Topological methods for Azumaya algebras

PI将在代数几何和代数拓扑学的边缘从事几个项目。三个项目的目标是使用拓扑方法来理解Brauer群、Azumaya代数以及更一般的格式上的扭子。(1)PI将研究Jackowski、McClure和Oliver关于复代数群的分类空间之间的映射的基本结果可以在多大程度上扩展到对这些分类空间的有限逼近。在这一问题上的进展将使一系列问题的解决成为可能，如复代数群的扭矩何时从方案的通用点扩展到整个方案。在低维方面，PI和Ben Williams利用这一问题的早期进展解决了Auslander和Goldman关于未分枝除代数中Azumaya极大序存在的一个老问题，其中揭示了存在对这些Azumaya极大序存在纯粹的拓扑障碍。(2)PI将用于计算特殊线性群的Chow群和各种中心子群的分类空间的奇异上同调。Vezzosi和Vistoli在特殊情况下已经做到了这一点。然而，大多数应用程序需要更大的通用性。这些Chow群是代数几何中的基本对象，控制着与在Brauer群研究中具有基本重要性的某些扭量相关的特征类。这些计算将直接对第一个项目和下一个项目有用。(3)PI和Ben Williams以前提出了拓扑期指标问题，并建立了第一个结果。他们将继续这一研究，特别是与代数周期指数猜想有关的研究。特别是，他们在低维方面的结果提出了一种方法来反驳周期指数猜想，这将是一个根本性的进步。遵循这一想法得出结论是第一套项目的主要愿望。第四个项目旨在继续在高级范畴理论和经典代数几何之间架起一座桥梁，将前者强大的技术应用于派生范畴算法中的各种问题。例如，PI正在开发一个使用更高范畴理论的工具箱，一旦知道这些空间的分裂形式上存在某些例外对象，它将允许对Panin的射影齐次空间的K理论的计算进行纯粹的派生范畴证明。PI建议在代数几何和代数拓扑这两个现代数学领域开展工作。代数几何是一门古老的学科，与现实世界的问题有许多联系。它的目标是了解多项式方程解集的几何，在不同学科中至关重要的方程，如理论物理、密码学和像天气这样的动力系统的建模。另一方面，代数拓扑学是在19世纪较新发展起来的，它的目的是研究形状的一般概念，比几何中研究的形状概念不那么僵硬。在过去十年里，它得到了惊人的应用，例如，在计算机视觉和癌症研究中出现的大型数据集的分析中，它经常发现更传统的数据分析方法无法找到的模式。PI的提出将把相当大的代数拓扑学的机械和洞察力应用到代数几何中的几个问题上，这些问题已经被社区确定为最重要的问题之一。

The PI will engage in several projects at the border of algebraic geometry and algebraic topology. Three projects aim to use topological methods to understand the Brauer group, Azumaya algebras, and more generally torsors on schemes. (1) The PI will study the extent to which the foundational results of Jackowski, McClure, and Oliver on maps between classifying spaces of complex algebraic groups can be extended to finite approximations to these classifying spaces. Progress on this problem will enable the solution of a host of problems about when torsors for complex algebraic groups extend from the generic point of a scheme to the entire scheme. In low dimensions, early progress on this problem has been used by the PI and Ben Williams to settle an old question of Auslander and Goldman on the existence of Azumaya maximal orders in unramified division algebras, where it transpires that there are purely topological obstructions to the existence of these Azumaya maximal orders. (2) The PI will work toward computing the Chow groups and singular cohomology of the classifying spaces of special linear groups by various central subgroups. This has been done in special cases by Vezzosi and Vistoli. However, greater generality is needed for most applications. These Chow groups are fundamental objects in algebraic geometry, controlling the characteristic classes associated to certain torsors of fundamental importance in the study of the Brauer group. The computations will be directly useful to the first project, and to the following project. (3) The PI and Ben Williams previously formulated the topological period-index problem and established first results. They will continue this study, especially as it relates to the algebraic period-index conjecture. In particular, their results in low dimensions suggest a method for disproving the period-index conjecture, which would be a fundamental advance. Following this idea to its conclusion is the major aspiration of the first set of projects. A fourth project aims to continue to build a bridge between higher category theory and classical algebraic geometry, bringing the formidable techniques of the former to bear on various questions in the arithmetic of derived categories. For example, the PI is developing a toolbox using higher category theory that will allow a purely derived-category proof of Panin's computations of the K-theory of projective homogeneous spaces, once the existence of certain exceptional objects on the split forms of these spaces is known.The PI proposes work in algebraic geometry and algebraic topology, two areas of modern mathematics. Algebraic geometry is an ancient subject with many connections to real-world problems. Its goal is to understand the geometry of solutions sets of polynomial equations, equations of central importance in various disciplines, such as theoretical physics, cryptography, and the modeling of dynamical systems like weather. Algebraic topology on the other hand developed more recently, in the 19th century, and aims to study a general notion of shape, less rigid than the idea of shape studied in geometry. It has found striking applications in the last decade, for instance to the analysis of large data sets that occur in computer vision and cancer research, frequently finding patterns that more traditional methods of data analysis fail to find. The proposal of the PI will bring the considerable machinery and insight of algebraic topology to bear on several questions in algebraic geometry which have been identified by the community as among the most important.