Mathematical Sciences: Connes Invariant X(M) and Related Invariants

9501515 Chen本研究涉及到Alain Connes在von Neumann代数上的一个不变量（称为chi不变量）的研究。这个不变量与群的上同调之间的联系将被探讨。特别有趣的是完美群的上同调理论。他还将使用连续群的上同调理论。目的之一是澄清与chi不变量相关的代数不变量之间的关系。另一个目标是实现适当的冯诺依曼代数的不变量。最后，研究了chi不变量在双模上的推广。这个不变量在子因子理论中也起着重要的作用。这个项目的理论基础是希尔伯特空间算子的代数。运算符可以被认为是有限或无限的复数矩阵。特殊类型的运算符通常放在一个代数中，自然地称为运算符代数。这些抽象对象有各种各样的应用。例如，它们在结理论中发挥着关键作用，而结理论目前正被用于研究DNA的结构，它们在非交换几何中具有重要的基础意义，而非交换几何在物理学中正变得越来越重要。＊＊＊

9501515 Chen This research involves the study of an invariant (called chi invariant) of Alain Connes on von Neumann algebras. The connection between this invariant and the cohomology of groups will be explored. Of special interest is the cohomology theory of perfect groups. He will also use cohomology theory of continuous groups. One objective is to clarify the relations between the algebraic invariants associated with the chi invariant. Another objective is to realize the invariants for appropriate von Neumann algebras. Finally, a study will be made of the generalization of the chi invariant to bimodules. This invariant also plays an important role in the theory of subfactors. The basis of the theory underlying this project is algebras of Hilbert space operators. Operators can be thought of as finite or infinite matrices of complex numbers. Special types of operators are often put together in an algebra, naturally called an operator algebra. These abstract objects have a variety of applications. For example, they play a key role in knot theory, which in turn is currently being used to study the structure of DNA, and they are of fundamental importance in noncommutative geometry, which is becoming increasingly important in physics. ***