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

低维拓扑学是对物体在表面（二维）、我们的空间（三维）和时空（四维）上建模的研究。这是一个中心领域的数学与强烈的当代利益。它是在许多数学子领域的十字路口，从这些方法有助于低维拓扑的发展，相反，在该领域的研究刺激了这些领域的进步。该奖项支持的研究项目解决了光滑四维拓扑中的基本问题，包括四维空间的自映射拓扑。将研究与全面理解三维空间的结构相关的其他主题。该项目的一部分是开拓适合本科生和研究生的研究问题。研究所需的背景材料以及新发现的想法将被纳入PI教授的课程中。PI的目标是开发解决光滑4维Schoenflies猜想的程序，并研究至少4维流形的同构群。项目包括拉紧叶理，横向定向基本层压，和左订单双曲三流形群之间的关系。PI还计划研究双曲三维流形上的马古利斯数，并解决低容量双曲三维流形的结构。PI和他的合作者发现了解决这些问题的新技术，并建议使用这些方法，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 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.