K-theory of Operator Algebras and Its Applications to Geometry and Topology

在经典几何中，人们研究坐标可交换的几何对象。非对易几何（英语：Noncommutative geometry）是一种数学理论，专门设计来处理“几何对象”，其坐标不对易，但在数学和物理中自然发生。在过去十年左右的时间里，在非对易几何新思想的帮助下，经典几何和拓扑学中长期存在的问题的解决取得了一系列进展。高次指数理论是非对易几何与经典几何、拓扑之间的桥梁。主要研究者和他的学生计划应用非交换几何方法来研究微分几何和拓扑学问题。某些算子代数的K群是椭圆微分算子的高指数的容器，在微分几何和拓扑学问题中有重要应用。流形的拓扑。这样的应用的例子包括估计的大小的模空间的所有黎曼度量与正标量曲率和问题的刚性或nonrigidity的一个流形。主要研究者打算将受控算子K理论，动态复杂性和有限嵌入性技术应用到Banach空间中，以研究算子代数的K理论和高指标理论。他还打算应用非交换几何方法来研究无限维空间的分析（例如，流形的循环空间）。

In classical geometry, one studies geometric objects whose coordinates commute. Noncommutative geometry is a mathematical theory specifically designed to handle "geometric objects" whose coordinates do not commute but which do occur naturally in mathematics and physics. In the last decade or so, with the help of new ideas from noncommutative geometry, there have been a series of advances towards the solutions of long-standing problems in classical geometry and topology. Higher index theory serves as a bridge between noncommutative geometry and classical geometry and topology. The principal investigator and his students plan to apply noncommutative geometry methods to study problems in differential geometry and topology.The K-groups of certain operator algebras are receptacles of higher indices of elliptic differential operators and have important applications to problems in differential geometry and in the topology of manifolds. Examples of such applications include estimation of the size of the moduli space of all Riemannian metrics with positive scalar curvature and questions concerning the rigidity or nonrigidity of a manifold. The principal investigator intends to apply the techniques of controlled operator K-theory, dynamic complexity, and finite embeddability into Banach spaces to study K-theory of operator algebras and higher index theory. He also intends to apply noncommutative geometry methods to study analysis on infinite-dimensional spaces (e.g., loop spaces of manifolds).