K-theory of Operator Algebras and its Applications to Geometry and Topology

算子代数的K理论及其在几何和拓扑中的应用

• 批准号: 0600216
• 负责人: Guoliang Yu
• 金额: $ 39.43万
• 依托单位: Vanderbilt University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-01 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0600216&HistoricalAwards=false
• 关键词: theory; Operator; Algebras; its; Applications

The investigator proposes to study the K-theory of group C*-algebras and its applications to geometry and topology of manifolds. The K-theory of group C*-algebras are receptacles of higher indices of elliptic differential operators and have important applications to problems in differential geometry and topology of manifolds such as the existence problem for Riemannian metrics with positive scalar curvature and the Novikov conjecture on homotopy invariance of higher signatures. The methods to be employed include group actions on Banach spaces, uniform embedding into Banach spaces, and infinite dimensional index theory of elliptic operators.Manifolds are spaces glued together by Euclidean spaces. Roughly speaking, these are spaces on which one can do calculus. Examples of manifolds inlcude spheres and tori. Manifold theory plays an important role in mathematics and physics. A fundamental problem in manifold theory is the classification of manifolds. By surgery theory, the classification problem for higher dimensional manifolds is essentially reduced to the Novikov conjecture. The investigator plans to study the Novikov conjecture using methods from infinite dimensional analysis such as noncommutative geometry. The investigator also intends to apply noncommutative geometry methods to study analysis on loop spaces of manifolds.

研究C*-代数群的k理论及其在流形几何和拓扑中的应用。群C*-代数的k理论是椭圆微分算子的高指标的载体，在微分几何和流形拓扑问题中有重要的应用，如正标量曲率黎曼度量的存在性问题和高特征同伦不变性的Novikov猜想。所采用的方法包括Banach空间上的群作用、Banach空间的一致嵌入以及椭圆算子的无限维指标理论。流形是由欧几里德空间粘合在一起的空间。粗略地说，这些空间可以用来做微积分。流形的例子包括球面和环面。流形理论在数学和物理中占有重要地位。流形理论中的一个基本问题是流形的分类。根据外科理论，高维流形的分类问题本质上归结为诺维科夫猜想。研究人员计划利用非交换几何等无限维分析方法来研究诺维科夫猜想。研究者还打算应用非交换几何方法来研究流形环空间的分析。