CAREER: Classical Problems in Differential Geometry, Topology, and Convexity

职业：微分几何、拓扑和凸性的经典问题

• 批准号: 0332333
• 负责人: Mohammad Ghomi
• 金额: $ 40万
• 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: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0332333&HistoricalAwards=false
• 关键词: CAREER; Classical; Problems; Differential; Geometry

AbstractAward: DMS-0332333Principal Investigator: Mohammad GhomiThe principal investigator is interested primarily in theinterplay between the geometry and topology of submanifolds,including low dimensional problems in Euclidean space (curves andsurfaces). These investigations often involve some notion ofconvexity, and include the following categories: (i) Shadows (orshades) on illuminated hypersurfaces, and their application togeometric variational problems; (ii) Embeddings of manifolds inEuclidean space without creating parallel or intersecting tangentlines (totally skew embeddings), and their relation to quadrichypersurfaces and nonsingular bilinear maps; (iii) Globalproperties of locally convex hypersurfaces with boundary,including connections with Monge-Ampere equations, and a newconvex hull property which is dual to that of negatively curvedsurfaces; (iv) Certain deformations of space curves (unfoldings),and their application to study of extremals of knot energies anddistortion.Curves and surfaces are to geometry what numbers are toalgebra. They form the basic ingredients of our visualperception, and inspire the development of far reachingmathematical tools. For instance, those aspects of the PI's workdealing with shadows on illuminated surfaces are motivated inpart by a study of soap films, and have connections to computervision (the ``shape from shading" problems). Further, theinvestigations on knot energies may be of interest in studyingDNA. Yet, despite an abundance of potential applications andcenturies of pure study, there are still numerous open problemsin submanifold geometry and topology which are strikinglyintuitive and elementary to state. The PI believes thatadvertising these problems at an early stage is an excellent toolfor sparking the interest of students in mathematicalresearch. With the aid of computer workshops, courses, seminars,and the lecture series proposed in this project, the PI plans tocommunicate the beauty and excitement of geometric problems to aswide an audience as possible.

摘要奖项：DMS-0332333 首席研究员：Mohammad Ghomi 首席研究员主要对子流形的几何和拓扑之间的相互作用感兴趣，包括欧几里得空间（曲线和曲面）中的低维问题。这些研究通常涉及一些凸性的概念，并包括以下类别：（i）照明超曲面上的阴影（或阴影）及其在几何变分问题中的应用； (ii) 欧几里得空间中流形的嵌入而不创建平行或相交的切线（完全倾斜嵌入），以及它们与四边曲面和非奇异双线性映射的关系； (iii) 有边界的局部凸超曲面的全局性质，包括与 Monge-Ampere 方程的联系，以及与负曲面对偶的新凸包性质； (iv) 空间曲线的某些变形（展开），以及它们在研究结能量和扭曲的极值中的应用。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。例如，PI 处理照明表面阴影的工作部分是受到肥皂膜研究的启发，并且与计算机视觉（“阴影形状”问题）有关。此外，对结能量的研究可能也很有趣然而，尽管有大量的潜在应用和几个世纪的纯粹研究，子流形几何和拓扑仍然存在许多悬而未决的问题，这些问题是非常直观和基本的。相信在早期阶段宣传这些问题是激发学生对数学研究兴趣的绝佳工具。借助本项目提出的计算机研讨会、课程、研讨会和系列讲座，PI 计划传播几何的美丽和兴奋。尽可能扩大受众范围的问题。