Geometry of Curves and Surfaces

曲线和曲面之于几何就像数字之于代数。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，尽管经过了几个世纪的研究和大量潜在的应用，在这一领域仍然存在许多基本的开放性问题，这些问题非常直观和简单。研究这些问题可能会刺激数学的有益发展，或者导致科学和技术的更广泛应用。例如，首席研究员对表面刚性问题的研究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。此外，首席研究员提议开发的各种技术可能在计算机辅助设计、数学物理和新兴的离散微分几何领域有用。最后，这些问题是向公众介绍令人兴奋的现代数学世界的理想选择，也是激发初学几何的学生的兴趣的理想选择。PI对曲线和曲面以及更广泛的黎曼子流形的研究涵盖了广泛的主题，包括等距嵌入、h原理理论、等距问题、几何结理论、多面体逼近以及与实际代数几何的联系。在这些研究中，一些反复出现的主题是凸性或优化的各种概念，几何和拓扑概念之间的相互作用，或子流形的局部与全局性质。更具体地说，一个典型的问题是曲率、固有度规或各种边界条件的限制如何影响曲线或超曲面的整体形状，甚至允许在低余维的欧几里得空间中嵌入该对象。该领域的一个基本问题是曲面的等距刚性问题：能否在不改变其固有度规的情况下连续变形欧几里得空间中的光滑封闭曲面？PI还考虑了一些相关的问题，包括闭合曲线的自连接数或顶点，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内轴线的研究。

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.