Operator K-theory and its applications to geometry and topology

这位研究人员和他的学生计划学习算子代数的K-理论及其在流形几何和拓扑中的应用。某些算子代数的K-群是高指数椭圆微分算子的容器，在微分几何和流形拓扑问题中有着重要的应用。这类应用的例子包括具有正数量曲率的黎曼度量的存在性问题和关于高签名同伦不变性的Novikov猜想。作者还打算应用非对易几何方法来研究流形的环空间上的分析。本课题所采用的方法包括Banach空间上的群作用、Banach空间的粗嵌入、扩张器的几何以及椭圆算子的无穷维索引理论。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”的坐标是不交换的，但在数学和物理中都是自然发生的。在过去的十年左右，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、拓扑学之间的桥梁。这位研究人员和他的学生计划应用非对易几何方法来研究微分几何和拓扑学中的问题。

The investigator and his students plan to study the K-theory of operator algebras and its applications to geometry and topology of manifolds. 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 topology of manifolds. Examples of such applications include the existence problem for Riemannian metrics with positive scalar curvature and the Novikov conjecture on homotopy invariance of higher signatures. The investigator also intends to apply noncommutative geometry methods to study analysis on loop spaces of manifolds. The methods to be employed in this project include group actions on Banach spaces, coarse embedding into Banach spaces, geometry of expanders, and infinite dimensional index theory of elliptic operators.In classic 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 both 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 classic geometry and topology. Higher index theory serves as a bridge between noncommutative geometry and classic geometry and topology. The investigator and his students plan to apply noncommutative geometry methods to study problems in differential geometry and topology.