Mathematical Sciences: Coarse Geometry of Homogeneous Spaces, Quantization and Asymptotic Homomorphisms

这项研究包括三个不同的项目：齐次空间上椭圆算子的粗几何和指数理论，量子力学系统的拓扑不变量，以及A.Connes和N.Higson定义的E-理论群的一个等变版本的构造。这三个概念都涉及到C*-代数的渐近同态的概念，它提供了K-同调的一种优雅的实现，即对偶于K-理论的广义同调理论。第一个是由P.Baum，R.Douglas和M.Taylor计算流形上一阶椭圆微分算子的K-同调类的像，流形上的边界在K-同调的边界映射下。第二种是基于这样的观察，即基于普朗克常数的连续参数的经典力学系统的量子化可以用来构造渐近态射，从而构成K-同调群的元素。第三个是与N·希格森和J·特劳特的联合项目。我们的等变群将在许多情况下被证明是有用的。特别是，它们可以应用于Baum-Connes猜想。所提出的研究是关于具有大量对称性的非紧复流形上的指数理论。这些流形边界的几何结构在指数理论中起着重要的作用。我建议利用粗几何的技巧和新发展的E-理论来研究它与流形上椭圆微分算子本身的解析性质之间的精确关系。第二个项目是基于这些流形上的椭圆微分算子与某些量子力学系统之间的密切关系。我的方法可以用来获得这些系统的新的拓扑不变量，并将它们与指数理论联系起来。这项研究应该会从数学物理上更好地理解一些量子力学系统，并与指数理论中的一些突出问题有关，包括Baum-Connes猜想。

Abstract Guentner The proposed research comprises three distinct projects; coarse geometry and index theory of elliptic operators on homogeneous spaces, topological invariants of quantum mechanical systems, and the construction of an equivariant version of the E-theory groups defined by A. Connes and N. Higson. All three involve the notion of asymptotic homomorphisms of C*-algebras which provide an elegant realization of K-homology, the generalized homology theory dual to K-theory. The first is motivated by work of P. Baum, R. Douglas and M. Taylor calculating the image of the K-homology class of a first order, elliptic differential operator on a manifold with boundary under the boundary map in K-homology. The second is based on the observation that a quantization of a classical mechanical system based on a continuous parameter of Planck's constant may be used to construct an asymptotic morphism and hence an element of a K-homology group. The third is a joint project with N. Higson and J. Trout. Our equivariant groups will prove useful in a number of contexts. In particular they have applications to the Baum-Connes conjecture. The proposed research deals with index theory on non-compact, complex manifolds possessing a large number of symmetries. The geometric structure of the boundary of these manifolds plays an important role in index theory. I propose to study its precise relationship to the analytic properties of elliptic differential operators on the manifolds themselves using the techniques of coarse geometry and the newly developed E-theory. The second project is based on the close relationship between elliptic differential operators on these manifolds to certain quantum mechanical systems. My methods can be used to obtain new topological invariants of these systems and relate them to index theory. The research should lead to a better understanding of a number of quantum mechanical systems from mathematical physics and also has bearing on a number of outstanding problems in index theory including the Baum-Connes Conjectupe.