Deformation spaces of hyperbolic 3-manifolds

双曲3流形的变形空间

• 批准号: 1006298
• 负责人: Richard Canary
• 金额: $ 19.96万
• 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: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1006298&HistoricalAwards=false
• 关键词: Deformation; spaces; hyperbolic; manifolds

Prof. Canary proposes to study the space AH(M) of marked hyperbolic 3- manifolds homotopy equivalent to a fixed compact 3-manifold M and its natural quotient, the space AI(M) of unmarked hyperbolic 3-manifolds homotopy equivalent to M. The space AH(M) may be viewed as the 3-dimensional analogue of the Teichmuller space of marked hyperbolic surfaces homeomorphic to a fixed closed surface, while the space AI(M) is the natural analogue of moduli space. Recent work on AH(M) has focussed on the classification of elements in AH (M), e.g. the Ending Lamination Theorem, and the difficulty in understanding the topology of the space, e.g. results establishing that AH(M) need not be locally connected. Prof. Canary proposes a number of projects which study, and attempt to circumscribe, the pathological behavior of the topology of AH(M). The study of AI(M) is still in its infancy, but preliminary results indicate that the topology of AI(M) reflects the topology of M. Prof. Canary proposes to study the topology of AI(M) and the dynamics of the action of the outer automorphism group of the fundamental group of M on AH(M) and, more generally, on the relevant character variety.Prof. Canary studies the deformation theory of geometric structures on 3-dimensional manifolds. A 2-dimensional manifold is a space which looks locally like the 2- dimensional plane. For example, one may consider surfaces of familiar 3-dimensional objects such as footballs or doughnuts. Similarly, a 3-manifold is a space that looks locally like 3-dimensional Euclidean space. For example, the universe we live in is a 3-dimensional manifold. A geometric structure on a manifold gives a way of measuring distances and angles in a manifold. The study of how geometric structures on manifolds may vary has been a prominent theme in recent work in both mathematics, for example in the solution of Thurston's Geometrization Conjecture, and physics, for example in string theory. Prof. Canary will study the variation of geometric structures in the setting of hyperbolic 3-manifolds. In addition, Prof. Canary will continue his commitment to undergraduate education, by pioneering a new course using inquiry-based methods, and his work mentoring graduate students and postdoctoral assistant professors.

Canary教授提议研究明显的双曲3-歧管的空间AH（m），同等的同质型，等效于固定的紧凑型3个manifold M及其自然人，无标记的双曲3个manifolds的空间AI（m）等于M. Space AH（M），可以将其视为3二二轴的Space AH（M）。固定的闭合表面，而空间AI（M）是模量空间的天然类似物。关于AH（M）的最新工作集中在AH（M）中的元素分类上，例如结束层压定理以及理解空间拓扑的困难，例如结果确定AH（M）不必局部连接。 Canary教授提出了许多研究并试图限制AH（M）拓扑的病理行为的项目。 对AI（M）的研究仍处于起步阶段，但初步结果表明，AI（M）的拓扑结构反映了Canary教授M. Canary教授的拓扑结构，建议研究AI（M）的拓扑结构，以及M on AH（M）基本基本组和相关特征的M of AH（M）基本组的外部自动化组的动态。金丝雀研究在三维流形上的几何结构的变形理论。二维歧管是一个局部看起来像2维平面的空间。例如，人们可能会考虑熟悉的三维物体（例如足球或甜甜圈）的表面。 同样，一个3个manifold是一个看起来像3维欧几里得空间的空间。例如，我们所生活的宇宙是一个三维流形。流形上的几何结构提供了一种测量歧管中距离和角度的方法。 关于流形的几何结构如何变化的研究一直是两种数学中最近工作的重要主题，例如在瑟斯顿的几何化猜想的解决方案中，例如在弦理论中的物理学。 Canary教授将研究在双曲线3个manifolds的设置中几何结构的变化。此外，Canary教授将通过使用基于询问的方法开创新课程以及他的工作指导研究生和博士后助理教授，继续他对本科教育的承诺。