This proposal concerns three problems regarding semisimple algebraic groups over an arbitrary field. The first problem is to classify the cohomological invariants of such a group. A classification is only known in a few cases, due to J-P. Serre, M. Rost, and the PI, and new cases would be of great interest. The second problem is the famous Kneser-Tits Problem relating the notions of simplicity for group schemes with the notion of simplicity for abstract groups. Previous work by J. Dieudonne, G. Prasad, M.S. Raghunathan, etc., has reduced this problem to a short list of particular cases. Third, what is the structure of a semisimple group as a variety? Is it rational? Is it even R-trivial? The PI will attempt to complete the proof of the conjecture that simply connected groups over number fields are R-trivial. The tools used to attack these problems include Galois and flat cohomology, Galois descent, root systems, and J. Tits's constructions of exceptional groups.
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 in the early 1900s -- with quadratic forms, which were studied by the Babylonians.
