K-Theory of Operator Algebras and Its Applications

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，专门用于研究坐标不交换但在数学和物理中自然存在的“几何物体”。近十年来，在非交换几何新思想的帮助下，经典几何和拓扑学在解决长期存在的问题方面取得了很大进展。k理论、高指标理论和次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和次指标不变量的容器，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。主要研究者和他的学生计划应用定量算子k理论和动态复杂性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。主要研究者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.