Linear algebraic groups and related topics in algebra

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The PI proposes to exploit recently developed techniques in the study of projective homogeneous varieties to pursue several questions in the theory of semisimple algebraic groups over an arbitrary field, their Galois cohomology, and their cohomological invariants. The new techniques have already produced some extremely strong results and the investigator proposes to push them further. Previous work on these topics has had applications to other areas of mathematics and to physics, and further such applications are expected.The family of semisimple groups includes familiar matrix groups like special linear and special orthogonal groups. These groups appear in many areas of mathematics, and may be viewed as an essential outgrowth of the linear algebra developed in the early 1800s and now taught to undergraduates. The groups became prominent objects of mathematical interest in the late 1800s via Sophus Lie's famous general theory of Lie groups. In algebra, the notion of semisimple group unifies various historically distinct areas of study. For example, it connects Jordan algebras -- discovered by physicists -- with quadratic forms and division algebras.