Structure and Deformation in Low-Dimensional Topology

二维和三维的几何学和拓扑学展示了一种特别丰富的结构，其中许多不同的数学领域走到了一起，包括经典的复分析，组合拓扑学，几何学，动力系统和群论。当三维空间（流形）沿其二维边界沿着粘合在一起时，所得空间的性质以复杂的方式依赖于粘合映射的性质。胶合映射本身形成了一个丰富的代数结构，称为映射类组;这个研究项目探讨了一些设置，这种结构发挥作用。一个特殊的作用是由一个特殊的几何性质的映射类组称为相对双曲性，这使得精细的几何结构的三维流形梳理和分析。在这一领域的一些以前的进展已经产生了一般性的声明，但没有具体的估计，并在一些领域的研究人员预计能够提高目前的理解和计算具体的界限。然而，重点是加深对二维和三维流形结合在一起的方式的概念性理解。该项目研究了曲面拓扑和三流形之间的相互作用，以及与几何结构的变形空间和Teichmuller空间中的测地线流的连接。在工作中的一个共同点是映射类组的层次结构和通过曲线复合体的相对双曲现象。在一个关于纤维双曲三流形的项目中，研究人员将探索所有纤维化集合的“地下剖面”，即与单值映射相关的稳定和不稳定叶理的纤维子表面的投影模式。这里的目标是获得统一（属独立）的描述，并将它们与转向三角剖分的组合结构。在Teichmuller理论中，研究人员将对表面的模空间上的Weil-Petersson测地线流进行结构描述，开发新的工具来分析测地线射线的组合模式或行程。研究了Thurston在双曲三流形边界的Teichmuller空间上定义的蒙皮映射。他将致力于一个更好的定量控制的直径范围已知的地图在acylinertia的情况下。在一般情况下，他将延长以前的工作，建立了相对有界图像的结果，以完成证明的索赔瑟斯顿迭代的皮肤和胶合地图的设置胶合产生一个atoroidal流形。本文的工作也将与建立双曲三维流形的新的统一双Lipschitz模型的项目相联系。

Geometry and topology in two and three dimensions display a particular richness of structure where many different fields of mathematics come together, including classical complex analysis, combinatorial topology, geometry, dynamical systems, and the theory of groups. When three-dimensional spaces (manifolds) are glued together along their two-dimensional boundaries, the properties of the resulting space depend in complex ways on the properties of the gluing maps. The gluing maps themselves form a rich algebraic structure known as a mapping class group; this research project explores a number of settings where this structure comes into play. A particular role is played by a special geometric property of the mapping class group known as relative hyperbolicity, which has enabled the fine geometric structure of three-dimensional manifolds to be teased out and analyzed. Some of the previous advances in this area have yielded general statements but not concrete estimates, and in a number of areas the investigator expects to be able to sharpen current understanding and compute concrete bounds. The focus, nevertheless, is to deepen conceptual understanding of the way that two-dimensional and three-dimensional manifolds fit together. The project investigates aspects of the interaction between topology of surfaces and three-manifolds, with connections to deformation spaces of geometric structures and geodesic flows in Teichmuller space. A common thread in the work is the hierarchical structure of mapping class groups and the phenomenon of relative hyperbolicity via curve complexes. In a project on fibered hyperbolic three-manifolds, the investigator will explore the "subsurface profiles" of the set of all fibrations, namely the pattern of projections to fiber subsurfaces of the stable and unstable foliations associated to the monodromy maps. The goal here is to obtain uniform (genus independent) descriptions and to relate them to the combinatorial structure of veering triangulations. In Teichmuller theory, the investigator will pursue a conjectural description of the Weil-Petersson geodesic flow on the moduli space of a surface, developing new tools with which to analyze the combinatorial patterns or itineraries of geodesic rays. The investigator will study the skinning map defined by Thurston on the Teichmuller space of the boundary of a hyperbolic three-manifold. He will work toward a better quantitative control of the diameter bounds known for such maps in the acylindrical case. In the general case he will extend previous work that established relative bounded image results, to complete the proof of a claim of Thurston on iterates of the skinning-and-gluing maps in the setting where a gluing yields an atoroidal manifold. This work will also connect to a project on establishing new uniform bi-Lipschitz models for hyperbolic three-manifolds.