Semisimple groups and cohomological invariants

本文提出了关于任意域上的半简单代数群的三个问题。第一个问题是对这类群的上同不变量进行分类。由于J-P，只有在少数情况下才知道分类。塞雷，罗斯特先生，私家侦探，还有一些新的案子会引起人们极大的兴趣。第二个问题是著名的Kneser-Tits问题，它将群方案的简单性概念与抽象群的简单性概念联系起来。J. Dieudonne， G. Prasad， M.S. Raghunathan等人之前的工作已经将这个问题简化为一个简短的特定案例列表。第三，作为品种的半单群的结构是怎样的？这是理性的吗？它是r平凡的吗？PI将尝试完成关于数域上的单连通群是r平凡的猜想的证明。用于解决这些问题的工具包括伽罗瓦和平面上同论、伽罗瓦下降、根系统和J. Tits的特殊群的构造。半单群族包括我们所熟悉的矩阵群，如特殊线性群和特殊正交群。这些群体出现在数学的许多领域，并且可以被视为19世纪早期发展起来的线性代数的重要产物，现在被教授给本科生。在19世纪晚期，通过李绪斯著名的李群的一般理论，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20世纪初由物理学家发现的约旦代数与巴比伦人研究过的二次型联系起来。

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.