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

数学科学：齐次空间的粗略几何、量化和渐近同态

• 批准号: 9706960
• 负责人: Erik Guentner
• 金额: $ 5.14万
• 依托单位: Dartmouth College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-01 至 1998-12-08
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706960&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Coarse; Geometry; Homogeneous

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.

摘要 Guentner 拟议的研究包括三个不同的项目：齐次空间上椭圆算子的粗略几何和指数理论、量子力学系统的拓扑不变量以及 A. Connes 和 N. Higson 定义的 E 理论群的等变版本的构造。 所有这三个都涉及 C* 代数的渐近同态概念，它提供了 K-同调（与 K 理论对偶的广义同调理论）的优雅实现。 第一个是由 P. Baum、R. Douglas 和 M. Taylor 的工作启发的，他们计算了流形上的一阶椭圆微分算子的 K-同调类图像，该流形上的边界位于 K-同调中的边界图之下。 第二个是基于这样的观察：基于普朗克常数的连续参数的经典机械系统的量化可用于构造渐近态射并因此构造K-同调群的元素。 第三个是与 N. Higson 和 J. Trout 的联合项目。 我们的等变组将在许多情况下证明是有用的。 特别是它们适用于鲍姆-康尼斯猜想。 拟议的研究涉及具有大量对称性的非紧复杂流形的指数理论。 这些流形边界的几何结构在指数理论中起着重要作用。 我建议使用粗几何技术和新开发的 E 理论来研究它与流形本身上的椭圆微分算子的解析性质的精确关系。 第二个项目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。 我的方法可用于获得这些系统的新拓扑不变量并将它们与索引理论联系起来。 这项研究应该有助于从数学物理角度更好地理解许多量子力学系统，并且还对指数理论中的许多突出问题（包括鲍姆-康尼斯猜想）产生影响。