Topological methods for Azumaya algebras

该PI将从事代数几何和代数拓扑边界的几个项目。三个项目旨在使用拓扑方法来理解Brauer群、Azumaya代数和更一般的方案上的torsor。(1) PI将研究Jackowski、McClure和Oliver关于复代数群的分类空间之间映射的基本结果在多大程度上可以推广到这些分类空间的有限近似。这一问题的进展将有助于解决一系列关于复杂代数群的向量何时从方案的一般点扩展到整个方案的问题。在低维中，PI和Ben Williams利用这一问题的早期进展解决了Auslander和Goldman关于非分支除法代数中Azumaya极大阶的存在性的老问题，其中Azumaya极大阶的存在性存在纯粹的拓扑障碍。(2)通过不同的中心子群计算特殊线性群的分类空间的周群和奇异上同调。这是Vezzosi和Vistoli在特殊情况下做的。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，控制着与某些在Brauer群研究中具有重要基础意义的体相关的特征类。这些计算将对第一个项目和接下来的项目直接有用。(3) PI和Ben Williams先前提出了拓扑周期指数问题，并建立了第一个结果。他们将继续这项研究，特别是因为它涉及到代数周期指数猜想。特别是，他们在低维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。将这个想法贯彻到底是第一组项目的主要愿望。第四个项目旨在继续在高等范畴论和经典代数几何之间建立一座桥梁，将前者的强大技术应用于派生范畴的算术中的各种问题。例如，PI正在开发一个使用更高范畴论的工具箱，一旦已知这些空间的分裂形式上存在某些特殊对象，该工具箱将允许对Panin的投影齐次空间的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。它的目标是理解多项式方程解集的几何，这些方程在理论物理、密码学和天气等动力系统建模等各个学科中都具有中心重要性。另一方面，代数拓扑学是在19世纪发展起来的，目的是研究形状的一般概念，比几何中研究的形状概念更不死板。在过去十年中，它发现了一些引人注目的应用，例如对计算机视觉和癌症研究中出现的大型数据集的分析，经常发现更传统的数据分析方法无法发现的模式。PI的提议将引入代数拓扑的大量机制和洞察力来承担代数几何中的几个问题，这些问题已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.