Applications of torsors in algebra and Lie theory

理解对称性，以及它是如何和为什么在自然界中产生的，在数学和物理学中都很重要。 测量对称性的数学对象称为群。李群，以挪威人的名字命名 在19世纪末发现并首次研究这些天体的数学家索菲斯·李，出现了 自然而然地成为物理学中许多理论的基本对称性。 20世纪中期见证了代数群的诞生，捕捉到Lie群精神的对象仍然是 更具普遍性。当代数学中许多最引人注目的结果都利用了代数 组。它们的起源可以追溯到Weil、Chvalley、Borel、Serre、Grothendieck、Demazure的基本工作，他们在20世纪40年代和50年代在代数几何的背景下系统地发展了Lie和Cartan的思想。在接下来的几十年里，代数群论被用来统一处理代数的几个关键概念，包括二次型和厄米特型、中心单代数、对合代数和非结合代数的理论。 该研究项目的中心是理解代数群本身的本质，以及它们在数学和物理学的几个领域中的应用。给定一个代数群，就可以将不同的几何对象联系在一起。特别感兴趣的是扭子和射影齐次簇。代数的最新结果表明，这一数学领域的未来进展取决于我们对它们几何性质的理解有多深入。我的项目是研究这些物体的性质。作为应用，我们计划将我们的结果应用于物理中出现的无限维李代数的分类。

Understanding symmetry, and how and why it arises in nature, is important in both Mathematics and Physics. The mathematical objects that measure symmetry are called Groups. Lie groups, named after the Norwegian mathematician Sophus Lie who discovered and first studied these objects late in the 19th century, arise naturally as the underlying symmetry of many theories in Physics. The mid 20th century saw the birth of Algebraic Groups, objects that capture the spirit of Lie groups yet are much more universal. Many of the most striking results in contemporary Mathematics make use of algebraic groups. Their origins go back the fundamental work of Weil, Chevalley, Borel, Serre, Grothendieck, Demazure, who in the 1940s and 50s systematically developed the ideas of Lie and Cartan in the context of algebraic geometry. Over the course of subsequent decades the theory of algebraic groups has been used to give a unified treatment of several key ideas of algebra, including the theories of quadratic and hermitian forms, central simple algebras, algebras with involutions and non-associated algebras. The research project centers on understanding the very nature of algebraic groups themselves, and their applications to several areas of Mathematics, as well as Physics. Given an algebraic group one can associate different geometric objects. Of special interest are torsors and projective homogeneous varieties. Recent results in algebra show that future progress in this area of mathematics depends how deeply we can understand their geometric properties. My project concerns on studying properties of these objects. As an application we plan to apply our results to classification of infinite dimensional Lie algebras which appear in Physics.