Semisimple groups and cohomological invariants

这个建议涉及三个问题，半单代数群在任意域。 第一个问题是对这样一个群的上同调不变量进行分类。 由于J-P. Serre，M.罗斯特，和PI，和新的案件将是非常感兴趣的。 第二个问题是著名的Kneser-Tits问题，它将组计划的简单性概念与抽象组的简单性概念联系起来。 作者：J. Dieudonne，G. Prasad，M.S. Raghunathan等，已经将这个问题简化为一个简短的特定案例列表。 第三，半单群作为一个簇的结构是什么？ 是理性的吗？ 它是R-平凡的吗？ PI将试图完成证明猜想，即数域上的单连通群是R-平凡的。 用来解决这些问题的工具包括伽罗瓦和平坦上同调，伽罗瓦下降，根系和J. Tits的例外群的构造。半单群族包括熟悉的矩阵群，如特殊线性群和特殊正交群。 这些群出现在数学的许多领域，并且可以被看作是19世纪早期发展起来的线性代数的重要产物，现在教授给本科生。 在1800年代后期，通过Sophus Lie著名的李群一般理论，李群成为数学兴趣的突出对象。 在代数中，半单群的概念统一了历史上不同的研究领域。 例如，它将物理学家在20世纪初发现的约旦代数与巴比伦人研究的二次型联系起来。

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.