Smooth 4-manifolds, hyperbolic 3-manifolds and diffeomorphism groups

Low dimensional topology is the study of objects modeled on surfaces (two dimensions), our space (three dimensions) and space-time (four dimensions). It is a central area of mathematics with intense contemporary interest. It is at the crossroads of many subfields of mathematics, methods from which have contributed to the development of low dimensional topology, and conversely, research in that field stimulates advances in those areas. The research project supported by this award addresses fundamental questions in smooth four-dimensional topology including the topology of self-mappings of four dimensional spaces. Additional topics related to structures for globally understanding three dimensional spaces, will be investigated. A part of the project is to carve out research problems suitable for undergraduate and beginning graduate students. Background material needed for the research as well as new ideas discovered will be incorporated into the courses the PI teaches. The PI aims to develop his program for resolving the smooth 4-dimensional Schoenflies conjecture and to study diffeomorphism groups of manifolds of dimension at least four. Projects include relationships between taut foliations, transversely orientable essential laminations, and left orders on hyperbolic three-manifold groups. The PI also plans to investigate Margulis numbers on hyperbolic three-manifolds and to address the structure of low volume hyperbolic 3-manifolds. The PI and his collaborators have discovered new techniques to address these problems and propose to use these methods and discover new ones to make further advances. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.