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.

自从Thurston提出他的几何化猜想以来，无限体积双曲三维流形的研究在低维拓扑学和几何学中已经上升到一个突出的位置。 在过去的30年里，有四个主要的猜想指导着这个领域：马尔登的驯服猜想，瑟斯顿的终结层压猜想，Bers-Thurston-Sullivan密度猜想和Ahlfors的测度猜想;所有这些都在过去的四年里得到了解决。 这些解决方案为该领域引入了新技术，并为更深入的研究和新方向的探索打开了大门。 在这个专题研究小组中，主要的研究人员建议使用这些新技术来加深他们对双曲三维流形几何的理解，包括无限和有限体积，进一步探索他们仍然神秘的变形理论，开拓该领域研究的新方向，并与低维几何和拓扑学的相关分支建立联系。拓扑学家一直在追求这样的想法，即某些称为三维流形的空间可以被简单地描述。 在20世纪70年代，瑟斯顿的几何化猜想向拓扑学家展示了将几何学应用于这个问题的力量，并为拓扑、几何和动力学特征之间的广泛联系开辟了可能性。 使用技术工具所产生的最新突破，PI希望进一步互连这些领域的不同观点，并暴露早期职业数学家和研究生从这个肥沃的领域出现的新问题。 重点研究小组将在其第一年和最后一年资助小型会议，重点是新兴的研究领域，介绍性研讨会将在会议开始前一天举行。 该项目还将支持主要研究人员的研究生的研究，并为他们在各机构之间的互动提供旅费。每一个这些努力将使年轻的geometers和topologists都了解令人兴奋的最新发展领域，并探讨新的方向开辟了这些发展。