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这个项目涉及Baum-Connes的猜想。该项目的重点不是证明猜想的新案例，而是试图更清楚地了解猜想与几何学和表示理论中可能相关的主题之间的联系。长期目标是使用猜想建议的观点，以对这些主题提供新的见解。利用猜想的极为广泛的范围（从诺维科夫的猜想到p-adic群体代表理论），人们可能希望在显然不同的地区之间建立新的联系。提出的工作代表了这个方向上一些非常适度的第一步。 该项目数学的一般领域在希尔伯特领域的运营商代数理论中具有其基础。可以将操作员视为复数的有限或无限矩阵。特殊类型的操作员通常将其放在代数中，自然称为操作员代数。这些看似抽象的对象具有令人惊讶的应用。例如，它们在结理论中起着关键作用，而这反过来又被用来研究DNA的结构。 ***