度量空间的粗几何为研究Baum-Connes高指标问题提供了许多有效的方法。本项目主要研究粗几何中的若干热点问题及其在高指标问题中的应用，具体内容包括：研究度量空间的几种渐近分解性质的关系及保持性问题；研究可粗嵌入条件下度量空间的Roe代数的刚性问题；与群的正则表示相类比，考虑Roe代数的正则表示问题；对带有等距、恰当群作用的度量空间，研究非等变粗嵌入情形下的等变高指标问题。这些问题在几何群论、算子代数和非交换几何等领域有重要应用。

The coarse geometry of metric spaces provides many powerful methods to study Baum-Connes higher index problems. In this project, we will study many interested problems in coarse geometry and some applications for higher index problems. The main contents are as follows: We shall study the relations of several asymptotic decomposition properties for metric spaces, and some permanence properties for them. We will study the rigidity of the Roe algebras for the coarse embeddable metric spaces. Compared with the regular representation of groups, we are going to study the regular representation problems for Roe algebras. For metric spaces with isometrical and proper actions, we will study the equivariant higher index theory in the case of non-equivariant coarse embedding. These topics will have many important applications in geometric group theory, operator algebras and non-commutative geometry.