Differential Geometry of Curves and Surfaces

AbstractAward：DMS 1308777，首席研究员：穆罕默德Ghomi首席研究员建议继续他的工作理论的曲线和曲面在欧几里德空间，更一般的黎曼子流形的低维或余维。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，而解决这些问题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，几何结理论，多面体近似，以及与真实的代数几何的联系。一些反复出现的主题在这些调查是各种概念的凸性或优化，几何和拓扑概念之间的相互作用，或局部与整体性质的子流形。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，甚至允许该对象嵌入低余维的欧几里得空间。 在这方面的一个基本问题是等距刚性的表面：可以连续变形的光滑闭曲面在欧氏空间，而不改变其内在的度量？ 我们还考虑了一些相关的问题，涉及自链接的数量或顶点的封闭曲线，球面图像的封闭表面，和各种变形的子流形保持的符号或大小的曲率。其他项目包括子流形的平均弦长不等式，真实的代数超曲面的正则性，凸多面体的展开。曲线和曲面对于几何就像数对于代数一样。它们构成了我们视觉感知的基本成分，并激发了深远的数学工具的发展。然而，尽管有几个世纪的纯研究和丰富的潜在应用，在这个领域仍然有许多基本的开放问题，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，或导致更广泛的应用在科学和技术。例如，PI在表面刚性问题上的工作可能会应用于现代建筑中复杂圆顶的稳定性，或各种物理框架。PI提出的多面体逼近技术在计算机辅助设计和离散微分几何的新兴领域可能是有用的。表面高斯映射的相关研究在计算机视觉和光学中可能是有用的，而研究等周问题一直是变分法和数学物理的重要来源。此外，折叠-展开问题有许多应用，从在太空中部署卫星碟形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，如PI在切锥上的工作，它结合了几何测度理论，代数几何和凸分析的概念。最后，这些问题是理想的向公众介绍现代数学的令人兴奋的世界，并引起初学者的兴趣几何。

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.