Mathematical Sciences: Geometries and Groups

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This project concerns characterizations of geometries related to the Lie incidence geometries in terms of local hypotheses on points and lines. The first part concerns geometries whose basic planar unit is an affine plane. The second part concerns extending the point-line characterization theory to affine and other non-spherical Lie incidence systems. 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 theory of finite groups, for algebraic coding theory, and for finite geometry.