Geometry of Curves and Surfaces

曲线和曲面之于几何就像数字之于代数。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，在这个领域仍然有许多基本的开放问题，这些问题非常直观，而且很简单。研究这些问题可以促进纯数学的有益发展，并在科学和技术中得到更广泛的应用。例如，等周不等式有许多应用，因为它与许多其他重要的不等式有联系，包括数学分析中的Sobolev不等式和光谱分析中的Faber-Krahn不等式。此外，表面的刚性可以应用于现代建筑中复杂圆顶的稳定性，而中间轴或切割距离函数轨迹是计算机图形学和数学生物学中感兴趣的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，并引起初学几何的学生的兴趣。该项目将开展一系列活动，包括公开讲座和文章，以促进这些主题。该项目涉及曲线和曲面，以及更广泛的黎曼子流形，涵盖了广泛的主题和工具，包括等距问题、等距嵌入、几何结理论、多面体近似、h原理理论和曲率流。在PI与他的学生和合作者共同进行的这些研究中，一些反复出现的主题是凸性或优化的各种概念以及几何和拓扑概念之间的相互作用，或者子流形的局部与全局性质。更具体地说，一个典型的问题是曲率、固有度规或各种边界条件的限制如何影响曲线或超曲面的整体形状，或允许在低余维空间中等距嵌入该对象。例如，PI最近建立了Zalgaller关于在其凸包中包含单位球的最短封闭曲线的长度和形状的猜想。其他项目包括等距嵌入的刚性，凸多面体的不可折叠性或丢勒猜想，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间轴的切割轨迹的研究。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。

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.