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

主要研究者：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。这些研究通常涉及一些凸性的概念，并包括以下类别：(i)光照超表面上的阴影（或阴影）及其应用几何变分问题；（ii）不产生平行或相交切线的流形嵌入（完全倾斜嵌入）及其与二次超曲面和非奇异双线性映射的关系；（iii）具有边界的局部凸超曲面的整体性质，包括与Monge-Ampere方程的连接，以及与负曲面对偶的一个新的凸壳性质；（iv）空间曲线的某些变形（展开）及其在结能极值和畸变研究中的应用。曲线和曲面之于几何，正如数字之于代数。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。例如，PI在处理照明表面上的阴影方面的工作部分是由对肥皂膜的研究激发的，并且与计算机视觉（“阴影形状”问题）有关。此外，对结能的研究可能会对dna的研究产生兴趣。然而，尽管有大量潜在的应用和几个世纪的纯粹研究，在子流形几何和拓扑中仍然有许多开放的问题，这些问题非常直观和基本。PI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绝佳工具。借助计算机研讨会、课程、研讨会和本项目中提出的系列讲座，PI计划向尽可能广泛的受众传达几何问题的美丽和令人兴奋之处。

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.