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形的渐近几何、拓扑和解析性质；以及由几何构造引起的算子代数的几何不变量和拓扑不变量的研究。关于开流形的渐近性质，我们提出研究：1)空间熵的结构及其应用；叶与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，我们建议研究使用基于几何构造的技术推导的各种不变量。我们建议考虑：5)算子的熵和谱性质；非紧黎曼叶理的指标理论；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组件-与研究生共同完成的工作-通过计算机建模来研究表面的几何和动力学。我们的研究涉及数学的许多领域，特别是微分拓扑、动力学和算子的谱理论。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”——包括数学物理问题的研究，自然现象的建模，以及对物理系统行为对称性的理解。该项目范围广泛，主题多样，反映了许多富有成效的研究项目和与其他数学家合作的成果，这些项目得到了国家科学基金会以前的资助。这种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，并增加了所获得结果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.