Mathematical Sciences: The Geometry of Configurations

数学科学：构型几何

• 批准号: 9322475
• 负责人: Jacob Goodman
• 金额: --
• 依托单位: CUNY City College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-01 至 1997-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9322475&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Geometry; Configurations

9322475 Goodman The investigator and Richard Pollack (Courant Institute of the Mathematical Sciences, NYU) will continue their study of the interplay between the geometry and the topology of Euclidean spaces by extending their recent results (1) on a generalization to Grassmann manifolds of the standard convexity structure of Euclidean spaces, (2) on the classification of ordered configurations and their generalizations, and (3) on related geometric and combinatorial problems. Building on their past work on allowable sequences and order types of configurations, as well as on ideas coming out of their solution of the Vincensini problem for hyperplane transversals to families of convex bodies and their recent extension of convexity to Grassmann manifolds, Goodman and Pollack plan to continue their work on configurations, arrangements of flats, and geometric transversal theory. One of the most elegant things that Goodman and Pollack have done recently is to extend convexity, as noted above, from sets of points in Euclidean spaces to sets of higher dimensional flats, for example lines in ordinary three-space. One obtains in this way "convex" sets of lines such as all the lines through a particular point, or all the lines in a particular plane, and there is nothing especially exciting about these. However, one also obtains such sets as all the lines which are surrounded by a particular hyperboloid of one sheet together with one set of rulings lying on the surface of the hyperboloid. Since the suggestive formalism of convexity theory becomes available in this new and broader context, it leads to some surprising new geometric conjectures, proofs, and theorems. ***

9322475 Goodman 研究员和 Richard Pollack（纽约大学库朗数学科学研究所）将继续研究欧几里得空间的几何和拓扑之间的相互作用，方法是扩展他们最近的结果（1）对欧几里得空间标准凸性结构的格拉斯曼流形的推广，（2）有序配置的分类及其推广，以及 (3)有关几何和组合问题。古德曼和波拉克基于他们过去在构型的允许序列和阶类型方面的工作，以及解决凸体族超平面横截的 Vincensini 问题以及最近将凸性扩展到格拉斯曼流形的想法，计划继续他们在构型、平面排列和几何横截理论方面的工作。古德曼和波拉克最近所做的最优雅的事情之一就是将凸性从欧几里得空间中的点集扩展到更高维的平面集，例如普通三空间中的线。人们以这种方式获得“凸”线组，例如通过特定点的所有线，或特定平面中的所有线，并且这些没有什么特别令人兴奋的。然而，人们还获得这样的集合：被一张纸的特定双曲面包围的所有线以及位于双曲面表面上的一组标线。由于凸性理论的暗示形式主义在这个新的、更广泛的背景下变得可用，它导致了一些令人惊讶的新几何猜想、证明和定理。 ***