Topological methods for Azumaya algebras

Azumaya 代数的拓扑方法

• 批准号: 1358832
• 负责人: David Antieau
• 金额: $ 10.13万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2014-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1358832&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将在代数几何和代数拓扑边界的几个项目中进行几个项目。三个项目旨在使用拓扑方法来了解Brauer群体，Azumaya代数以及方案上更一般的Torsors。 （1）PI将研究Jackowski，McClure和Oliver对复杂代数基团的分类空间之间的地图的基础结果的程度，可以扩展到对这些分类空间的有限近似值。关于此问题的进展将使有关复杂代数组的Torsors何时从方案的通用点延伸到整个方案的问题。在低维度中，PI和Ben Williams已经使用了此问题的早期进展，以解决澳大利亚和高盛在未遭受的分区代数中存在Azumaya Maximal订单的一个旧问题，在该代数中，它彻底使这些Azumaya Maximal Maximal订单存在纯粹的拓扑障碍。 （2）PI将致力于计算各种中央亚组的特殊线性群体分类空间的ChOW群体和奇异共同体。这是在Vezzosi和Vistoli的特殊情况下进行的。但是，大多数应用程序都需要更大的一般性。这些食物组是代数几何形状中的基本对象，控制着与Brauer组研究中与某些基本重要性相关的特征类别。这些计算将直接对第一个项目和以下项目有用。 （3）PI和Ben Williams先前提出了拓扑时期 - 指数问题，并确定了首先结果。他们将继续这项研究，尤其是与代数周期索引猜想有关的。特别是，它们在低维度中的结果提出了一种反驳时期索引猜想的方法，这将是一个基本进步。遵循这个想法得出的结论是第一组项目的主要愿望。第四个项目旨在继续在较高类别理论和经典代数几何形状之间建立桥梁，这使前者的强大技术在派生类别的算术中涉及各种问题。例如，PI正在使用高级类别理论开发工具箱，该工具箱将允许panin对投射均匀空间的K理论的计算进行纯粹派生的类别证明，一旦知道这些空间的某些特殊对象的存在。代数几何形状是一个古老的主题，与现实世界中的问题有许多联系。它的目标是了解多项式方程的解决方案集的几何形状，在各种学科中的核心方程，例如理论物理，密码学以及天气等动态系统的建模。另一方面，代数拓扑是在19世纪发展的，旨在研究形状的一般概念，比几何形状所研究的形状概念不如几何。它在过去十年中发现了引人注目的应用程序，例如对计算机视觉和癌症研究中发生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的提议将使代数拓扑的大量机制和见解在代数几何学中的几个问题上都持续存在，这些几何形状已被社区确定为最重要的问题之一。