Smooth 4-Manifold Topology, 3-Manifold Group Actions, the Heegaard Tree, and Low Volume Hyperbolic 3-Manifolds

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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 those areas. This research project addresses topics in hyperbolic geometry and foliation theory, which are structures for globally understanding three dimensional spaces, and will also address fundamental questions in smooth 4-dimensional topology. 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 graduate courses. The award provides funds to support graduate students.The investigator will continue his approach to the smooth 4-dimensional Schoenflies conjecture through the investigation of knotted three-balls in four-dimensional manifolds and diffeomorphisms of four-dimensional manifolds. He plans to discover new constructions of foliations and laminations and to address the structure of low volume hyperbolic 3-manifolds. The PI and collaborators have discovered new techniques to address these problems and plan 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.