Applications of Fourier analysis to convex geometry

首席研究员计划将傅里叶分析方法应用于凸几何、泛函分析和概率。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。基于傅立叶分析方法的凸体截面的新方法，最近由研究者开发。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，然后用谐波分析的方法来解决几何问题。研究者计划将这种方法应用于不同类型物体的特征描述，从其截面和投影的数据确定凸体，Busemann-Petty型的几何不等式。交叉体是凸几何中的主要对象之一，研究者最近发现了它与功能空间之间的联系。研究者计划利用这一联系，利用功能分析方法建立关于相交体的新结果。概率论中的一个老问题是描述所有随机向量的特征，这些向量具有所有坐标的线性组合具有相同分布的性质，直到一个常数。经典的例子是稳定随机向量。研究者计划描述所有具有此属性的随机向量。本建议所考虑的问题属于数学的三个领域：凸几何、泛函分析和概率论。然而，对于大多数结果，解决的策略是常见的-问题被转换成傅里叶变换的语言，然后作为谐波分析的问题来处理。在凸几何中，主要研究者根据与凸体的剖面和投影相关的数据来研究凸体的性质。这个方向通常被称为几何断层扫描，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嵌入理论有关，该理论在计算机科学中有许多应用。在概率论中，研究者计划研究在数学和统计的几个领域中起重要作用的稳定过程的概括。分布的傅里叶变换将作为主要的技术工具，并且工作的重要部分将依赖于计算傅里叶变换的新技术。这些技术具有独立的价值，并已应用于数学、信号处理和统计学的其他领域。该项目的一个重要组成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.