Mathematical Sciences: Operator Algebras and Noncommutative Topology

本课题是关于算子代数理论的数学研究。这门学科与拓扑学关系的起点是这样一种情况，即交换C*-代数正是作为局部紧拓扑空间上的连续复值函数的代数出现的。(后一个代数包含了足够的信息，可以完全恢复底层空间。)其一般程序是将思想和结构从拓扑的交换环境扩展到算子代数的非对易领域。要研究的具体问题是：C*-代数范畴上的同调和上同调理论的存在性和性质；K-理论中的环结构；奇异空间上的算子场；以及非稳定的K-理论，特别是对于单C*-代数。

This project is mathematical research in the theory of operator algebras. The starting point for the relationship of this subject to topology is the circumstance that the commutative C*-algebras are precisely those that arise as the algebra of continuous complex-valued functions on a locally compact topological space. (The latter algebra contains enough information to recover completely the underlying space.) The general program is to extend ideas and constructions from the commutative setting of topology to the noncommutative realm of operator algebras. Specific topics to be investigated are: existence and properties of homology and cohomology theories on the category of C*-algebras; ring structure in K-theory; operator fields over singular spaces; and nonstable K-theory, especially for simple C*-algebras.