Tits theory of spherical buildings gives a uniform geometric context to study all finite simple groups (except the alternating groups and the sporadic simple groups) and simple algebraic groups. More generally, the theory of buildings is central to the Lie theory associated with infinite root systems. These structures are `built ofâ€™ two basic objects: Coxeter complexes and Moufang generalized polygons. The generalized polygons (which includes projective planes) are rank 2 geometries (incidence geometries with 2 kind of objects - points and lines - and an incidence relation among them) whose classification is fundament, difficult and perhaps impossible.

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