CAREER: Higher Brauer Groups and Topological Azumaya Algebras

职业：高等布劳尔群和拓扑 Azumaya 代数

• 批准号: 1552766
• 负责人: David Antieau
• 金额: $ 45.49万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-06-01 至 2021-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1552766&HistoricalAwards=false
• 关键词: CAREER; Higher; Brauer; Groups; Topological

This award is focused on algebraic geometry and algebraic topology, two areas of modern mathematics. Algebraic geometry has ancient origins 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 number theory. 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 robotics and to the analysis of large data sets, and a recent revolution in the foundations of algebraic topology has broadened its applicability to other fields of pure mathematics. The proposal of the PI will bring the considerable machinery and insight of algebraic topology to bear on several well-known questions and conjectures in algebraic geometry. Some of these questions will be investigated jointly with students as part of undergraduate research projects in the Mathematical Computing Laboratory at UIC, which the PI founded in 2015 with David Dumas and Jan Verschelde. This integration will provide valuable opportunities for students to engage in research and impact ongoing research.The PI will work on several projects at the border of algebraic geometry and algebraic topology. Three projects aim to use various topological methods to understand the Brauer group and Azumaya algebras as well as the role of higher algebraic structures in algebraic geometry. (1) The PI will study (higher) algebraic representatives of étale cohomology classes, generalizing the connection between the Picard and Brauer groups and low-degree cohomology groups. (2) The PI will explore separate applications of motivic homotopy theory and persistent homology to the period-index conjecture with the aim of finding algebraic counterexamples to the conjecture. (3) The PI will explore the Hochschild-Kostant-Rosenberg theorem in characteristic p, specifically focusing on the question of whether or not the local-global spectral sequence for Hochschild homology degenerates at the E_2-page for smooth projective surfaces in characteristic 2.

该奖项的重点是代数几何和代数拓扑，这是现代数学的两个领域。代数几何形状具有古老的起源，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，在各个学科中的核心方程，例如理论物理学，密码学和数理论。另一方面，代数拓扑结构在19世纪发展，旨在研究形状的一般通知，而不是几何形状所研究的形状概念的僵化。它在过去十年中发现了惊人的应用程序，例如对机器人技术和大型数据集的分析，而代数拓扑基础的近期革命扩大了其对其他纯数学领域的适用性。 PI的提议将带来对代数拓扑的考虑机制和洞察力，以实现几种众所周知的问题和猜想的代数几何形状。其中一些问题将与学生共同研究，这是UIC数学计算实验室的本科研究项目的一部分，PI于2015年与David Dumas和Jan Verschelde成立。这种整合将为学生提供研究和影响正在进行的研究的宝贵机会。PI将在代数几何和代数拓扑边界的几个项目上工作。三个项目旨在使用各种拓扑方法来了解Brauer组和Azure代数以及较高代数结构在代数几何形状中的作用。 （1）PI将研究（较高）代表代表étalecoomomology类别的代数，从而推广Picard和Brauer组与低度同胞组之间的联系。 （2）PI将探索基序同义理论的单独应用，并探索对周期 - 索引猜想的持续同源性，目的是找到该概念的代数反例。 （3）PI将在特征P中探索Hochschild-Kostant-Rosenberg定理，特别关注Hochschild同源性的局部 - 全球光谱序列在e_2页上脱离了e_2页，以在特征性2中进行平滑的射弹性表面。