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。椭圆算子在同质空间，量子机械系统的拓扑不变性以及A. Connes和N. Higson定义的E理论组的构造的粗糙几何学和索引理论。 这三个都涉及C* - 代数的渐近均形概念的概念，这些概念提供了K-理论的广义同源性理论二的k-词素实现。 第一个是由Baum，R。Douglas和M. Taylor的作品进行的，计算了一阶的K-总体学类别的图像，椭圆差分运算符在K--学中边界图下的边界的歧管上。 第二个是基于这样的观察结果：基于普朗克常数连续参数的经典机械系统的量化可用于构建渐近态度，从而构建K-同学组的元素。 第三是与N. Higson和J. Trout的联合项目。 我们的模棱两可的群体将在许多上下文中被证明是有用的。 特别是它们对鲍姆 - 康涅狄格州的猜想有应用。 拟议的研究涉及有关具有大量对称性的非紧凑，复杂流形的索引理论。 这些歧管边界的几何结构在索引理论中起着重要作用。 我建议使用粗糙的几何形状和新开发的电子理论的技术来研究其与椭圆形差分运算符本身上椭圆形差分运算符的分析特性的精确关系。 第二个项目是基于这些歧管上椭圆差分运算符与某些量子机械系统之间的密切关系。 我的方法可用于获取这些系统的新拓扑不变，并将其与索引理论联系起来。 这项研究应该使对数学物理学的许多量子机械系统有更好的了解，并且还与包括鲍姆 - 康涅斯（Baum-Connes）猜想在内的指数理论中的许多杰出问题有关。