Harmonic Analysis in Convex Geometry

凸几何中的调和分析

• 批准号: 1101636
• 负责人: Artem Zvavitch
• 金额: $ 37.5万
• 依托单位: Kent State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-05-01 至 2017-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101636&HistoricalAwards=false
• 关键词: Harmonic; Analysis; Convex; Geometry

The goal of the proposed research is to develop methods of Harmonic Analysis to solve problems from Convex Geometry including problems from Geometric Tomography and questions concerning duality and volume. Many conjectures stipulate that there must exist direct duality connections between projections and sections of convex bodies. In order to gain an understanding of these connections, it is important to try to obtain a description of the duality phenomena involved. This is where the Fourier Analysis comes into play. A major component of this proposal is to study problems, which arise naturally from the recent work of the investigators. They will use Fourier Transform and other Harmonic Analysis tools as a link between volumes of a convex body and its polar. They will also address and study other duality problems about projections and sections of convex and star bodies. Convex Geometry 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 and Harmonic Analysis. In addition, the advantage of the problems addressed in this proposal is that most of them are "intuitively clear" not only to undergraduate students, but also to children in high school or even middle school. Even the geometric notion of duality can be explained to a child in the high school. On the other hand, the answers are (very often) counter-intuitive, different in the plane and in the space, and this stimulates an interest of students to the subject. The investigators believe that those problems will help to attract more students to Harmonic Analysis and Convexity as well as to Mathematics and Science in General.

拟议研究的目标是开发调和分析方法来解决凸几何问题，包括几何层析成像问题以及有关二元性和体积的问题。许多理论都规定凸体的投影与截面之间必须存在直接的对偶联系。为了理解这些联系，重要的是试图获得对所涉及的二元现象的描述。这就是傅立叶分析发挥作用的地方。这项建议的一个主要组成部分是研究调查人员最近工作中自然产生的问题。他们将使用傅立叶变换和其他谐波分析工具作为凸体体积与其极坐标之间的联系。 他们还将解决和研究有关凸体和星星体的投影和截面的其他对偶问题。凸几何在真实的生活中有很多应用。它的目标与许多相关的、往往是实际的领域的目标相似。最著名的例子之一是经典计算机辅助断层扫描，其目的是通过线积分重建物体的密度。其他的例子是晶体学、机器人学、体视学和电子显微镜学。该方案的一个思想是通过几何层析成像和调和分析将凸几何的理论结果与实际领域联系起来。此外，这份建议书所针对的问题，其优点在于，不仅对本科生，对高中甚至初中阶段的孩子，大多“直观明了”。 甚至对偶的几何概念也可以解释给一个高中生听。另一方面，答案（通常）是反直觉的，在平面和空间中不同，这激发了学生对该主题的兴趣。研究人员认为，这些问题将有助于吸引更多的学生谐波分析和凸性，以及数学和科学一般。