CAREER: Higher Brauer Groups and Topological Azumaya Algebras

该奖项集中在现代数学的两个领域--代数几何和代数拓扑学。代数几何有着古老的起源，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，这些方程在不同的学科中非常重要，如理论物理、密码学和数论。另一方面，代数拓扑学是在19世纪较新发展起来的，它的目的是研究形状的一般概念，比几何中研究的形状概念不那么僵硬。在过去的十年里，它得到了惊人的应用，例如在机器人学和大型数据集的分析中，最近在代数拓扑基础上的一场革命将它的适用性扩大到了纯数学的其他领域。PI的提出将把代数拓扑学的相当大的机械性和洞察力应用于代数几何中的几个众所周知的问题和猜想。其中一些问题将作为UIC数学计算实验室本科生研究项目的一部分与学生共同调查，该实验室是PI于2015年与David Dumas和Jan Verschelde共同创立的。这一整合将为学生提供宝贵的研究机会，并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开展几个项目。三个项目旨在使用不同的拓扑方法来理解Brauer群和Azumaya代数，以及更高代数结构在代数几何中的作用。(1)PI将研究上同调类的(更高)代数表示，推广Picard和Brauer群与低次上同调群之间的联系。(2)PI将探索Motivic同伦理论和持久同调在周期指数猜想中的不同应用，目的是找到与猜想相反的代数反例。(3)PI将探索特征p中的Hochschild-Kostant-Rosenberg定理，特别是特征2中光滑射影曲面的Hochschild同调的局部-整体谱序列是否在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.