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

小行星9500898 该项目涉及的研究指标理论的非紧黎曼流形的循环上同调和K-理论的方法。非紧完备黎曼流形上的几何算子是广义的Fredholm，在这个意义上，它们是模所有具有有界传播的局部可迹算子的理想可逆的，称为Roe代数。几何算子的K-理论指标存在于Roe代数的K-理论中。一个重要而有趣的特征是K-理论指数的计算仅依赖于黎曼流形的粗糙几何。几何算子的K-理论指标在几何学、拓扑学和非紧黎曼流形的分析中有重要的应用。 这个项目处于几何、分析和代数之间的界面。Connes的现代方法使用非交换代数结构研究几何不变量。这些不变量只在少数情况下被明确地实现。该项目的完成将提供新的 这些不变量的显式例子从非交换几何。 这些工作将有助于流形的几何分类。 ***

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. ***