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.
