Diffeomorphisms of 4-manifolds, essential laminations and geometric constraints on hyperbolic 3-manifolds

Low-dimensional topology, the study of objects modeled on surfaces (two dimensions), our space (three dimensions) and space-time (four dimensions) is a central area of mathematics of intense contemporary interest. It is at the crossroad of many subfields of mathematics. Methods from combinatorial topology, geometry, minimal surface theory, analysis, group theory, number theory, dynamical systems, and theoretical computer science have contributed to the development of low dimensional topology, and conversely, research in that field stimulates advances in other areas. The research project addresses topics in smooth 4-manifold topology, foliation theory and hyperbolic geometry which are structures for globally understanding three and four dimensional spaces. Parts of the research are suitable for undergraduate and beginning graduate student projects. Background material needed for the research as well as new ideas discovered will be incorporated into the PI's graduate courses. The PI and his collaborators have discovered new techniques to address these problems and plan to use these methods and discover new ones, e.g. artificial intelligence to attack problems in hyperbolic geometry, to make further advances. The PI plans to develop his program for resolving the smooth 4-dimensional Schoenflies conjecture and discovering an exotic smooth diffeomorphism of the 4-sphere. He plans to investigate relations between taut foliations, transversely orientable essential laminations and left orders on hyperbolic 3-manifold groups. He also plans to investigate Margulis numbers on hyperbolic 3-manifolds via methods to which AI may contribute. 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.