Differential Geometry of Curves and Surfaces

曲线曲面的微分几何

• 批准号: 1308777
• 负责人: Mohammad Ghomi
• 金额: $ 17.6万
• 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: 2013
• 资助国家: 美国
• 起止时间: 2013-09-01 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1308777&HistoricalAwards=false
• 关键词: Differential; Geometry; Curves; Surfaces

AbstractAward: DMS 1308777, Principal Investigator: Mohammad Ghomi The principal investigator proposes to continue his work on the theory of curves and surfaces in Euclidean space, and more generally on Riemannian submanifolds of low dimension or codimension. He specializes in applying contemporary methods such as curvature flows and h-principle theory to solve classical problems which often have simple intuitive statements, while their solutions may require sophisticated techniques. The PI's research in this area spans a wide range of topics including isometric embeddings, isoperimetric problems, geometric knot theory, polyhedral approximations, and connections with real algebraic geometry. Some recurring themes throughout these investigations 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 problem is how restrictions on curvature, intrinsic metric, or various boundary conditions, influence the global shape of a curve or a hypersurface, or even allow an embedding of that object in a Euclidean space of low codimension. A fundamental problem in this area is that of isometric rigidity of surfaces: can one continuously deform a smooth closed surface in Euclidean space without changing its intrinsic metric? We also consider a number of related problems involving the self-linking number or vertices of closed curves, spherical images of closed surfaces, and various deformations of submanifolds which preserve the sign or magnitude of the curvature. Other projects include inequalities for mean chord lengths of submanifolds, regularity of real algebraic hypersurfaces, and unfoldings of convex polyhedra.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 despite centuries of pure study, and an abundance of potential applications, there are still many fundamental open problems in this area which are strikingly intuitive and elementary to state. Studying these problems may stimulate useful developments in pure mathematics, or lead to wider applications in science and technology. For instance, the PI's work on rigidity problem for surfaces may have applications for stability of complicated domes in modern architecture, or various physical frameworks. The polyhedral approximation techniques which the PI is proposing could be useful in computer aided design, and the emerging field of discrete differential geometry. The related studies of the Gauss maps of surfaces could be useful in computer vision and optics, while studying isoperimetric problems has been a significant source of enrichment in calculus of variations and mathematical physics. Further, folding-unfolding problems have numerous applications ranging from deployment of satellite dishes in space to implantation of stents in human arteries. Another impact of the proposed activity would be development of connections between various fields, as in the PI's work on tangent cones, which combines concepts from geometric measure theory, algebraic geometry, and convex analysis. Finally, these problems are ideal for introducing the general public to the exciting world of modern day mathematics, and arousing the interest of beginning students in Geometry.

摘要奖：DMS 1308777，首席研究员：Mohammad Ghomi主要研究员建议继续他在欧氏空间中曲线和曲面理论方面的工作，更广泛地说，他在低维或余维的黎曼子流形上的工作。他擅长应用现代方法，如曲率流和h-原理理论来解决经典问题，这些问题通常有简单的直观陈述，而它们的解决方案可能需要复杂的技术。PI在这一领域的研究涵盖了广泛的主题，包括等距嵌入、等周问题、几何纽结理论、多面体逼近以及与实代数几何的联系。在这些研究中，一些反复出现的主题是各种凸性或最优化概念，以及几何和拓扑概念之间的相互作用，或者子流形的局部和全局性质。更具体地说，一个典型的问题是对曲率、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，甚至允许将对象嵌入到低余维欧氏空间中。这一领域的一个基本问题是曲面的等距刚性问题：在欧氏空间中，人们是否可以在不改变其固有度量的情况下连续地变形光滑的闭曲面？我们还考虑了一些相关的问题，涉及闭合曲线的自链接数或顶点，闭合曲面的球面像，以及保持曲率符号或大小的子流形的各种变形。其他项目包括子流形的平均弦长的不等式，实代数超曲面的正则性，以及凸多面体的展开。曲线和曲面对于几何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，并启发了影响深远的数学工具的发展。然而，尽管经过了几个世纪的纯粹研究和丰富的潜在应用，这一领域仍然有许多基本的开放问题，这些问题对于国家来说是惊人的直观和基本的。研究这些问题可能会刺激纯数学的有益发展，或者导致更广泛的科学技术应用。例如，PI关于曲面刚度问题的工作可能适用于现代建筑中复杂穹顶的稳定性，或各种物理框架。PI提出的多面体逼近技术可用于计算机辅助设计和离散微分几何的新兴领域。曲面的高斯映射的相关研究在计算机视觉和光学中是有用的，而研究等周问题一直是变分和数学物理中丰富的重要来源。此外，折叠-展开问题有许多应用，从在太空部署卫星天线到在人体动脉植入支架。拟议活动的另一个影响是发展不同领域之间的联系，如PI关于切锥的工作，该工作结合了几何测度论、代数几何和凸分析的概念。最后，这些问题是向普通大众介绍现代数学令人兴奋的世界，并激发几何初学者的兴趣的理想选择。