Finite linearly representable geometries and symmetry

有限几何与对称理论有着深刻的数学联系。有限几何和对称的进步在历史上导致了代数、计算和理论物理等不同领域的进步。该项目旨在利用它们的对称性来描述基本的几何物体，称为“投影平面”和“广义多边形”。为了实现这些目标，必须发展经典对称群中某些元素与几何平面和多边形之间的概念联系。这些特定元素的密度对概率几何算法有重要的应用。

Finite geometry has profound mathematical connections to the theory of symmetry. Advances in finite geometry and in symmetry have historically led to advances in diverse areas such as algebra, computing, and theoretical physics. The project aims to characterise basic geometric objects called "projective planes'' and "generalised polygons'' using their symmetry properties. To achieve these aims, conceptual links between certain elements in classical symmetry groups and geometric planes and polygons must be developed. The density of these certain elements has important applications to probabilistic geometric algorithms.