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

摘要Guentner 拟议的研究包括三个不同的项目;粗糙的几何和指数理论的椭圆算子在齐次空间，拓扑不变量的量子力学系统，和建设的一个等变版本的E-理论群定义的A。Connes和N.希格森 所有这三个涉及的概念渐近同态的C *-代数提供了一个优雅的实现K-同调，广义同调理论对偶的K-理论。 第一个是由P.Baum，R.道格拉斯和M. Taylor计算了一阶椭圆微分算子在K-同调的边界映射下的K-同调类在有边界流形上的象。 第二个是基于这样的观察，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于构造渐近态射，因此是K-同调群的元素。 第三个是与N. Higson和J. Trout。 我们的等变群将在许多情况下证明是有用的。 特别是他们有应用的鲍姆-康纳斯猜想。 建议的研究涉及非紧的，复杂的流形具有大量的对称性的指数理论。 这些流形的边界的几何结构在指标理论中起着重要的作用。 我建议研究其精确的关系，椭圆微分算子的分析性质的流形本身使用的技术粗糙的几何和新发展的E-理论。 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。 这项研究将有助于从数学物理学的角度更好地理解一些量子力学系统，并对指数理论中的一些突出问题（包括Baum-Connes猜想）产生影响。

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.