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

数学科学：C* 代数的 K 理论、群表示和粗略几何

• 批准号: 9500977
• 负责人: Nigel Higson
• 金额: $ 10.1万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-01 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9500977&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Theory; Algebras; Group

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. ***

9500977 Higson 该项目涉及 Baum-Connes 猜想。该项目的重点并不是证明猜想的新案例，而是试图更清楚地理解猜想与几何和表示论中可能相关的主题之间的联系。长期目标是利用猜想提出的观点为这些主题提供新的见解。利用该猜想的极其广泛的范围（从诺维科夫猜想到 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。拟议的工作代表了朝这个方向迈出的一些非常温和的第一步。 该项目的一般数学领域以希尔伯特空间上的算子代数理论为基础。运算符可以被认为是复数的有限或无限矩阵。特殊类型的运算符通常放在代数中，自然称为运算符代数。这些看似抽象的物体有着令人惊讶的多种应用。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来研究 DNA 的结构。 ***