Mathematical Sciences: Index Theory for Noncompact Riemannian Manifolds and Coarse Geometry

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9500898 Yu The project involves the study of index theory for noncompact Riemannian manifolds by cyclic cohomology and K-theory methods. Geometric operators on noncompact complete Riemannian manifolds are generalized Fredholm in the sense that they are invertible modulo the ideal of all locally traceable operators with bounded propagations, called the Roe algebra. The K-theoretic indices of geometric operators live in the K-theory of the Roe algebra. An important and interesting feature is that the computation of K-theoretic indices depends only on the coarse geometry of Riemannian manifolds. The K-theoretic indices of geometric operators have important applications in geometry, topology, and the analysis of noncompact Riemannian manifolds. This project lies at the interface between geometry, analysis and algebra. A modern approach of Connes studies geometrical invariants using non-commutative algebraic structures. These invariants have been explicitly realized only in a few cases. The completion of this project should provide new explicit examples of these invariants from non-commutative geometry. The work will contribute to geometric classification of manifolds. ***