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

在经典几何中，人们研究坐标互换的几何对象。非对易几何是一门专门为处理“几何对象”而设计的数学理论，这些“几何对象”的坐标是不可交换的，但在数学和物理中是自然出现的。在过去的十多年里，在非对易几何的新思想的帮助下，在解决经典几何和拓扑学中长期存在的问题方面取得了一系列进展。高指数理论是非对易几何与经典几何和拓扑学之间的桥梁。主要研究人员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中的问题。某些算子代数的K-群是高指数椭圆微分算子的容器，在微分几何和流形的拓扑中有重要的应用。这种应用的例子包括估计所有具有正数量曲率的黎曼度量的模空间的大小，以及关于流形的刚性或非刚性的问题。本文的主要研究内容是利用受控算子K-理论、动态复杂性和有限可嵌入性等技巧研究算子代数的K-理论和高指标理论。他还打算应用非对易几何方法来研究无限维空间(例如，流形的环空间)上的分析。

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).