K-theory of operator algebras and invariants of elliptic operators

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In real life, one seeks to understand objects in the universe by studying various characteristic quantities associated with them, such as shape, size, temperature, and so on. These characteristics allow one to distinguish one object from another. Some characteristics may change drastically under the influence of very small external forces, and some remain resistant to change under such forces. The former are unstable and usually difficult to measure, while the latter are more stable and easier to detect, hence much more useful. In mathematics, those stable characteristics are called invariants. They provide some of the most fundamental tools in almost all branches of mathematics. In this project, the principal investigator will study a certain class of invariants of differential equations and apply them to study problems in classical geometry and topology. Index theoretical invariants of elliptic operators are important for understanding the geometry of their underlying spaces. The famous Atiyah-Singer index theorem and its noncommutative geometric generalizations have many applications to geometry and topology. All these index theoretical invariants live naturally in the K-theory of certain operator algebras. The principal investigator will use methods developed in the studies of K-theory of operator algebras to investigate various invariants of elliptic operators on manifolds and spaces with singularities. In particular, he is interested in secondary invariants such as the higher rho invariant. The principal investigator proposes to use these secondary invariants to study the structure group of a closed topological manifold and the homotopy groups of the space of positive scalar curvature metrics on a given manifold. The principal investigator also plans to explore the connections of these problems to the Novikov conjecture and the Baum-Connes conjecture.