K-Theory of Operator Algebras and Its Applications

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研究“几何对象”而设计的数学理论，这些“几何对象”的坐标不交换，但在数学和物理中是自然出现的。在过去的十几年里，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很大的进展。K-理论、高指数理论和次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-群是椭圆微分算子的高指数和次指数不变量的容器，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。主要研究人员和他的学生计划应用定量算子K-理论和动态复杂性的技术来研究算子代数的K-理论、高指标理论和二级指标理论。主要研究人员和他的学生还打算应用定量技术来研究代数K-理论中的同构猜想。

In classical geometry, one studies geometric objects whose coordinates commute. Noncommutative geometry is a mathematical theory specifically designed to study "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, great advances have been made towards the solutions of long-standing problems in classical geometry and topology. K-theory, higher index theory, and secondary index theory serve as bridges 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 and secondary index invariants 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 and his students plan to apply the techniques of quantitative operator K-theory and dynamic complexity to study K-theory of operator algebras, higher index theory, and secondary index theory. The principal investigator and his students also intend to apply quantitative techniques to study the isomorphism conjectures in algebraic K-theory.