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

摘要奖：DMS-0332333首席研究员：Mohammad Ghomi主要研究子流形的几何和拓扑之间的相互作用，包括欧氏空间(曲线和曲面)中的低维问题。这些研究往往涉及一些凸性的概念，包括以下几类：(I)照亮超曲面上的阴影(或阴影)及其应用；(Ii)不产生平行或相交切线的流形在欧氏空间中的嵌入(完全斜嵌入)，以及它们与四次超曲面和非奇异双线性映射的关系；(Iii)局部凸超曲面的全局性质，包括与Monge-Ampere方程的联系，以及一个新的凸壳性质，它对偶于负曲曲面的性质；(Iv)空间曲线的某些变形(展开)，及其在研究结点能量和变形极值方面的应用。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，并启发了深远的数学工具的发展。例如，PI处理照亮表面上的阴影的工作的这些方面部分是由一项肥皂片研究激发的，并与计算机视觉有关(从阴影中形成形状的问题)。此外，对结能的研究可能会对DNA的研究产生兴趣。然而，尽管有丰富的潜在应用和数百年的纯研究，子流形几何和拓扑学中仍有许多公开问题，这些问题对状态来说是非常直观和初级的。国际数学联合会认为，在早期阶段宣传这些问题是激发学生对数学研究兴趣的一个很好的工具。借助计算机工作坊、课程、研讨会和本项目中提出的系列讲座，PI计划将几何问题的美丽和兴奋传达给尽可能广泛的受众。

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.