Some Problems in Analysis

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This mathematics research project will support the investigation of several important problems in mathematical analysis. One of these problems is in Fourier analysis and concerns the restrictions of Fourier transforms to curves in Euclidean spaces. The other problems in this project are in geometric measure theory. These fall into two categories. The problems in the first of these two categories are all specific instances of the following: what can be said about the dimension of a set in d-dimensional space if that set is known to be the union of a particular class of sets of a certain type. The most famous of such problems is the so-called Kakeya conjecture that a set containing a unit line segment in each possible direction must have full dimension. These problems lead directly to the study of the boundedness of operators known as Radon transforms. The problems in the second of the two above-mentioned categories have to do with the extent to which certain given patterns can occur in sets of a certain fixed size. A prototypical example here is the unit distance problem. That problem asks about the number of line segments of length one which can be formed by joining pairs of points chosen from a given set.This mathematics research project in the areas of Fourier analysis and geometric measure theory has the potential to impact a quite diverse collection of science and technology-related disciplines. Here are some examples: Fourier analysis is an important tool in the study of wavelets, and these in turn have applications ranging from data compression and image analysis to communications theory; the study of the restriction theory of Fourier transforms has applications to disciplines that are concerned with waves and wave-like phenomena, such as fluid mechanics and quantum physics; the Radon transforms are the basic mathematical tools which make possible today's sophisticated medical imaging techniques; sets called fractals, which are one of the objects of study in geometric measure theory, provide patterns for the antennae in some cellular telephones.