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 首席研究员建议继续研究欧几里得空间中的曲线和曲面理论，更广泛地研究低维或余维的黎曼子流形。他擅长应用曲率流和 h 原理理论等当代方法来解决经典问题，这些问题通常具有简单的直观陈述，而其解决方案可能需要复杂的技术。 PI 在该领域的研究涵盖了广泛的主题，包括等距嵌入、等周问题、几何结理论、多面体近似以及与实代数几何的联系。这些研究中反复出现的一些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属性。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，或者甚至允许将该对象嵌入到低余维的欧几里德空间中。 该领域的一个基本问题是曲面的等距刚性：能否在不改变其内在度量的情况下使欧几里得空间中的平滑闭合曲面连续变形？ 我们还考虑了许多相关问题，涉及闭合曲线的自链接数或顶点、闭合曲面的球面图像以及保留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学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，尽管经过几个世纪的纯粹研究和大量的潜在应用，该领域仍然存在许多基本的开放问题，这些问题非常直观且易于陈述。 研究这些问题可能会刺激纯数学的有益发展，或者导致科学和技术的更广泛应用。例如，PI 在表面刚性问题方面的工作可能适用于现代建筑中复杂圆顶或各种物理框架的稳定性。 PI 提出的多面体近似技术可用于计算机辅助设计和新兴的离散微分几何领域。曲面高斯图的相关研究可用于计算机视觉和光学，而研究等周问题是丰富变分法和数学物理的重要来源。此外，折叠展开问题具有多种应用，从在太空中部署卫星天线到在人体动脉中植入支架。拟议活动的另一个影响是发展各个领域之间的联系，例如 PI 关于切锥的工作，它结合了几何测度论、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，并激发初学几何的学生的兴趣。