Dynamics, Geometry, and K-Theory

动力学、几何和 K 理论

• 批准号: 1401126
• 负责人: Rufus Willett
• 金额: $ 12.6万
• 依托单位: University of Hawaii
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1401126&HistoricalAwards=false
• 关键词: Dynamics; Geometry; Theory

Manifolds are geometric objects that can be parametrized using the standard numerical variables of algebra and calculus. They are fundamental objects for much of modern mathematics and its applications to the other sciences, where they model physical space as well as many important data sets. The principal investigator is planning to study properties of higher dimensional manifolds. Here "higher dimensional" means that at least five variables are needed to parametrize the manifold (think of a data set depending on five or more variables). The extra flexibility inherent in a larger number of variables means that many more tools can be brought to bear on problems. The geometry of the expanding networks that are important in communications theory and the theory of dynamical systems (i.e.,change under symmetries of a system) are particularly important tools for the project. Throughout the period of the project the principal investigator intends to mentor young scientists through their involvement in the research, and to continue to work on structural improvements to mathematical education and research in Hawaii.One of the most important invariants of a manifold is its fundamental group, which governs the way loops can be formed inside the manifold. Associated to fundamental groups are algebraic group rings and analytic group operator algebras. The theory of linear algebra over group rings and group operator algebras is organized by their K-theory, and understanding the associated K-groups is a difficult problem that is fundamental to the understanding of higher dimensional manifolds via such problems as the Novikov and Borel conjectures in topological classification and the Gromov-Lawson conjecture in differential geometry. The Baum-Connes and Farrell-Jones conjectures postulate techniques for computing K-theory and anchor the current approach to the problems in manifold theory mentioned earlier. The principal investigator intends to use techniques from index theory of elliptic differential operators, operator algebras, topological dynamical systems, metric geometry of expanding networks, and representation theory to study these conjectures and related issues.

流形是可以使用代数和微积分的标准数值变量参数化的几何对象。它们是许多现代数学及其在其他科学中的应用的基本对象，在这些科学中，它们对物理空间以及许多重要的数据集进行建模。 主要研究者计划研究高维流形的性质。这里的“高维”意味着至少需要五个变量来参数化流形（考虑一个依赖于五个或更多变量的数据集）。大量变量所固有的额外灵活性意味着可以使用更多的工具来解决问题。在通信理论和动力系统理论中很重要的扩展网络的几何形状（即，系统对称下的变化）对于该项目来说是特别重要的工具。在整个项目期间，首席研究员打算指导年轻科学家，通过他们参与研究，并继续致力于结构改进的数学教育和研究在夏威夷。流形的最重要的不变量之一是它的基本群，它支配着流形内部可以形成循环的方式。与基本群相关的是代数群环和解析群算子代数。群环和群算子代数上的线性代数理论是由它们的K-理论组织的，而理解相关的K-群是一个困难的问题，它是通过拓扑分类中的诺维科夫和波莱尔猜想以及微分几何中的格罗莫夫-劳森猜想等问题理解高维流形的基础。Baum-Connes和Farrell-Jones假设了计算K理论的技术，并锚了前面提到的流形理论中问题的当前方法。主要研究人员打算使用的技术，从指数理论的椭圆微分算子，算子代数，拓扑动力系统，度量几何的扩展网络，和表示理论来研究这些结构和相关问题。