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，即等价于M的未标记双曲3-流形同伦空间AI（M）。空间AH（M）可以被看作是与固定闭曲面同胚的标记双曲曲面的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）的工作集中在AH（M）中元素的分类上，例如结束层合定理，以及理解空间拓扑的困难，例如确定AH（M）不需要局部连通的结果。Canary教授提出了许多研究项目，并试图限制AH（M）拓扑的病理行为。 AI（M）的研究尚处于起步阶段，但初步结果表明AI（M）的拓扑反映了M的拓扑。Canary教授建议研究AI（M）的拓扑结构，以及M的基本群的外自同构群在AH（M）上的作用的动力学，更一般地，研究相关的特征变量。Canary教授研究三维流形上几何结构的变形理论。二维流形是局部看起来像二维平面的空间。例如，可以考虑熟悉的三维物体（诸如足球或甜甜圈）的表面。 类似地，三维流形是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。 流形上几何结构如何变化的研究是近期数学和物理学中的一个重要主题，前者是瑟斯顿几何化猜想的解决方案，后者是弦论的解决方案。Canary教授将研究双曲三维流形中几何结构的变化。此外，Canary教授将继续他对本科教育的承诺，通过开创一门使用探究式方法的新课程，以及他指导研究生和博士后助理教授的工作。

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.