Mathematical Sciences: Geometries and Groups

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This research is concerned with showing that hyperplanes of the major embeddable Lie incidence geometries all arise from embeddings. The cases that remain are dual polar spaces, geometries associated with exceptional Lie groups, and certain exceptional fields for the half-spin geometries. A class of geometries crucial to forming general theorems about Veldkamp spaces will also be considered. The principal investigator will continue work on ım!-ovoids, which are generalizations of both ovoids and spreads 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.