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

作者建议研究群C*-代数的K-理论及其在流形几何和拓扑中的应用。群C*-代数的K-理论是高指数椭圆微分算子的容器，在流形的微分几何和拓扑学中有着重要的应用，如具有正标量曲率的黎曼度量的存在性问题和关于高签名同伦不变性的Novikov猜想。所采用的方法包括Banach空间上的群作用、一致嵌入到Banach空间和椭圆算子的无穷维指标理论。流形是由欧氏空间粘合在一起的空间。粗略地说，这些是人们可以在上面做微积分的空间。流形的例子包括球面和环面。流形理论在数学和物理中起着重要的作用。流形理论中的一个基本问题是流形的分类。通过外科理论，高维流形的分类问题本质上归结为Novikov猜想。研究人员计划使用无限维分析的方法来研究诺维科夫猜想，比如非对易几何。作者还打算应用非对易几何方法来研究流形的环空间上的分析。

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.