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

这个项目有两个主题：研究完全开流形和叶的渐近几何、拓扑和分析性质；以及研究由几何结构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：1)空间的熵的结构和应用；2)叶和映射空间的粗上同调；3)叶的遍历理论和渐近不变量；4)几何算子代数的指标不变量。对于由几何模型产生的算子代数，我们建议研究基于几何结构的技巧得到的各种不变量。我们建议考虑：5)算子的熵和谱性质；6)非紧黎曼叶的指标理论；7)叶的微分拓扑和算子代数的结构。最后，这个项目包括一个计算机辅助研究部分--与一名研究生共同完成--通过计算机建模研究表面的几何和动力学。我们的研究涉及到许多数学领域，特别是与微分拓扑学、动力学和算子谱理论有关的课题。这些都是纯数学的领域，在过去已经被证明可以应用于“现实生活中的问题”--包括研究数学物理中的问题，对自然界中的现象进行建模，以及理解物理系统行为中的对称性。这个项目的范围广泛，主题多样，反映了许多富有成效的研究项目的结果，以及与其他数学家在以前国家科学基金会资助下的合作。这一广度确保了拟议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广泛适用的可能性。

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.