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.
This project involved the study of geometric structures on manifolds. A manifold is a space which looks locally like Euclidean space of some dimension. Examples of manifolds include surfaces of 3-dimensional objects (which locally look like the plane) and our universe (which looks locally like 3-dimensional space). A geometric structure on a manifold is a way of measuring distances and angles on a manifold. One may think of the geometric structure as the ``shape'' of the manifold. It is then natural to study the space of all geometric structures of a certain type on a manifold. Teichmuller theory studies the space of hyperbolic geometric structures on a surface and has had important applications in physics (particularly in string theory) and in several fields of mathematics (including topology, geometry and dynamics). Many of the projects in this proposal were motivated by the beautiful classical results obtained in Teichmuller theory. In this project, the PI and his co-authors established significant new results concerning the structure of various spaces of geometric structures on manifolds. This brief report will mention three highlights. (1) In collaboration with Brock, Bromberg and Lecuire, the PI characterized algebraic convergence of sequences of hyperbolic geometric structures on manifolds which are the product of a surface and the line. (2) In collaboration with Magid, the PI studied dynamics on deformation spaces of hyperbolic geometric structures and the character varieties they live on. (3) In collaboration, with Bridgeman, Labourie and Sambarino the PI developed a metric on the Hitchin component, which is a higher rank analogue of Teichmuller space.
