Linear algebraic groups and related topics in algebra

PI建议利用最近发展的研究射影齐次变的技术来研究任意域上的半简单代数群理论中的几个问题，它们的伽罗瓦上同调，以及它们的上同调不变量。新技术已经产生了一些非常有力的结果，研究人员建议进一步推动它们。以前关于这些主题的工作已经应用到数学和物理的其他领域，并且期望进一步的应用。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，它将物理学家发现的约旦代数与二次型和除法代数联系起来。

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.