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

自从瑟斯顿提出几何猜想以来，无穷大体积双曲三维流形的研究在低维拓扑学和几何学中占据了突出的地位。在过去的30年里，四个主要的猜想一直指导着这个领域：马登的驯服猜想、瑟斯顿的结束分层猜想、Bers-瑟斯顿-沙利文密度猜想和Ahlfors的测度猜想；所有这些猜想都在过去的四年里得到了解决。这些猜想的解决为这一领域引入了新的技术，并为更深入的研究和探索新的方向打开了大门。在这个重点研究小组中，主要研究人员建议使用这些新技术来加深他们对无限和有限体积的双曲三维流形几何的理解，进一步探索他们仍然神秘的形变理论，开拓该领域研究的新方向，并发展与低维几何和拓扑学的相关分支的联系。自庞加莱时代以来，拓扑学家们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年代的S，瑟斯顿的几何化猜想向拓扑学家展示了将几何运用到这个问题上的力量，并为拓扑、几何和动力学特征之间的广泛联系打开了可能性。利用最近的突破所产生的技术工具，私人投资机构希望进一步将该领域的不同观点联系起来，并使早期职业数学家和研究生接触到这一肥沃领域出现的一系列新问题。重点研究小组将在第一年也是最后一年资助以新兴研究领域为重点的小型会议，并在会议开始前一天举办介绍性讲习班。该项目还将支持主要研究人员研究生的研究，并为他们在各机构之间的互动提供旅费。每一项努力都将使年轻的几何学家和拓扑学家了解该领域令人振奋的最新发展，并探索这些发展开辟的新方向。

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.