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

项目名称：算子代数的K-理论及其在流形拓扑中的应用主要研究者：余国良摘要：研究者提出研究与度量空间和群相关的算子代数的K-理论及其在流形拓扑中的应用。这类算子代数的K-理论是椭圆型微分算子的高指数的容器，在微分几何和流形拓扑中有重要的应用，如具有正标量曲率的黎曼度量的存在性问题，高阶签名同伦不变性的Novikov猜想.所采用的方法包括控制算子K-理论，无限维几乎平坦丛，几何群论。流形是由欧几里得空间粘在一起的空间。流形的例子包括球面和环面。在微分几何中，人们研究流形如何弯曲。例如，一张平面纸的曲率为零，而球面的曲率为正。这就是为什么我们不能把一张纸弯曲成一个球体。 微分几何中的一个基本问题是确定流形何时具有正的数量曲率.数学中的另一个重要问题是流形的分类。利用外科手术理论，高维流形的分类问题可以归结为诺维科夫猜想。某些椭圆型微分算子的K-理论高阶指数可以用来解决正数量曲率问题和Novikov猜想。在这种分析方法中的一个关键步骤是计算的K-理论的高指标，这可以本质上减少到计算的问题与度量空间和群的算子代数的K-理论。

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.