Fourier analytic methods in convex geometry and geometric tomography

凸几何和几何断层扫描中的傅立叶分析方法

• 批准号: RGPIN-2019-06013
• 负责人: Yaskin, Vladyslav
• 金额: $ 1.38万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=715899
• 关键词: Fourier; analytic; methods; convex; geometry

The proposed research program belongs to the areas of convex geometry, geometric functional analysis, geometric tomography, and discrete tomography. These are very rich and active areas. Results and methods from these areas find applications in many other branches of mathematics, as well as other sciences, such as materials science, medicine, geophysics, etc. For example, geometric tomography provides a mathematical foundation for CT-scanners, the study of seismic activity, the search for oil, to name a few. The general objective of this proposal is to study convex bodies (and other objects) by looking at the sizes of their sections or projections. The main tools that I plan to use to attack various geometric problems are those of harmonic analysis. Several longstanding problems, including the Busemann-Petty problem about central sections of convex bodies and Klee's problem about maximal sections, have been solved with the help of Fourier analysis. One of the goals of this proposal is to further develop and refine such techniques, as well as to find new applications of Fourier analysis to geometric problems. These methods will be used in combination with other methods, especially those coming from geometric functional analysis, discrete geometry, and algebra. There are three main topics in this proposal. The first part is dedicated to Arnold's problem about algebraically integrable domains. The second part is about generalizations and applications of Gruenbaum-type inequalities. In the last part I will describe some uniqueness problems in geometric tomography and discrete tomography.

拟议的研究项目属于凸几何、几何泛函分析、几何断层扫描和离散断层扫描领域。这些都是非常丰富和活跃的领域。这些领域的结果和方法在数学的许多其他分支以及其他科学中都有应用，例如材料科学、医学、地球物理学等。例如，几何断层扫描为 CT 扫描仪、地震研究提供了数学基础。活动，例如寻找石油等等。 该提案的总体目标是通过观察凸体（和其他物体）的截面或投影的大小来研究它们。我计划用来解决各种几何问题的主要工具是调和分析。几个长期存在的问题，包括关于凸体中心截面的布斯曼-佩蒂问题和关于最大截面的克利问题，已经在傅立叶分析的帮助下得到了解决。该提案的目标之一是进一步开发和完善此类技术，并找到傅里叶分析在几何问题上的新应用。这些方法将与其他方法结合使用，特别是来自几何泛函分析、离散几何和代数的方法。该提案共有三个主要议题。第一部分专门讨论阿诺德关于代数可积域的问题。第二部分是格伦鲍姆型不等式的概括和应用。在最后一部分中，我将描述几何层析成像和离散层析成像中的一些唯一性问题。