Mathematical Sciences: K-Theory of C*-Algebras, Group Representations, and Coarse Geometry

9500977 Higson这个项目处理Baum-Connes猜想。这个项目的重点不是证明这个猜想的新情况，而是试图更清楚地理解这个猜想与几何和表示理论中可能相关的主题之间的联系。长期目标是利用这一猜想提出的观点，对这些主题提出新的见解。利用这个猜想极其广泛的范围（从诺维科夫猜想延伸到p进群表示理论），人们可能希望在明显不同的领域之间建立新的联系。所提出的工作是朝着这个方向迈出的一些非常温和的第一步。这个项目的一般数学领域在希尔伯特空间上的算子代数理论中有它的基础。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。这些看似抽象的对象有着各种各样的应用。例如，它们在结理论中起着关键作用，而结理论目前正被用于研究DNA的结构。＊＊＊

9500977 Higson This project deals with the Baum-Connes conjecture. The emphasis of the project is not so much on proving new cases of the conjecture as on trying to understand more clearly the connections between the conjecture and the presumably related topics in geometry and representation theory. The long term goal is to use points of view suggested by the conjecture to give new insights into these subjects. Taking advantage of the extremely broad scope of the conjecture (which reaches from the Novikov conjecture to p-adic group representation theory) one might hope to forge new links between apparently dissimilar areas. The work proposed represents some very modest first steps in this direction. The general area of mathematics of this project has its basis in the theory of algebras of operators on Hilbert space. 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 seemingly abstract objects have a surprising 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. ***