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

在经典几何中，人们研究的是坐标可交换的几何物体。非交换几何是一种数学理论，专门用于处理坐标不交换但在数学和物理中自然存在的“几何对象”。在过去十年左右的时间里，在非交换几何新思想的帮助下，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。高指标理论是连接非交换几何与经典几何和拓扑学的桥梁。首席研究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问题。某些算子代数的k群是椭圆微分算子的高指标的容器，在微分几何和流形拓扑问题中有重要的应用。这类应用的例子包括估计所有具有正标量曲率的黎曼度量的模空间的大小，以及有关流形的刚性或非刚性的问题。本课题拟将控制算子k理论、动态复杂性和有限可嵌入技术应用于巴拿赫空间，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（如流形的循环空间）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).