associates of the National Academy of Sciences Fellows of the American Mathematical Society Recipients of the Lomonosov Gold Medal Swedish expatriates in the United States Members of the Royal Swedish Academy of Sciences Presidents of the International Mathematical Union Uppsala University faculty