Topological methods for Azumaya algebras

PI将在代数几何和代数拓扑的边界从事几个项目。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torsors方案。(1)PI将研究Jackowski、麦克卢尔和奥利弗关于复代数群的分类空间之间的映射的基本结果在多大程度上可以扩展到这些分类空间的有限近似。在这个问题上的进展将使一系列问题的解决方案时torsors复代数群从一般点的计划，整个计划。在低维，早期的进展，这个问题已被用于PI和本威廉姆斯解决一个老问题的Auslander和高盛的存在阿祖马亚最大的订单在unramified司代数，在那里它透露，有纯粹的拓扑障碍的存在，这些阿祖马亚最大的订单。(2)PI将致力于计算Chow群和奇异上同调的分类空间的特殊线性群的各种中心子群。Vezzosi和Vistoli在特殊情况下已经这样做了。然而，对于大多数应用来说，需要更大的通用性。这些周群是代数几何中的基本对象，控制与某些在Brauer群研究中具有根本重要性的torsors相关的特征类。这些计算将直接用于第一个项目，并用于下一个项目。(3)PI和本威廉姆斯以前制定的拓扑周期指数问题，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。第一批项目的主要愿望是将这一想法贯彻到底。第四个项目的目的是继续建立一个桥梁之间的高级范畴理论和经典代数几何，使强大的技术，前者承担各种问题的算术派生类别。例如，PI正在开发一个工具箱，使用更高的范畴理论，将允许一个纯粹的导出范畴证明Panin的计算的K-理论的射影齐性空间，一旦存在某些例外对象的分裂形式，这些空间是已知的。PI提出的工作代数几何和代数拓扑，两个领域的现代数学。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的解集的几何，这些方程在各个学科中具有核心重要性，例如理论物理学，密码学以及天气等动力系统的建模。另一方面，代数拓扑学是在世纪发展起来的，它的目的是研究形状的一般概念，比几何学中研究形状的概念更不严格。在过去的十年中，它已经发现了惊人的应用，例如分析计算机视觉和癌症研究中发生的大型数据集，经常发现传统数据分析方法无法发现的模式。该提案的PI将带来相当大的机械和洞察力的代数拓扑承担几个问题的代数几何已确定由社会作为最重要的。

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.