K-theory of Operator Algebras and Its Applications to Topology of Manifolds

算子代数的K理论及其在流形拓扑中的应用

• 批准号: 0101561
• 负责人: Guoliang Yu
• 金额: $ 28.82万
• 依托单位: Vanderbilt University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-06-01 至 2007-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0101561&HistoricalAwards=false
• 关键词: theory; Operator; Algebras; Its; Applications

Project Title: K-theory of operator algebras and its applications to topology of manifolds Principal Investigator: Guoliang Yu Abstract: The investigator proposes to study the K-theory of operator algebras associated to metric spaces and groups, and its applications to topology of manifolds. The K-theory of such operator 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 positivescalar curvature, the Novikov conjecture on homotopy invariance ofhigher signatures. The methods to be employed include controlled operator K-theory, infinite dimensional almost flat bundles, and geometric group theory. Manifolds are spaces glued together by Euclidean spaces. Examples of manifolds include spheres and tori. In differential geometry one studies how manifolds are curved. For example a flat piece of paper has zero curvature while the sphere has positive curvature. This is why we can not bend a piece of paper into a sphere. A basicproblem in differential geometry is to determine when a manifold can have positive scalar curvature. Another important problem in mathematics is the classification of manifolds. By surgery theory the classification problem for higher dimensional manifolds can be essentially reduced to the Novikov conjecture. The K-theoretic higher indices of certain elliptic differential operators can be usedto attack the positive scalar curvature problem and the Novikov conjecture. A key step in this analytic approach is the computation of the K-theoretic higher indices, which can be essentiallyreduced to the problem of computing the K-theory of operator algebras associated to metric spaces and groups.

项目标题：K-算子代数理论及其在流形拓扑学中的应用主要研究人员：余国良摘要：研究与度量空间和群相关的算子代数K-理论及其在流形拓扑学中的应用。这类算子代数的K-理论是高指数椭圆微分算子的容器，在微分几何和流形拓扑问题中有着重要的应用，如具有正数量曲率的黎曼度量的存在性问题，关于高签名同伦不变性的Novikov猜想等。所采用的方法包括受控算子K-理论、无限维几乎平坦丛和几何群论。流形是由欧几里得空间粘合在一起的空间。流形的示例包括球体和圆环。在微分几何中，人们研究流形是如何弯曲的。例如，一张平面纸的曲率为零，而球体的曲率为正值。这就是为什么我们不能把一张纸弯成球体的原因。微分几何中的一个基本问题是确定流形何时可以有正的标量曲率。数学中的另一个重要问题是流形的分类。根据外科理论，高维流形的分类问题本质上可以归结为Novikov猜想。某些椭圆微分算子的K-理论高指数可以用来攻击正标量曲率问题和Novikov猜想。这种分析方法的一个关键步骤是K-理论高指标的计算，它本质上可以归结为计算与度量空间和群相关的算子代数的K-理论问题。