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

9800869 Yu建议研究Roe代数和群C*代数的K-理论。Roe代数和群C*代数的K-理论都是椭圆算子高指数的容器，并在微分几何和拓扑学中有重要的应用，如具有正标量曲率的黎曼度量的存在性问题，关于高签名同伦不变性的Novikov猜想，关于拉普拉斯谱的谱中零猜想等。所采用的方法包括受控算子K-理论、无限维几乎平坦丛、循环上同调和组合Chern-Connes特征标。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Roe代数和群C*代数的K-理论的计算。

9800869 Yu The investigator proposes to study the K-theory of Roe algebras and group C* algebras. The K-theory of Roe algebras and group C* algebras both serve as receptacles of higher indices of elliptic operators and have important applications to problems in differential geometry and topology such as the existence problem for Riemannian metrics with positive scalar curvature, the Novikov conjecture on the homotopy invariance of higher signatures, and the zero-in-the-spectrum conjecture on the spectra of Laplacians. The methods to be employed include controlled operator K-theory, infinite dimensional almost flat bundles, cyclic cohomology, and combinatorial Chern-Connes characters. 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 basic problem 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 used to attack the positive scalar curvature problem and the Novikov conjecture. A key step in this analytic approach is the computation of the K-theory of Roe algebras and group C* algebras.