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

Vanderbilt University, Nashville TN

Investigators

Abstract

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 positive scalar curvature, the Novikov conjecture on homotopy invariance of higher 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 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-theoretic higher indices, which can be essentially reduced to the problem of computing the K-theory of operator algebras associated to metric spaces and groups.