Fourier Analytic Approach to the Geometric Tomography

几何断层扫描的傅立叶分析方法

• 批准号: 0504049
• 负责人: Artem Zvavitch
• 金额: --
• 依托单位: Kent State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-06-01 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0504049&HistoricalAwards=false
• 关键词: Fourier; Analytic; Approach; Geometric; Tomography

ABSTRACT:The main goal of this proposal is to study problems in Convex Geometry and Geometric Tomography using methods of Harmonic Analysis. Geometric Tomography is the area of mathematics where one investigates properties of solids based on the information about sections and projections of these solids. It borrows ideas and methods from many fields of Theoretical Mathematics, such as integral and differential geometry, statistics, and Fourier analysis. But perhaps its biggest overlap is with Convex Geometry. A major component of this proposal is to study problems which arise naturally from the recent work of the PI on Fourier analytic approach to the case of the most general measure of sections of star bodies. This method provided a number of new links between harmonic analysis, probability theory and convex geometry. PI intends to continue his work in this area and to consider a number of questions motivated by Slicing Problem and Isomorphic Busemann-Petty problem. PI also proposes a new approach to the study of the geometric properties of convex sets with respect to the Gaussian measure. As an application PI consider problems connected to the Gaussian Correlation Conjecture. The idea, of this method, is to restate the conjecture in the language of Geometric Tomography and to apply recently developed Fourier Analytic techniques. Geometric Tomography has a lot of real life applications. It has goals similar to those of many related and often practical areas. One of the best known examples is Classical Computer Aided Tomography, which aims to reconstruct the density of objects by means of their line integrals. Other examples are crystallography, robotics, stereology, and electron microscopy. One of the ideas of this proposal is to connect theoretical results from Convex Geometry to those practical areas via Geometric Tomography. Another component of the broader impact of proposed research lies in the training of graduate and undergraduate students. Indeed, a lot of problems addressed in this proposal stated in such a way that they are intuitively clear not only to graduate but to undergraduate students. On the other hand, the answers for many of those problems are quite counterintuitive which stimulates an interest of students to the subject and Mathematics in general!

摘要：该提案的主要目标是使用调和分析方法研究凸几何和几何断层扫描中的问题。 几何断层扫描是数学领域，人们根据有关固体截面和投影的信息来研究固体的性质。它借鉴了理论数学许多领域的思想和方法，例如积分和微分几何、统计学和傅立叶分析。但也许它最大的重叠是与凸几何。该提案的一个主要组成部分是研究 PI 最近针对星体截面最一般测量情况的傅立叶分析方法工作中自然产生的问题。该方法在调和分析、概率论和凸几何之间提供了许多新的联系。 PI 打算继续他在这一领域的工作，并考虑由切片问题和同构 Busemann-Petty 问题引发的许多问题。 PI还提出了一种新的方法来研究凸集相对于高斯测度的几何性质。作为应用程序 PI，请考虑与高斯相关猜想相关的问题。这种方法的想法是用几何断层扫描的语言重述猜想并应用最近开发的傅立叶分析技术。 几何断层扫描在现实生活中有很多应用。它的目标与许多相关且通常是实用领域的目标相似。最著名的例子之一是经典计算机辅助断层扫描，其目的是通过物体的线积分来重建物体的密度。其他例子包括晶体学、机器人学、体视学和电子显微镜。该提案的想法之一是通过几何断层扫描将凸几何的理论结果与这些实际领域联系起来。 拟议研究更广泛影响的另一个组成部分在于对研究生和本科生的培训。事实上，该提案中解决的许多问题都以这样一种方式表述，不仅对研究生而且对本科生来说都是直观清晰的。另一方面，其中许多问题的答案都非常违反直觉，这激发了学生对这门学科和数学的兴趣！