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

低维拓扑是研究在表面（二维）、我们的空间（三维）和时空（四维）上建模的对象。它是数学的中心领域，具有强烈的当代兴趣。它处于许多数学子领域的交叉点，这些子领域的方法为低维拓扑的发展做出了贡献，反过来，该领域的研究也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。将研究与结构相关的其他主题，以全面了解三维空间。这个项目的一部分是为本科生和刚毕业的研究生找出适合的研究问题。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。PI的目标是开发他的程序来解决光滑的四维舍恩菲猜想，并研究至少四维流形的微分同构群。项目包括紧叶理之间的关系，横向可定向的基本层合，以及双曲三流形群上的左序。PI还计划研究双曲三流形上的马古利数，并解决小体积双曲三流形的结构问题。PI和他的合作者已经发现了解决这些问题的新技术，并建议使用这些方法和发现新的方法来取得进一步的进展。该奖项反映了美国国家科学基金会的法定使命，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.