Applications of Fourier analysis to convex geometry

首席研究员计划将傅里叶分析的方法应用于凸几何、泛函分析和概率论。基于凸体的截面和投影信息来研究凸体的几何性质在数学和科学的许多领域都有重要的应用。这位研究者最近发展了一种基于傅立叶分析方法的凸体截面的新方法。这种方法的思想是用傅里叶变换来表示物体的不同截面特征，然后用调和分析的方法来解决几何问题。研究人员计划将这种方法应用于不同类别物体的刻画，从关于其截面和投影的数据确定凸体，以及Busemann-Petty类型的几何不等式。作为凸几何中的主要对象之一，相交体与函数空间之间的联系最近被研究者发现了。研究人员计划利用这种联系，利用泛函分析的方法，建立关于相交物体的新结果。概率论中的一个老问题是刻画所有随机向量的性质，即坐标的所有线性组合都具有相同的分布，直到一个常数。经典的例子是稳定的随机向量。研究人员计划刻画所有具有这种性质的随机向量。该建议中考虑的问题属于三个数学领域：凸几何、泛函分析和概率。然而，对于大多数结果，求解策略是共同的--问题被转化为傅里叶变换的语言，然后作为调和分析的问题来处理。在凸几何学中，主要研究人员基于与凸体的截面和投影相关的数据来研究凸体的性质。这一方向通常被称为几何层析成像，在工程和医学上有许多应用。泛函分析中的问题与度量空间的嵌入理论有关，该理论在计算机科学中有许多应用。在概率学中，这位研究人员计划研究在数学和统计学的几个领域中发挥重要作用的稳定过程的推广。分布的傅里叶变换将作为主要的技术工具，而这项工作的很大一部分将取决于计算傅里叶变换的新技术。这些技术具有独立的价值，并已被应用于数学、信号处理和统计的其他领域。该项目的一个重要部分是研究生的参与和培训。

The principal investigator plans to apply methods of Fourier analysis to convex geometry, functional analysis and probability. The study of geometric properties of convex bodies based on information about sections and projections of these bodies has important applications to many areas of mathematics and science. A new approach to sections of convex bodies, based on methods of Fourier analysis, has recently been developed by the investigator. The idea of this approach is to express different cross-sectional characteristics of a body in terms of the Fourier transform and then use methods of harmonic analysis to solve geometric problems. The investigator plans to apply this approach to characterizations of different classes of bodies, determination of convex bodies from data about their sections and projections, geometric inequalities of the Busemann-Petty type. A connection between intersection bodies, one of the main objects in convex geometry, and functional spaces has recently been found by the investigator. The investigator plans to use this connection to establish new results about intersection bodies using methods of functional analysis. An old problem in probability is to characterize all random vectors having the property that all linear combinations of coordinates have the same distribution, up to a constant. The classical examples are stable random vectors. The investigator plans to characterize all random vectors with this property.The problems considered in this proposal belong to three areas of mathematics: convex geometry, functional analysis and probability. However, the strategy of solution is common for most of the results - the question is translated into the language of the Fourier transform and then treated as a problem from harmonic analysis. In convex geometry, the principal investigator studies properties of convex bodies based on data related to sections and projections of these bodies. This direction, often named geometric tomography, has numerous applications to engineering and medicine. The problems in functional analysis are related to the theory of embedding of metric spaces that has many applications to computer science. In probability, the investigator plans to study generalizations of stable processes that play important role in several areas of mathematics and statistics. The Fourier transform of distributions will serve as the main technical tool, and a significant part of the work will depend on new techniques for calculating the Fourier transform. These techniques have independent value and have already been applied to other areas of mathematics, signal processing and statistics. An important part of the project is the involvement and training of graduate students.