Geometry and Combinatorics of Hyperbolic 3-manifolds

在这个项目中，明斯基将研究双曲三维流形的几何结构和曲面的组合结构之间的联系。该项目的主要焦点是解决瑟斯顿的结束分层猜想的程序，该猜想指出，双曲三维流形由其拓扑类型和描述其渐近几何的不变量列表唯一确定。这个猜想是这个领域的中心，并将解决关于给定流形上所有双曲结构的变形空间结构的一些悬而未决的问题。这种方法的主要工具是曲线的复数，它是一个组合对象，它描述了给定曲面上的所有基本同伦类的集合。这样的复数被视为度量空间，在Cannon和Gromov意义下是双曲的，描述三维流形几何的不变量就是复数的“无穷大点”。通过研究曲线复数中的测地线，我们就可以为三维流形的几何构造一个模型，该模型只依赖于它的末端不变量。该计划的一部分是与布罗克和金丝雀的联合工作。曲线的复数在研究曲面上双曲结构的TeichMuller空间及其对称群，即所谓的映射类群中也是有用的。复形的双曲性蕴含着TeichMuller空间和映射类群的一种相对双曲性性质，Minsky希望利用这一性质来研究TeichMuller空间的几何性质，并回答关于映射类群的一些群论和算法问题。拓扑空间出现在数学中的不同抽象水平。例如，地球可以被认为是一个围绕其轴心旋转的球体。这个球体的所有位置的集合，也许就像纽约一天中的时间所参数化的那样，本身可以被可视化为一个圆。地球在围绕太阳的轨道上的位置也构成了一个圆，所有这些信息加在一起可以用一个称为环面的二维表面来描述，因此环面上的每个点都恰好对应于地-太阳系统的一个位置，我们在太空中的运动产生了绕环面的轨迹。一旦一个曲面被研究，我们通常会给它不同的结构：它可以有一个度量，而不是测量其中的距离；它也可以用环以不同的方式填充。所有可能的度量值的集合本身可以被研究为一个“空间”，在该空间中的运动对应于表面形状的变化。曲面中的所有回路的集合将上升到另一个空间，具有更强的组合性。后关联现象在数学中并不少见，它是这些不同抽象层次之间的类比现象。例如，曲面上单个度量的一个性质称为“双曲性”，它以另一种形式再次出现，它是曲面上所有度量空间和所有环的空间的一个性质。这些不同层次之间的相互作用产生了有趣的现象和工具，可以用来解决以下几个问题

DMS-0203976Yair MinskyIn this project Minsky will study connections between the geometricstructure of hyperbolic 3-manifolds, and the combinatorial structureof surfaces. The main focus of the project is a program to solveThurston's Ending Lamination Conjecture, which states that ahyperbolic 3-manifold is uniquely determined by its topological typeand a list of invariants describing the asymptotic geometry of itsends. This conjecture is central in the field and would settle anumber of outstanding questions about the structure of the deformationspace of all hyperbolic structures on a given manifold. The principaltool in this approach is the Complex of Curves, a combinatorial objectwhich describes the set of all essential homotopy classes of simpleloops on a given surface. Such a complex, viewed as a metric space,is hyperbolic in the sense of Cannon and Gromov, and the invariantswhich describe the geometry of a 3-manifold turn out to be "points atinfinity" for the complex. A study of geodesics within the complex ofcurves then allows us to construct a model for the geometry of the3-manifold which depends only on its end invariants. Part of theprogram is joint work with Brock and Canary. The complex of curves isalso instrumental in studying the Teichmuller space of hyperbolicstructures on a surface, and its symmetry group, known as the MappingClass Group. Hyperbolicity of the complex implies a type of relativehyperbolicity property for the Teichmuller space and the Mapping ClassGroup, and in collaboration with Brock and Masur, Minsky hopes toapply this to study the geometric properties of Teichmuller space, andto answer a number of group-theoretic and algorithmic questions aboutthe mapping class group.Topological spaces appear in mathematics at varying levels ofabstraction. The earth, for example, can be thought of as a sphere,spinning around its axis. The set of all positions of this sphere,perhaps as parameterized by the time of day in New York, can itself bevisualized as a circle. The positions of the earth in its orbitaround the sun also make up a circle, and all this informationtogether can be described by a 2-dimensional surface, called a torus,so that each point in the torus corresponds to exactly one position ofthe earth-sun system, and our motion in space gives rise to atrajectory winding around the torus. Once a surface is being studied,we often give it various structures: it can have a metric, which is away of measuring distance in it; it can also be filled up in differentways by loops. The set of all possible metrics can itself be studiedas a "space", where motion in this space corresponds to changingshapes of the surface. The set of all loops in the surface givesrise to another space, of a more combinatorial character. Afascinating phenomenon, not uncommon in mathematics, is the appearanceof analogies between these different levels of abstraction. Forexample, a property of a single metric on a surface called "hyperbolicity"appears again, in another guise, as a property of the space of allmetrics on the surface and the space of all loops. The interactionbetween these different levels yields interesting phenomena and toolswhich can be applied to solve a number of problems in