Mathematical Sciences: Noncommutative Geometric Methods in Operator Algebras

9500886计用非对易微分几何的方法研究与离散群和C*-动力系统有关的算子代数。研究的重点是构造相应的交叉积C*-代数的光滑子代数和稠密子代数，并对光滑子代数进行相关分析。所使用的主要技巧是Connes关于结合代数的循环理论和研究者介绍的关于离散群的Schwartz上同调。这些项目的完成有助于阐明算子代数、代数拓扑和微分几何的深层性质，例如，进一步理解非交换Toeplitz指标理论；解决Baum-Connes映射、Novikov猜想和Kadison-Kaplansky猜想对快速衰减离散群的有理内射性。这项研究位于几何、分析和代数的交界处。Connes的一种现代方法使用非对易代数结构来研究几何不变量。非对易结构的一个很好的例子是固定大小的方阵集合上的乘法结构。这个项目将使用Connes机制来阐明其他重要的非交换代数，称为算子代数，它可以被视为无限矩阵的集合。结果将对代数、几何和分析产生影响。***

9500886 Ji This project involves the study of operator algebras associated with discrete groups and C*-dynamical systems by the method of noncommutative differential geometry. Central to this study is the construction of smooth and dense subalgebras of the corresponding crossed product C*-algebras and the related analysis to the smooth subalgebras. The main techniques employed are Connes' cyclic theory for associative algebras and the Schwartz cohomology for discrete groups introduced by the investigator. Completion of the projects described in this proposal help elucidate deep properties of operator algebras, algebraic topology and differential geometry, such as, a further understanding of noncommutative Toeplitz index theory; a solution to the rational injectivity of the Baum-Connes map, the Novikov conjecture and the Kadison-Kaplansky conjecture for the class of rapidly decaying discrete groups. This research lies at the interface between geometry, analysis and algebra. A modern approach of Connes studies geometrical invariants using non-commutative algebraic structures. A good example of a noncommutative structure is the multiplicative structure on the collection of square matrices of fixed size. This project will employ Connes machinery to elucidate other important non-commutative algebras called operator algebras which can be viewed as collections of infinite matrices. The results will impact algebra, geometry and analysis. ***