Mathematical Sciences: Geometries and Groups

本研究是关于李氏入射几何和其他与单群有关的几何的特征。一部分是关于以仿射平面为基本平面单位的几何。该研究的一个关键附带问题是李氏入射几何的几何超平面的确定。在这方面，交替k-线性形式的几何表征似乎迫在眉睫。这将表明，所有的超平面的格拉斯曼空间产生于这样的形式。第二部分涉及一个更长远的努力，将李相关系统的特征概括为一个系统的整体。第三部分涉及与单群有关的“非抛物”几何的一些新方向。第四个是关于极空间的子结构。本项目的研究涉及有限维几何与这些几何上的变换群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影响。

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.