Topological methods for Azumaya algebras

Azumaya 代数的拓扑方法

• 批准号: 1461847
• 负责人: David Antieau
• 金额: $ 4.53万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1461847&HistoricalAwards=false
• 关键词: Topological; methods; Azumaya; algebras

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.

PI 将参与代数几何和代数拓扑边界的多个项目。三个项目旨在使用拓扑方法来理解 Brauer 群、Azumaya 代数以及更一般的图式上的 Torors。 (1) PI 将研究 Jackowski、McClure 和 Oliver 在复杂代数群的分类空间之间的映射上的基础结果可以在多大程度上扩展到这些分类空间的有限近似。这个问题的进展将能够解决一系列关于复杂代数群的 Torors 何时从方案的通用点扩展到整个方案的问题。在低维中，PI 和 Ben Williams 已经利用这个问题的早期进展来解决 Auslander 和 Goldman 的一个老问题，即无分支代数中 Azumaya 最大阶的存在性，其中发现这些 Azumaya 最大阶的存在存在纯粹的拓扑障碍。 (2) PI 将致力于计算由各种中心子群组成的特殊线性群的分类空间的 Chow 群和奇异上同调。 Vezzosi 和 Vistoli 在特殊情况下就这样做了。然而，大多数应用需要更大的通用性。这些 Chow 群是代数几何中的基本对象，控制着与布劳尔群研究中具有根本重要性的某些扭转相关的特征类。计算将直接用于第一个项目和后续项目。 (3) PI 和 Ben Williams 先前提出了拓扑周期指数问题并得出了第一个结果。他们将继续这项研究，特别是与代数周期指数猜想相关的研究。特别是，他们在低维方面的结果提出了一种反驳周期指数猜想的方法，这将是一个根本性的进步。遵循这一想法并得出结论是第一组项目的主要愿望。第四个项目旨在继续在高级范畴论和经典代数几何之间架起一座桥梁，将前者的强大技术应用于派生范畴算术中的各种问题。例如，PI 正在开发一个使用更高范畴论的工具箱，一旦知道这些空间的分割形式上某些特殊对象的存在，就可以对 Panin 射影齐次空间 K 理论的计算进行纯派生范畴证明。PI 提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很多联系。其目标是理解多项式方程解集的几何形状，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具有核心重要性。另一方面，代数拓扑是在 19 世纪才发展起来的，旨在研究形状的一般概念，不像几何学中研究的形状概念那么严格。它在过去十年中得到了惊人的应用，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找到的模式。 PI 的提议将带来代数拓扑的大量机制和洞察力，以解决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。