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，尽管有几个世纪的研究和丰富的潜在应用，在这一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数学的有益发展，或导致更广泛的应用在科学和技术。例如，首席研究员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或各种物理框架的稳定性。此外，各种技术的主要研究者建议开发可能是有用的计算机辅助设计，数学物理，和离散微分几何的新兴领域。 最后，这些问题是理想的介绍一般公众到现代数学的令人兴奋的世界，并引起初学者在几何的兴趣。PI的研究曲线和曲面，更一般的黎曼子流形，跨越了广泛的主题，包括等距嵌入，h-原理理论，等周问题，几何纽结理论，多面体逼近，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？PI还考虑了一些相关的问题，包括闭曲线的自链接数或顶点，闭曲面的球面图像，以及保持曲率的符号或大小的子流形的各种变形。其他项目包括凸多面体的展开，以及欧几里得空间中可收缩区域的切割轨迹或中轴的研究。

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 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 mathematics, or lead to wider applications in science and technology. For instance, the Principal Investigator's work on rigidity problem for surfaces may have applications for stability of complicated domes in modern architecture, or various physical frameworks. Further, various techniques which the Principal Investigator is proposing to develop could be useful in computer aided design, mathematical physics, and the emerging field of discrete differential geometry. 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.The PI's research on curves and surfaces, and more generally Riemannian submanifolds, spans a wide range of topics including isometric embeddings, h-principle theory, 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? The PI also considers 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 unfoldings of convex polytopes, and the study of the cut locus or medial axis of contractible regions in Euclidean space.