This project is concerned with the theory of modular representations of the finite groups of Lie type, in which case the natural geometry is the Tits building. Earlier work has established techniques for describing important modules via generators and relations determined by the geometry. This research will extend the class of modules to which the methods apply. These techniques will be used to study the modular representations of the highly exceptional sporadic simple groups. The study of groups arises from consideration of the symmetry of physical and theoretical objects. The expression of these symmetry transformations by means of matrices leads to group representation theory. This project will develop representation theory by exploiting techniques from combinatorics and discrete geometry.
