Structure and Deformation in Low-Dimensional Topology

低维拓扑中的结构和变形

• 批准号: 1610827
• 负责人: Yair Minsky
• 金额: $ 37万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-08-01 至 2020-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1610827&HistoricalAwards=false
• 关键词: Structure; Deformation; Low; Dimensional; Topology

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.

两个和三个维度的几何和拓扑表现出特殊的结构丰富性，其中许多不同的数学领域都聚集在一起，包括经典的复杂分析，组合拓扑，几何学，动力学系统和群体理论。当三维空间（流形）沿其二维边界粘合在一起时，所得空间的特性以复杂的方式取决于胶合图的性质。胶合地图本身形成了一个丰富的代数结构，称为映射类组。该研究项目探讨了这种结构发挥作用的许多设置。映射类组的特殊几何特性被称为相对双曲线，这使得三维流形的精细几何结构可以被嘲笑并分析。该领域的一些先前进步已经产生了一般性声明，但没有提出具体的估计，并且在许多领域，研究人员希望能够增强当前的理解并计算混凝土界限。然而，重点是加深对二维和三维流形融合在一起的方式的概念理解。该项目调查了表面和三个manifolds拓扑之间相互作用的各个方面，并与Teichmuller空间中的几何结构和地球流量的变形空间进行了连接。工作中的一个共同线程是映射类组的分层结构，以及通过曲线复合物的相对双曲线的现象。在一个关于纤维双曲线三个序列的项目中，研究者将探索所有纤维组集的“地下剖面”，即投影模式到与单构图相关的稳定且不稳定叶子的纤维子表面的模式。这里的目的是获得统一的（独立属）描述，并将其与弯曲三角形的组合结构联系起来。在Teichmuller理论中，研究人员将猜想对表面模量空间上的Weil-Petersson测量流进行猜想，从而开发新工具，以分析地球射线的组合模式或行程。研究人员将研究由瑟斯顿定义的皮肤图，在双曲线三个manifold边界的Teichmuller空间上。他将致力于在酰基偶然的情况下对以这种地图闻名的直径界限进行更好的定量控制。在一般情况下，他将扩展以前的工作，以建立相对有限的图像结果，以完成瑟斯顿在斜线上迭代术的索赔的证明，在胶水产生粘合剂的情况下，粘膜和镀膜图。这项工作还将连接到一个项目，以建立针对双曲线三个manifolds的新统一Bi-Lipschitz模型。