Why has he been awarded the Abel Prize for 2006?

The basis in short for the committee's decision is:

for his deep, fundamental contributions to harmonic analysis and to the theory of smooth dynamic systems.

Somewhat (extremely) simplified, we can divide mathematicians into two categories, theory builders and problem solvers. Most mathematicians have a little bit of both in them, but some are more typically one than the other. Carleson fits well into the category of “problem solver”. He has made his reputation taking on old, difficult problems, which he has then managed to solve, partly by means of extremely complicated methods. The committee has a rather poetic way of explaining this: "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 new kingdom he has discovered, and he moves on to even wilder and more remote domains of Science."

The committee singles out three special problems that Carleson has solved, one of which is ranked above the other two. That is a problem in the field of harmonic analysis, formulated by the Frenchman Jean Baptiste Joseph Fourier in 1807. It concerns the possibility of describing random functions by means of simple wave functions. Fourier was often rather vague and imprecise in his formulations, and it was the Russian mathematician Lusin who precisely defined the problem. He wrote in a work in 1913 that he assumed that the result was true, but that he was unable to prove this assertion. The problem was therefore given the name of Lusin's conjecture. Despite persistent efforts, no one managed to proof this conjecture until Carleson made his breakthrough in 1966 and converted Lusin's conjecture into Carleson's theorem about “the convergence almost everywhere of Fourier series of quadratic integrable functions”.

The other two problems that the committee emphasizes in its decision are the “Corona problem” and a problem in dynamic systems related to the Hénon map. The Corona problem is a pure mathematics problem that deals with functions defined on a circular disk. To what extent can we describe what will happen with these functions on the edge of the disk, when we know what happens in the inner area? The name Corona problem refers to the ring of light that is seen around the eclipsed solar disk during a total solar eclipse. Carleson's result has nothing to do with astronomy; mathematicians (in this case the Japanese mathematician Kakutani in the beginning of the 1940s) gave the problem its name by association with a better-known phenomenon.

The Hénon map is named after the astronomer Michel Hénon and refers to work from 1976. In both astronomy and meteorology, it has proven to be expedient to describe phenomena with a system of mathematical models known as dynamic systems. A difficult problem in this theory is to determine whether a system has a so-called “strange attractor”.

The Hénon map describes a way of jumping from point to point in a plane. When we start at a point, several things can happen. We can jump more and more in the direction of a particular point, we can end up on a course where we jump around and around among a finite number of points or we can disappear into infinity. However, it is also possible in the Hénon map that we end up in an area where we do not escape again, but that within this area we experience an apparently chaotic behaviour, we jump here and there and back and forth, with the only regularity being that we actually remain within this area. Such an area is called a strange attractor: attractor because the jumps have a tendency to remain within the area, strange because the map displays a strange or chaotic behaviour after we have entered the area.

A computer can easily make the (thousands of) calculations that are necessary in order to visualise the problem, but the computer can never come up with a formal and theoretical proof of the existence of an attractor. In 1991, Carleson and his countryman Benedicks proved that the Hénon map has a strange attractor. This was in fact the first proof that was given for the existence of a strange attractor.

When Carleson began to show interest in dynamic systems in the 1980s, this was a new field of research for him. That a mathematician after the age of 50 was willing to throw himself into something completely new is already rather unusual, but that in the course of a relatively short period of time he was able to solve one of the most challenging problems must be regarded as nothing short of sensational. It does not exactly lend credence to the myth that mathematics is a young man’s game!

The committee's decision is also partly based on Carleson's scientific policy work, though this is not one of the main reasons why he is being awarded the Abel Prize. Throughout his entire career as a mathematician, Carleson has shown great interest in the role of mathematics in the society-at-large. He has thrown himself into the debate on mathematics in school and like many of his colleagues in other countries has expressed concern over the decline in mathematical skills. As president of the International Mathematical Union (IMU) in the period 1978-82, he worked hard to have the People's Republic of China represented in the international society of mathematicians. Back home in Sweden in the 1970s, he built up the Institut Mittag-Leffler into one of the world's most attractive mathematics laboratories, a centre where mathematicians from all over the world can come together in a secluded atmosphere and work on mathematical problems of current interest.

The committee concludes their explanation by noting Carleson's broad range of interests and his important role as both an expert in his field and a spokesman on scientific policy matters: Lennart Carleson is a brilliant scientist with a broad vision for mathematics and for the role of mathematics in the global community.

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