This project is concerned with a number of interrelated questions about the cohomology and modular representations of semisimple algebraic groups, along with their Lie algebras and related finite subgroups of Lie type. Special attention will be given to the cohomology of line bundles on flag varieties, where there are many open questions about module structure and vanishing behavior. For the finite groups, the research will center on injective modules and on the decomposition behavior mod p of ordinary representations. A group is an algebraic structure with a multiplication defined on it. These structures occur commonly in many areas of mathematics, as well as, chemistry, physics and computer science. Finite groups may be viewed as algebraic sets of transformations of vector spaces and through these representations, it is possible to determine their properties and structure.
