Harmonic Analysis in Convex Geometry

这个项目使用谐波分析的方法来研究凸几何中的问题。凸性的定义是基本的：一个集合是凸的，如果它包含任意两个元素之间的直线路径。然而，对这些集合的研究导致了现代几何的一个丰富而复杂的分支，展示了与纯数学和应用数学不同领域的意想不到的联系，包括概率论，数论，线性规划，计算几何和断层摄影。该项目的重要部分将依赖于构建凸体的新技术，并使用傅里叶变换暴露其几何特性，傅里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提供研究机会。此外，主要研究人员将继续组织研讨会和其他会议，包括非正式分析研讨会，这是每年两次的会议，来自世界各地的研究人员针对早期职业研究人员进行讲座。该项目涉及使用调和分析和微分几何工具研究凸几何中的问题。提出的一些研究问题是在主要研究人员早期工作的背景下产生的，其中傅立叶分析方法用于研究涉及体积和混合体积的不等式。这些方法已被证明是解决凸几何和几何层析成像中长期存在的问题所不可或缺的。凸几何中的许多自然问题涉及凸集对的体积之间的关系；其中有一个是马勒在1939年提出的开放性问题。另一个灵感来源是所谓的“苏格兰书”，这是一本起源于20世纪30年代的大约200个数学问题的非正式文集。斯坦尼斯拉夫·乌拉姆（Stanislav Ulam）的第19个问题，取自《苏格兰书》（Scottish Book），要求描述密度固定（小于1）的凸集的特征，这些凸集在密度为1的每个方向（在水中）都处于平衡位置。令人惊讶的是，由于该项目的一位主要研究人员最近的结果，圆球并不是唯一的这样的集合。目前项目的一部分涉及进一步研究乌拉姆第19问题，该问题与凸体的截面和投影的性质有关。第二个研究方向涉及三维凸体体积的马勒猜想。已知这个猜想在一个额外的对称假设下成立，该项目将在三维空间中考虑这个猜想的一般版本。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参与研究的充分机会。该奖项反映了美国国家科学基金会的法定使命，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。

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.