K-theory of operator algebras and invariants of elliptic operators

在现实生活中，人们试图通过研究与之相关的各种特征量（如形状、大小、温度等）来理解宇宙中的物体。这些特征使人们能够将一个物体与另一个物体区分开来。有些特性在很小的外力的影响下会发生剧烈的变化，而有些特性在这种外力的作用下仍然保持不变。前者不稳定，通常难以测量，而后者更稳定，更容易检测，因此更有用。在数学中，这些稳定的特征被称为不变量。它们为几乎所有数学分支提供了一些最基本的工具。在本项目中，主要研究者将研究一类微分方程的不变量，并将其应用于经典几何和拓扑问题的研究。椭圆算子的指数理论不变量对于理解其基础空间的几何是重要的。著名的Atiyah-Singer指数定理及其非交换几何推广在几何和拓扑学中有着广泛的应用。所有这些指标理论不变量自然存在于某些算子代数的k理论中。主要研究者将使用在算子代数的k理论研究中发展的方法来研究流形和奇异空间上椭圆算子的各种不变量。他特别对次级不变量感兴趣，比如更高的不变量。作者提出利用这些次不变量来研究闭拓扑流形的结构群和给定流形上正标量曲率度量空间的同伦群。首席研究员还计划探索这些问题与诺维科夫猜想和鲍姆-康纳斯猜想之间的联系。

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.