Mathematical Sciences: Asymptotic Topology, Analysis and Dynamics of Spaces and Foliations

该项目有两个主题：研究完全开流形和叶理的渐近几何、拓扑和分析性质;研究由几何构造产生的算子代数的几何和拓扑不变量。 关于开流形的渐近性质，我们建议研究： 第一章 空间熵的结构及其应用; 2）空间的粗上同调 叶理和映射空间; 3）叶理叶的遍历理论与渐近不变量; 4）几何算子代数的指数不变量。 关于算子代数产生的几何模型，我们建议研究各种不变量使用技术的基础上几何 建筑。 我们建议考虑：5）熵和谱特性的 算子; 6）非紧黎曼叶理的指数理论; 7） 叶理的微分拓扑和算子代数的结构。 最后，该项目包括一个计算机辅助研究组件-与 工作要做的共同与研究生-研究几何和 通过计算机建模的表面动力学。 我们的研究涉及数学的许多领域，特别是在 微分拓扑学、动力学和算子谱理论。 这些是 所有的纯数学领域，过去已经证明， 应用于“真实的生活问题”-包括调查 数学物理问题，自然现象的建模，以及理解物理系统行为中的对称性。这个项目是广泛的范围和各种主题，反映了许多富有成效的研究项目和合作的成果与其他数学家 由国家科学基金会先前的资助支持。这种广度确保了所提出的项目将找到与其他数学领域的新联系，使研究生和本科生能够参与研究，并增加了所获得的结果的广泛适用性的可能性。

This project has two themes: the study of the asymptotic geometrical, topological and analytical properties of complete open manifolds and foliations; and the study of geometrical and topological invariants for operator algebras arising from geometric constructions. Regarding the asymptotic properties of open manifolds, we propose to investigate: 1) the structure and applications of entropy of spaces; 2) coarse cohomology for foliations and mapping spaces; 3) ergodic theory of leaves of foliations and asymptotic invariants; and 4) index invariants of geometric operator algebras. Regarding operator algebras arising from geometric models, we propose to study a variety of invariants derived using techniques based on geometric constructions. We propose to consider: 5) entropy and spectral properties of operators; 6) index theory of non-taut Riemannian foliations; and 7) differential topology of foliations and the structure of operator algebras. Finally, this project includes a computer aided research component - with the work to be done jointly with a graduate student - to study the geometry and dynamics of surfaces via computer modeling. Our research is related to many areas of mathematics, especially to topics in differential topology, dynamics, and spectral theory of operators. These are all areas of pure mathematics which have proven in the past to have applications to "real -life problems" - including the investigations of questions in mathematical physics, the modeling of phenomenon in nature, and for understanding symmetries in the behavior of physical systems. This project is broad in its scope and variety of topics, reflecting outgrowths of many fruitful research projects and collaborations with other mathematicians supported by previous funding of the National Science Foundation. This breadth ensures that the proposed projects will find new connections with other areas of mathematics, enables the participation of graduate and undergraduate students in the research, and increases the probability of the broad applicability of the results obtained.