Dynamics, Geometry, and K-Theory

流形是可以使用代数和微积分的标准数值变量进行参数化的几何对象。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，在其他科学中，它们对物理空间和许多重要的数据集进行建模。首席研究员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对流形进行参数化(想想依赖于五个或更多变量的数据集)。更多变量所固有的额外灵活性意味着可以使用更多的工具来处理问题。在通信理论和动力系统理论(即在系统对称性下的变化)中非常重要的扩展网络的几何形状是该项目的特别重要的工具。在整个项目期间，首席研究员打算通过年轻科学家参与研究来指导他们，并继续致力于夏威夷数学教育和研究的结构改进。流形最重要的不变量之一是它的基本群，它支配着流形内部形成回路的方式。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-理论组成的，通过拓扑分类中的Novikov猜想和Borel猜想以及微分几何中的Gromov-Lawson猜想等问题来理解高维流形是一个困难的问题。Baum-Connes和Farrell-Jones猜想了计算K理论的公设技术，并锚定了目前解决前面提到的流形理论问题的方法。本文的主要研究目的是利用椭圆微分算子指数论、算子代数、拓扑动力系统、扩张网络的度量几何和表示论等方法来研究这些猜想及相关问题。

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.