Some Problems in Analysis

这个数学研究项目将支持对数学分析中几个重要问题的研究。其中一个问题是傅里叶分析，涉及到欧几里德空间中曲线的傅里叶变换的限制。本课题的其他问题是几何测量理论。这分为两类。这两个范畴中的第一个问题都是以下问题的具体实例：如果一个集合在d维空间中已知是某一类特定类型集合的并集，那么该集合的维数是多少？这类问题中最著名的是所谓的Kakeya猜想，即在每个可能方向上包含单位线段的集合必须具有全维。这些问题直接导致了对称为Radon变换的算子的有界性的研究。上述两个类别中的第二个问题与某些给定模式可以在某种固定大小的集合中出现的程度有关。一个典型的例子是单位距离问题。这个问题问的是长度为1的线段的个数，这些线段可以由从给定集合中选择的点对连接而成。这个在傅里叶分析和几何测量理论领域的数学研究项目有可能影响相当多样化的科学和技术相关学科。这里有一些例子：傅里叶分析是小波研究中的一个重要工具，而这些分析反过来又有从数据压缩和图像分析到通信理论的应用；傅里叶变换限制理论的研究应用于与波和类波现象有关的学科，如流体力学和量子物理学；氡变换是基本的数学工具，使当今复杂的医学成像技术成为可能；分形集是几何测量理论的研究对象之一，它为某些移动电话的天线提供了图形。

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.