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项目负责人：余国良摘要：拟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标的集合，在微分几何和流形拓扑中有重要的应用，如正标量曲率黎曼度量的存在性问题、高特征同伦不变性的诺维科夫猜想等。所采用的方法包括控制算子k理论、无限维几乎平坦束和几何群论。流形是由欧几里德空间粘合在一起的空间。流形的例子包括球面和环面。在微分几何中，人们研究流形是如何弯曲的。例如，一张平坦的纸的曲率为零，而球体的曲率为正。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可以具有正的标量曲率。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诺维科夫猜想。某些椭圆型微分算子的k理论高指标可以用来攻击正标量曲率问题和诺维科夫猜想。这种分析方法的一个关键步骤是k理论高指标的计算，它本质上可以简化为计算与度量空间和群相关的算子代数的k理论问题。

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.