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

点击翻译按钮获取中文摘要

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.