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

光滑 4 流形、双曲 3 流形和微分同胚群

• 批准号: 2304841
• 负责人: David Gabai
• 金额: $ 53.77万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2028-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2304841&HistoricalAwards=false
• 关键词: Smooth; manifolds; hyperbolic; diffeomorphism; groups

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.

低维拓扑是对在表面（二维）、我们的空间（三维）和时空（四维）上建模的对象的研究。它是数学的中心领域，具有当代的浓厚兴趣。它处于数学许多子领域的十字路口，这些子领域的方法促进了低维拓扑的发展，相反，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平滑四维拓扑中的基本问题，包括四维空间自映射的拓扑。将研究与全局理解三维空间的结构相关的其他主题。该项目的一部分是找出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发现的新想法将纳入 PI 教授的课程中。PI 的目标是开发解决平滑 4 维 Schoenflies 猜想的程序，并研究维度至少为 4 的流形的微分同胚群。 。项目包括拉紧的叶状结构、横向可定向的基本叠层以及双曲三流形群上的左序之间的关系。 PI 还计划研究双曲三流形上的 Margulis 数，并解决低体积双曲 3 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，并提议使用这些方法并发现新方法以取得进一步进展。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的评估进行评估，被认为值得支持。影响审查标准。