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

平滑 4 流形拓扑、3 流形组动作、Heegaard 树和低容量双曲 3 流形

基本信息

批准号：
2003892
负责人：
David Gabai
金额：
$ 50.46万
依托单位：
Princeton University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2003892&HistoricalAwards=false
关键词：
Smooth Manifold Topology Group Actions

项目摘要

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.

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，是当代数学研究的一个中心领域。它处在数学的许多子领域的十字路口。组合拓扑学、几何学、极小曲面论、分析学、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，反过来，该领域的研究也促进了这些领域的进步。这项研究项目涉及双曲几何和分层理论，这是全球理解三维空间的结构，也将解决光滑4维拓扑的基本问题。这项研究的一部分适用于本科生和研究生入门项目。研究所需的背景材料以及发现的新想法将被纳入研究生课程。该奖项为研究生提供资助。研究人员将通过研究四维流形中的纽结三球和四维流形的微分同胚来继续他对光滑的四维Schoenfys猜想的研究。他计划发现新的叶状和层状结构，并解决低体积双曲3-流形的结构问题。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步。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。