CAREER: Higher Brauer Groups and Topological Azumaya Algebras

该奖项侧重于代数几何和代数拓扑，现代数学的两个领域。代数几何有着古老的起源，与现实世界的问题有着许多联系。其目标是了解多项式方程解集的几何形状，这些方程在理论物理、密码学和数论等各个学科中具有核心重要性。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，例如机器人技术和大型数据集的分析，最近代数拓扑基础的革命已经将其适用性扩展到纯数学的其他领域。PI的建议将带来相当大的机械和洞察力的代数拓扑承担在几个著名的问题和代数几何。其中一些问题将作为UIC数学计算实验室的本科生研究项目的一部分与学生共同研究，该实验室由PI与大卫大仲马和Jan Verschelde于2015年创立。这种整合将为学生提供宝贵的机会，从事研究和影响正在进行的研究。PI将在代数几何和代数拓扑的边界上的几个项目的工作。三个项目旨在使用各种拓扑方法来理解Brauer群和Azumaya代数以及代数几何中高等代数结构的作用。(1)PI将研究étale上同调类的（高等）代数代表，推广Picard和Brauer群与低度上同调群之间的联系。(2)PI将探索动机同伦理论和持久同源性的周期指数猜想的单独应用，目的是找到该猜想的代数反例。(3)PI将探索特征p中的Hochschild-Kostant-Rosenberg定理，特别关注Hochschild同调的局部-全局谱序列是否在特征2中的光滑射影曲面的E_2-page处退化的问题。

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.