Geometry of Curves and Surfaces

曲线和曲面之于几何，就像数之于代数。它们构成了我们视觉感知的基本成分，并激发了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悬而未决的问题，这些问题是非常直观和基本的。 研究这些问题可能会促进纯数学的有益发展，并导致更广泛的应用在科学和技术。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不等式和谱分析中的Faber-Krahn不等式。此外，表面的刚度可以应用于现代建筑中复杂穹顶的稳定性，而中轴或距离函数的切割轨迹是形状识别中的重要工具，这在计算机图形学和数学生物学中是感兴趣的。这些问题非常适合向公众介绍令人兴奋的现代数学世界，并引起初学者对几何的兴趣。本项目将通过一系列的活动，包括公开讲座和文章来推广这些主题。本项目关注曲线和曲面，更广泛地说是黎曼子流形，涵盖广泛的主题和工具，包括等周问题，等距嵌入，几何纽结理论，多面体近似，h-原理理论和曲率流。在这些研究中，PI与他的学生和合作者共同进行的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，或者局部与全局性质的子流形。更具体地说，一个典型的问题是如何限制曲率，内在度量，或各种边界条件影响曲线或超曲面的整体形状或允许等距嵌入该对象在低余维空间。例如，PI最近建立了Zalgaller猜想的长度和形状的最短封闭曲线，包含单位球在其凸船体。其他项目包括等距嵌入的刚性、凸多面体或丢勒猜想的可展开性、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，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估而被认为值得支持。

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.