Harmonic Analysis in Convex Geometry

凸几何中的调和分析

• 批准号: 2247771
• 负责人: Dmitry Ryabogin
• 金额: $ 36.51万
• 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: 2023
• 资助国家: 美国
• 起止时间: 2023-06-01 至 2026-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2247771&HistoricalAwards=false
• 关键词: Harmonic; Analysis; Convex; Geometry

This project investigates questions in convex geometry using methods originating in harmonic analysis. The definition of convexity is elementary: a set is convex if it contains the straight-line path between any two of its elements. Yet the study of such sets leads to a rich and intricate branch of modern geometry, exhibiting unexpected connections to diverse areas of pure and applied mathematics, including probability, number theory, linear programming, computational geometry, and tomography. Significant parts of the project will rely on new techniques for constructing convex bodies and exposing their geometric properties using the Fourier transform, which represents mathematical objects via their frequency decomposition. The project will provide research opportunities for students and postdoctoral fellows. In addition, the principal investigators will continue their organization of seminars and other meetings including the Informal Analysis Seminar, a twice yearly meeting where researchers from around the world present lectures aimed at early career researchers.The project involves the study of questions in convex geometry using tools from harmonic analysis and differential geometry. Some of the proposed research questions arose in the context of earlier work of the principal investigators, where Fourier analytic approaches were used to study inequalities involving volumes and mixed volumes. These approaches have proved indispensable in the solution of longstanding problems in convex geometry and geometric tomography. Many natural questions in convex geometry involve relationships between volumes of pairs of convex sets; among these is an open question posed by K. Mahler in 1939. Another source of inspiration is the so-called `Scottish Book’, an informal collection of about 200 problems in mathematics which originated in the 1930s. Stanislav Ulam’s 19th Problem, taken from the Scottish Book, asks for a characterization of the convex sets of a fixed density (less than one) which float in an equilibrium position in every orientation (in water, of density one). Surprisingly, round balls are not the only such sets, due to a recent result of one of the principal investigators of this project. Part of the current project involves a further study of Ulam’s 19th Problem in relation to the properties of sections and projections of convex bodies. A second research direction involves the Mahler conjecture for volumes of three-dimensional convex bodies. This conjecture is known to hold under an additional symmetry assumption, and the project will consider the general version of this conjecture in three dimensions. The project will provide ample opportunities for participation in research by early-career researchers, including graduate students and postdocs.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目使用源自谐波分析中的方法在凸几何形状中研究了问题。凸的定义是基本的：如果它包含其任何两个元素之间的直线路径，则集合为凸。然而，对此类集合的研究导致了现代几何形状的丰富而复杂的分支，与纯数学和应用数学的潜水区进行了意外的联系，包括概率，数字理论，线性编程，计算几何形状和组织。该项目的重要部分将依靠新技术来构建凸形物体并使用傅立叶变换来揭示其几何特性，该变换通过其频率分解代表数学对象。该项目将为学生和博士后研究员提供研究机会。此外，首席研究人员将继续他们的半人类和其他会议组织，包括非正式分析研讨会，这是一次每年一次的一次会议，来自世界各地的研究人员旨在针对早期职业研究人员进行讲座。该项目涉及使用谐波分析和不同几何学和不同几何学的工具中的问题中的问题研究。一些提出的研究问题是在主要研究人员早期工作的背景下出现的，其中使用傅立叶分析方法研究涉及体积和混合体积的不平等现象。这些方法在解决凸几何和几何层析成像中长期存在的问题方面提供了必不可少的。凸几何形状中的许多自然问题都涉及成对凸组的关系之间的关系。其中是K. Mahler在1939年提出的一个公开问题。灵感的另一个来源是所谓的“苏格兰书”，这是一个非正式的收集，该集合涉及1930年代的数学中约200个问题。 Stanislav Ulam的第19个问题，摘自苏格兰书籍，要求对固定密度（少于一个）的凸组进行表征，该凸组在每个方向（在水中，密度为一）中浮动平衡位置相等。令人惊讶的是，由于该项目的主要研究人员之一的最新结果，圆球并不是唯一的组合。当前项目的一部分涉及对ULAM关于凸形机构部分和项目性质的第19个问题的进一步研究。第二个研究方向涉及三维凸体体积的Mahler猜想。已知该猜想是在额外的对称假设下持有的，该项目将在三个维度上考虑该猜想的一般版本。该项目将为早期研究人员（包括研究生和博士学位）提供良好的参与研究机会。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响评估标准，被认为是通过评估而被视为珍贵的支持。