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.

AbstractAward：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。这些研究往往涉及一些概念的凸性，并包括以下几类：（i）阴影（orshades）在照明超曲面，及其应用几何变分问题;（ii）嵌入的流形在欧氏空间，而不产生平行或相交切线（全斜嵌入）及其与二次超曲面和非奇异双线性映射的关系;（iii）带边界局部凸超曲面的整体性质，包括与Monge-Ampere方程的联系，以及与负曲面对偶的一个新的凸船体性质;（iv）空间曲线的某些变形（展开）及其在研究纽结能和畸变的极值中的应用。曲线和曲面之于几何，就像数之于代数一样。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。例如，PI的工作中处理照明表面上阴影的那些方面部分是由肥皂膜的研究激发的，并且与计算机视觉（“阴影中的形状”问题）有关。 此外，结能的研究可能对DNA的研究有意义。然而，尽管有丰富的潜在应用和几个世纪的纯研究，仍然有许多悬而未决的问题，在子流形几何和拓扑是惊人的直观和基本的状态。 PI认为，在早期阶段宣传这些问题是激发学生对药物研究兴趣的一个很好的工具。借助计算机讲习班，课程，研讨会，并在这个项目中提出的系列讲座，PI计划沟通的美丽和兴奋的几何问题，尽可能广泛的观众。