Mathematical Sciences: Geometries and Groups

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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.