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

低维拓扑学，研究物体在表面（二维）、我们的空间（三维）和时空（四维）上的建模，是当代数学的一个中心领域。它处于数学许多子领域的交叉点。从组合拓扑学、几何学、极小曲面理论、分析、群论、数论、动力系统和理论计算机科学的方法促进了低维拓扑学的发展，相反，在该领域的研究刺激了这些领域的进步。该研究项目涉及双曲几何和叶理理论的主题，这些理论是全面理解三维空间的结构，并且还将解决光滑四维拓扑中的基本问题。部分研究适合本科生和研究生项目。研究所需的背景材料以及发现的新想法将纳入研究生课程。该奖项提供资金支持研究生。研究人员将继续他的方法，通过研究四维流形中的打结三球和四维流形的非同构，顺利四维Schoenflies猜想。 他计划发现新的结构的叶理和层压，并解决结构的低容量双曲3流形。PI和合作者已经发现了解决这些问题的新技术，并计划使用这些方法，并发现新的方法以取得进一步的进展。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.