Deformation spaces of hyperbolic 3-manifolds

Canary教授提出研究等价于固定紧致3-流形M的标记双曲3-流形同伦空间AH(M)及其自然商，等价于M的非标记双曲3-流形同伦空间AI(M)。空间AH(M)可以看作是等价于固定紧致3-流形M的标记双曲曲面的Teichmuller空间的三维模拟，而空间AI(M)是模空间的自然模拟。最近关于AH(M)的工作主要集中在AH(M)中元素的分类，例如结束分层定理，以及理解空间拓扑的困难，例如建立AH(M)不需要局部连接的结果。Canary教授提出了一些研究项目，并试图限制AH(M)拓扑结构的病理行为。对AI(M)的研究还处于起步阶段，但初步结果表明AI(M)的拓扑结构反映了M的拓扑结构。Canary教授建议研究AI(M)的拓扑结构以及M的基本群的外部自同构群对AH(M)的作用动力学，更一般地说，对相关的特征变化。Canary研究三维流形上几何结构的变形理论。二维流形是一个局部看起来像二维平面的空间。例如，人们可以考虑熟悉的三维物体的表面，如足球或甜甜圈。类似地，三维流形是一个局部看起来像三维欧几里得空间的空间。例如，我们生活的宇宙是一个三维流形。流形上的几何结构给出了一种测量流形中距离和角度的方法。流形上的几何结构如何变化的研究已经成为近年来数学（例如瑟斯顿几何猜想的解）和物理（例如弦理论）工作中的一个突出主题。Canary教授将研究双曲型3流形下几何结构的变化。此外，Canary教授将继续致力于本科教育，开创一门使用探究性方法的新课程，并指导研究生和博士后助理教授。

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.