Geometry of Curves and Surfaces

曲线和曲面的几何

基本信息

批准号：
2202337
负责人：
Mohammad Ghomi
金额：
$ 31.5万
依托单位：
Georgia Tech Research Corporation
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-05-15 至 2025-04-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2202337&HistoricalAwards=false
关键词：
Geometry Curves Surfaces

项目摘要

Curves and surfaces are to geometry what numbers are to algebra. They form the basic ingredients of our visual perception and inspire the development of far-reaching mathematical tools. Yet there are still many fundamental open questions in this area that are strikingly intuitive and elementary to state. Studying these questions may stimulate useful developments in pure mathematics and lead to wider applications in science and technology. For instance, the isoperimetric inequality has numerous applications due to its connections with a host of other important inequalities, including the Sobolev inequality in mathematical analysis and the Faber-Krahn inequality in spectral analysis. Furthermore, the rigidity of surfaces may have applications for stability of complicated domes in modern architecture, while the medial axis, or cut locus of distance functions, is an important tool in shape recognition, which is of interest in computer graphics and mathematical biology. These questions are ideal for introducing the public to the exciting world of modern mathematics and arousing the interest of beginning students in geometry. This project will engage in a range of activities, including accessible public lectures and articles, to promote these topics.This project is concerned with curves and surfaces, and more broadly Riemannian submanifolds, spanning a wide range of topics and tools including isoperimetric problems, isometric embeddings, geometric knot theory, polyhedral approximations, h-principle theory, and curvature flows. Some recurring themes throughout these investigations, which the PI conducts in joint work with his students and collaborators, are various notions of convexity or optimization and the interaction between geometric and topological concepts, or local versus global properties of submanifolds. More specifically, a typical question is how restrictions on curvature, intrinsic metric, or various boundary conditions influence the global shape of a curve or a hypersurface or allow an isometric embedding of that object in a space of low codimension. For instance, the PI recently established Zalgaller’s conjecture on the length and shape of the shortest closed curve that contains the unit sphere in its convex hull. Other projects include rigidity of isometric embeddings, unfoldability of convex polyhedra or Durer’s conjecture, optimization problems for space curves, and the study of the cut locus of distance functions, or medial axis, of contractible regions in Riemannian manifolds.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.

曲线和曲面对于几何学来说就像数字对于代数一样，它们构成了我们视觉感知的基本成分，并激发了影响深远的数学工具的发展，然而，在这个领域仍然存在许多非常直观和基本的基本问题。研究这些问题可能会刺激纯数学的有益发展，并导致科学和技术领域的更广泛应用，例如，等周不等式由于其与许多其他重要的不等式（包括数学分析中的索博列夫不等式）的联系而具有广泛的应用。此外，表面刚度可用于现代建筑中复杂圆顶的稳定性，而中轴或距离函数的切割轨迹是形状识别的重要工具。这些问题非常适合向公众介绍令人兴奋的现代数学世界，并激发初学者对几何的兴趣。该项目将开展一系列活动，包括易于理解的公开讲座和文章。 , 来推广这些该项目涉及曲线和曲面，以及更广泛的黎曼子流形，涵盖广泛的主题和工具，包括等周问题、等距嵌入、几何结理论、多面体近似、h 原理理论和一些重复出现的曲率流。在 PI 与他的学生和合作者共同进行的这些研究中，涉及凸性或优化的各种概念以及几何和几何之间的相互作用。拓扑概念，或者子流形的局部与全局属性更具体地说，一个典型的问题是曲率、内在度量或各种边界条件的限制如何影响曲线或超曲面的全局形状或允许该对象在等距嵌入中。例如，PI 最近建立了 Zalgaller 的关于凸包中包含单位球体的最短闭合曲线的长度和形状的猜想。等距嵌入、凸多面体的可展开性或杜勒猜想、空间曲线的优化问题以及黎曼流形中可收缩区域的距离函数或中轴的切割轨迹的研究。该奖项反映了 NSF 的法定使命，并被认为是值得的通过使用基金会的智力优势和更广泛的影响审查标准进行评估来获得支持。