K-theory of operator algebras and invariants of elliptic operators

算子代数的 K 理论和椭圆算子不变量

• 批准号: 1500823
• 负责人: Zhizhang Xie
• 金额: $ 12万
• 依托单位: Texas A&M University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1500823&HistoricalAwards=false
• 关键词: theory; operator; algebras; invariants; elliptic

In real life, one seeks to understand objects in the universe by studying various characteristic quantities associated with them, such as shape, size, temperature, and so on. These characteristics allow one to distinguish one object from another. Some characteristics may change drastically under the influence of very small external forces, and some remain resistant to change under such forces. The former are unstable and usually difficult to measure, while the latter are more stable and easier to detect, hence much more useful. In mathematics, those stable characteristics are called invariants. They provide some of the most fundamental tools in almost all branches of mathematics. In this project, the principal investigator will study a certain class of invariants of differential equations and apply them to study problems in classical geometry and topology. Index theoretical invariants of elliptic operators are important for understanding the geometry of their underlying spaces. The famous Atiyah-Singer index theorem and its noncommutative geometric generalizations have many applications to geometry and topology. All these index theoretical invariants live naturally in the K-theory of certain operator algebras. The principal investigator will use methods developed in the studies of K-theory of operator algebras to investigate various invariants of elliptic operators on manifolds and spaces with singularities. In particular, he is interested in secondary invariants such as the higher rho invariant. The principal investigator proposes to use these secondary invariants to study the structure group of a closed topological manifold and the homotopy groups of the space of positive scalar curvature metrics on a given manifold. The principal investigator also plans to explore the connections of these problems to the Novikov conjecture and the Baum-Connes conjecture.

在现实生活中，人们试图通过研究与形状，大小，温度等相关的各种特征数量来理解宇宙中的物体。这些特征允许一个人将一个对象与另一个对象区分开。 在很小的外力的影响下，某些特征可能会发生巨大变化，而有些特征在这种力量下仍能抵抗变化。前者是不稳定的，通常很难测量，而后者更稳定，更易于检测，因此更有用。在数学中，这些稳定的特征称为不变性。它们在几乎所有数学分支中提供了一些最基本的工具。在该项目中，首席研究人员将研究一定类别的微分方程不变，并将其应用于古典几何和拓扑中的问题。椭圆算子的索引理论不变性对于理解其潜在空间的几何形状很重要。著名的Atiyah-Singer索引定理及其非共同的几何概括具有许多用于几何和拓扑的应用。所有这些索引理论不变性自然生活在某些操作员代数的K理论中。首席研究者将使用在操作员代数研究中开发的方法来研究各种椭圆运算符的各种不变性，这些方法在具有奇异性的歧管和空间上。特别是，他对较高的Rho不变性等次要不变性感兴趣。主要研究者建议使用这些二级不变式研究封闭拓扑歧管的结构组和给定歧管上正标度曲率指标空间的同型组。首席研究人员还计划探讨这些问题与诺维科夫猜想和鲍姆 - 康涅狄格州的猜想的联系。