Focused Research Group: Collaborative Research: Geometry and Deformation Theory of Hyperbolic 3-Manifolds

重点研究组：合作研究：双曲3流形的几何与变形理论

• 批准号: 0554239
• 负责人: Richard Canary
• 金额: $ 13.19万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0554239&HistoricalAwards=false
• 关键词: Focused; Research; Group; Collaborative; Geometry

Since Thurston formulated his geometrization conjecture, the study of infinite volume hyperbolic 3-manifolds has risen to a prominent position in low-dimensional topology and geometry. For the past 30 years four major conjectures have guided this area: Marden's Tameness Conjecture, Thurston's Ending Lamination Conjecture, the Bers-Thurston-Sullivan Density Conjecture and Ahlfors' Measure Conjecture; all have been resolved in the last four years. The solutions of these conjectures have introduced new techniques into the field and opened the door to deeper investigation and the exploration of new directions. In this Focused Research Group, the principalinvestigators propose to use these new techniques to deepen their understanding of the geometry of hyperbolic 3-manifolds, both of infinite and of finite volume, to explore further their still mysterious deformation theory, to pioneer new directions for research in the field, and to develop connections with related branches of low-dimensional geometry and topology.Since the time of Poincare, topologists have pursued the idea that certain spaces called 3-manifolds might be simply described. In the 1970's, Thurston's geometrization conjecture showed topologists the power of bringing geometry to bear on this problem, and opened the possiblity for broad connections between topological, geometric and dynamical features that arise. Using technical tools arising from recent breakthroughs, the PIs hope to interconnect further these different perspectives on the field, and expose early career mathematicians and graduate students to the new range of problems emerging from this fertile area. The Focused Research Group will fund small conferences during its first and final year focused on emerging research areas, with introductory workshops to be run on the day prior to the beginning of the conference. This project will also support the research of the principal investigators' graduate students and provide travel funding for their interaction across institutions. Each of these efforts will allow young geometers and topologists both to learn about the exciting recent developments in the field and to explore the new directions opened up by these developments.

自瑟斯顿（Thurston）提出了他的几何化猜想以来，无限体积的双曲线3序列的研究已上升到低维拓扑和几何学中的显着位置。 在过去的30年中，四个主要的猜想指导了这一领域：Marden的驯服猜想，Thurston的结局层压猜想，Bers-Thurston-Sullivan密度的猜想和AHLFORS的衡量猜想；在过去的四年中，所有这些都得到了解决。 这些猜想的解决方案已将新技术引入了现场，并为更深入研究和探索新方向打开了大门。 In this Focused Research Group, the principalinvestigators propose to use these new techniques to deepen their understanding of the geometry of hyperbolic 3-manifolds, both of infinite and of finite volume, to explore further their still mysterious deformation theory, to pioneer new directions for research in the field, and to develop connections with related branches of low-dimensional geometry and topology.Since the time of Poincare, topologists have追求这样的想法，即可能简单地描述了某些称为3个manifolds的空间。 在1970年代，瑟斯顿（Thurston）的几何化猜想向拓扑师展示了将几何形状带入这个问题的力量，并为出现的拓扑，几何和动态特征之间的广泛连接打开了可能的可能性。 PIS使用最近突破性的技术工具，希望将这些不同的观点进一步互连，并使早期的职业数学家和研究生揭露了从这个肥沃地区出现的新问题。 专注的研究小组将在其第一年和最后一年的小型会议上为新兴研究领域提供资金，并将在会议开始前一天举办入门研讨会。 该项目还将支持主要研究人员的研究生的研究，并为其在机构之间的互动提供旅行资金。这些努力中的每一个都将使年轻的几何学家和拓扑学家都可以了解该领域令人兴奋的最新发展，并探索这些发展的新方向。