Operator K-theory and its applications to geometry and topology

研究者和他的学生计划研究算子代数的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-群是椭圆型微分算子的高指数的容器，在微分几何和流形拓扑中有重要的应用。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阶签名同伦不变性的诺维科夫猜想。研究者还打算应用非交换几何方法来研究流形的回路空间的分析。在这个项目中使用的方法包括Banach空间上的群作用，Banach空间的粗嵌入，扩张器的几何，椭圆算子的无穷维指数理论。非对易几何（英语：Noncommutative geometry）是一种数学理论，专门设计来处理“几何对象”，其坐标不对易，但在数学和物理中自然发生。在过去十年左右的时间里，借助于非对易几何的新思想，经典几何和拓扑学中的一些长期存在的问题的解决取得了一系列进展。 高次指数理论是非交换几何与经典几何、拓扑之间的桥梁。研究者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。

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.