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The Abel Prize |
Laureate 2010 |
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ContentsIntroduction Who is Lennart Carleson? Why has he been awarded the Abel Prize for 2006? The Abel Prize for 2006 in a broader perspective Precise (mathematical) formulations of Carleson's results Popular presentations of Carleson's results Convergence of Fourier series The Corona theorem and Carleson measure The Hénon map Kakeya's needle problem Background material Jean Baptiste Joseph Fourier (1768-1830) Institut Mittag-Leffler Fourier analysis Discrete dynamic systems References to the illustrations
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Popular presentations of Carleson's resultsKakeya's needle problem
We dip a needle in ink and lay it on a sheet of paper. The problem involves rotating the needle 180 degrees without lifting it from the paper. We are allowed to push it back and forth while we rotate it, roughly as we do when we turn a car around in a parking space. The question is how large the area on the sheet that is coloured with ink will be when we are finished, and particularly if there is a lower limit to this area when we choose an optimal way to carry out the rotation. This problem is known as Kakeya's needle problem, formulated by the Japanese mathematician Kakeya in 1917. Let us assume that the needle has a length of 1. If we rotate the needle around its midpoint, we will have coloured a space with an area equal to π•0,52≈0,78. This is clearly not optimal; we can definitely manage to turn inside an equilateral triangle with a height of 1 and with an area approximately equal to 0.58. Better yet would be a hypocycloid (the curve that is traced by a point on a small wheel that rolls around inside a larger wheel) with a diameter of 1. This has an area of 0.39. However, all of these answers are wrong. The correct answer is that the surface can be made as small as we like! This was proved by the Russian mathematician Besicovitch in 1928. His set consisted of a large number of very long and narrow triangles, almost like the branches on a Christmas tree in a child's drawing. It involves driving back and forth a great many times and only turning a little each time. Current interest is focused on the generalisations of this problem, and this is where Carleson-Sjölin's result for Fourier multipliers enters in as a standard tool. |
HomeNews ArchiveCalendar Editor: Anne Marie Astad The Norwegian Academy of Science and Letters E-mail: dnva@online.no
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The fourth problem that the committee mentions in its decision is not a problem for which Carleson has contributed an epoch-making breakthrough as was the case in the three above-mentioned problems. However, in collaboration with his student Sjölin, Carleson has proved a result that has turned out to be important in the study of this problem and especially its generalisations. Since the nature of the problem is fairly easy to explain, we shall include a description of it here.