Mathematical Sciences: Geometries and Groups

数学科学：几何与群

• 批准号: 9001693
• 负责人: Ernest Shult
• 金额: $ 6.19万
• 依托单位: Kansas State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1990
• 资助国家: 美国
• 起止时间: 1990-06-01 至 1992-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9001693&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Geometries; Groups

This research is concerned with characterizing geometries related to the Lie incidence geometries and other geometries related to simple groups. One part concerns geometries whose basic planar unit is an affine plane. A critical side issue of that study has been the determination of the geometric hyperplanes of the Lie incidence geometries. In this connection a geometric characterization of alternating k-linear forms seems imminent. This would show that all hyperplanes of Grassman spaces arise from such forms. A second part concerns a more long-range effort to generalize the characterizations of Lie incidence systems into a systematic whole. The third concerns some new directions into "non-parabolic" geometries related to simple groups. The fourth concerns substructures of polar spaces. The research in this project involves the interplay between finite dimensional geometry and the actions of groups of transformations on these geometries. This work has implications for the structure of finite groups, for algebraic coding theory, and for finite geometry.

这项研究涉及刻划与Lie关联几何有关的几何以及与单群有关的其他几何。其中一部分涉及基本平面单位是仿射平面的几何。这项研究的一个关键边际问题是确定Lie入射几何的几何超平面。在这方面，交替的k-线性形式的几何特征似乎迫在眉睫。这将表明Grassman空间的所有超平面都是由这样的形式产生的。第二部分涉及更长远的努力，将Lie关联系统的特征概括为一个系统的整体。第三个涉及到与单群有关的“非抛物”几何的一些新方向。第四，关于极地空间的子结构。这个项目的研究涉及有限维几何和变换群在这些几何上的作用之间的相互作用。这项工作对有限群的结构、代数编码理论和有限几何都有意义。