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.

本计画利用调和分析的方法探讨凸几何的问题。凸性的定义是基本的：一个集合是凸的，如果它包含任何两个元素之间的直线路径。然而，对这类集合的研究导致了现代几何学的一个丰富而复杂的分支，与纯数学和应用数学的各个领域（包括概率论、数论、线性规划、计算几何和断层摄影）有着意想不到的联系。该项目的重要部分将依赖于构建凸体的新技术，并使用傅立叶变换揭示其几何特性，傅立叶变换通过频率分解表示数学对象。该项目将为学生和博士后研究员提供研究机会。此外，主要研究者将继续组织研讨会和其他会议，包括非正式分析研讨会，每年两次的会议，来自世界各地的研究人员为早期职业研究人员提供讲座。该项目涉及使用调和分析和微分几何工具研究凸几何问题。一些提出的研究问题出现在背景下的早期工作的主要调查人员，傅立叶分析方法被用来研究不平等涉及体积和混合体积。这些方法已被证明是必不可少的，在凸几何和几何层析成像的长期问题的解决方案。凸几何中的许多自然问题都涉及到凸集对的体积之间的关系，其中有一个由K。1939年的马勒。 灵感的另一个来源是所谓的“苏格兰书”，这是一本非正式的集合，大约有200个数学问题，起源于20世纪30年代。斯坦尼斯拉夫·乌拉姆的第19个问题，取自苏格兰书，要求描述一个固定密度（小于1）的凸集，这些凸集在每个方向上都漂浮在平衡位置（在水中，密度为1）。令人惊讶的是，由于该项目的主要研究人员之一最近的结果，圆球并不是唯一的此类集合。目前的项目的一部分涉及进一步研究乌拉姆的第19个问题有关的性质的部分和投影的凸体。第二个研究方向涉及马勒猜想体积的三维凸体。这个猜想在一个额外的对称性假设下成立，该项目将考虑这个猜想在三维空间中的一般版本。该项目将为包括研究生和博士后在内的早期职业研究人员提供充分的参与研究的机会。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。