A foliation F of a manifold M is a regular decomposition of the space into connected submanifolds, usually with some assumptions that the decomposition is smooth, and the decomposition is uniform -- either there are no singular strata, or the singular strata themselves have a foliated structure. The breadth of the study of foliations includes all dynamical systems, an immensely broad topic in itself, so truly classifying all foliations is quite impossible. The goal of this project is to investigate three approaches to classification: 1) classify foliations via categorical (in the sense of Lusternik and Schnirelmann) decompositions and other "foliated cell" structures; 2) classify foliations up to homotopy of their transverse Haefliger structure; 3) classify foliations in terms of the ergodic theory and asymptotic properties of their leaves. The PI's study of the relations between the second and third of these topics -- dynamical systems and ergodic theory, and geometric and topological invariants of group actions and foliations -- has been ongoing for more than 20 years with support of the NSF, and resulted in many publications and advances in the field. In the past three years, there have been further decisive advances in each topic. The study of Lusternik and Schnirelmann category of foliations, and its related ideas, is a very new field, with great potential for advances. The fundamental connections between LS category theory and Morse theory, motivate the study of the connections between their foliated versions, and the classification of transverse Haefliger structures. Success with this part of the proposed research would result in truly fundamental new understanding of the structure of foliations.
The study of foliations of manifolds began as a subject approximately fifty years ago. The concept arises very naturally in many geometric and topological problems, so it is not surprising that understanding properties of foliations has become a fundamental aspect of the study of many other subjects in geometry, topology, dynamical systems, analysis and various areas of applied and theoretical physics. The intellectual merit of the proposed activity includes advancing our understanding of the structure of foliations, and developing new techniques with broad applications in many fields. Success with this research project will not just be measured by solving the proposed problems, but also in opening the field up with new questions. The broader impact resulting from the proposed activity includes enhanced training of undergraduate and graduate students in mathematics, collaborative research projects with postdoctoral investigators, and international collaborations promoting research in mathematics. The PI intends to continue working with graduate students and postdocs on projects related to this proposal, and to promote the field via conference talks and via the internet.
