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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758370
• 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.

拟议的研究计划属于凸几何，几何功能分析，几何层析成像和离散层析成像领域。这些是非常丰富和活跃的地区。这些领域的结果和方法在许多其他数学分支以及其他科学的分支中发现了应用，例如材料科学，医学，地球物理学等。该提案的一般目标是通过查看其部分或预测的大小来研究凸形机构（和其他对象）。我计划用来攻击各种几何问题的主要工具是谐波分析。在傅立叶分析的帮助下，解决了一些长期存在的问题，包括有关凸体的中央部分的Busemann-Petty问题以及Klee关于最大部分的问题。该提案的目标之一是进一步开发和完善此类技术，并找到傅立叶分析到几何问题的新应用。这些方法将与其他方法结合使用，尤其是来自几何功能分析，离散几何形状和代数的方法。该提案中有三​​个主要主题。第一部分专门解决了阿诺德关于代数集成域的问题。第二部分是关于Gruenbaum型不平等的概括和应用。在最后一部分中，我将描述几何层析成像和离散断层扫描中的一些唯一性问题。