The Topology, Geometry and Algebra of Projective Linear Groups

Azumaya代数是矩阵代数的扭曲形式。由于矩阵本身是无处不在的，因此这些对象在不同的数学环境中存在并值得注意。***在域上的代数中，Azumaya代数是在扩展到某个（可分离的）代数扩展后与矩阵代数同构的代数。这里的典型例子是Hamilton在实域上的四元数，扩展到复域后，它变成了2x2复矩阵的同构。域上的Azumaya代数是具有经典重要性的中心简单代数。****在拓扑上下文中，空间X上的Azumaya代数是由X参数化的代数族，它们局部同构于nxn复矩阵的平凡族。***介于上述两种情况之间的是Azumaya代数在变种上的情况，因为变种跨越了拓扑空间和代数在域上的世界。******该项目旨在将适用于拓扑空间上的Azumaya代数的拓扑工具用于与算术或代数问题相关的特定情况，从而提供一系列示例和反例。这些例子将有助于改进Azumaya代数在代数背景下的研究，通过反驳或支持现有的猜想，并通过提供一个对高维变量及其函数场的行为的指导，这可能很难直接使用代数技术来解决。***它还旨在将在拓扑上下文中进行的计算转置到代数上下文中，通过以太上同调和A1或动机同伦理论，并通过这种方法在代数理论中建立正结果**

An Azumaya algebra is a twisted form of a matrix algebra. Since matrices themselves are ubiquitous, these objects exist and are noteworthy in different mathematical contexts.***In the context of algebras over a field, an Azumaya algebra is simply an algebra that becomes isomorphic to a matrix algebra upon extension to some (separable) algebraic extension. The prototypical example here is Hamilton's Quaternions over the real field, which becomes isomorphic to 2x2 complex matrices upon extension to the complex field. Azumaya algebras over fields are the Central Simple Algebras of classical importance.****In the topological context, an Azumaya algebra over a space X is a family of algebras parametrized by X that is locally isomorphic to the trivial family of nxn complex matrices.*** Intermediate between the two contexts above is that of Azumaya algebras over varieties, since varieties straddle the worlds of topological spaces and of algebras over fields.******This project aims to bring topological tools applicable to Azumaya algebras over topological spaces to bear in specific cases that relate to arithmetic or algebraic questions, and consequently to furnish a stream of examples and counterexamples. These examples will help to refine the study of Azumaya algebras in the algebraic context by either disproving or supporting existing conjectures, and by providing a guide to the behaviour one expects over high-dimensional varieties and their function fields, which can be hard to address directly using algebraic techniques.***It also aims to transpose the calculations made in the topological context to an algebraic context, via etale cohomology and A1 or motivic homotopy theories, and to establish positive results in the algebraic theory by this method.**