Geometry and Combinatorics of Hyperbolic 3-manifolds

在这个项目中，Minsky将研究双曲型3流形的几何结构与曲面组合结构之间的联系。该项目的主要焦点是解决thurston 's Ending Lamination猜想的程序，该猜想指出，非双曲3流形是由其拓扑类型和描述其渐近几何的不变量列表唯一决定的。这一猜想是该领域的核心，它将解决关于给定流形上所有双曲结构的变形空间结构的许多悬而未决的问题。这种方法的主要工具是曲线复合体，它是一个组合对象，描述了给定曲面上所有基本同伦类的简单环的集合。这样一个复合体，被看作一个度量空间，在Cannon和Gromov的意义上是双曲的，描述3流形几何的不变量是复合体的“无穷点”。对曲线复合体内测地线的研究使我们能够为3流形的几何构造一个模型，该模型仅依赖于它的末端不变量。该计划的一部分是与布洛克和加那利的联合工作。曲线复合体也有助于研究曲面上双曲结构的Teichmuller空间及其对称群，即映射类群。复体的双曲性暗示了Teichmuller空间和映射类群的一种相对双曲性，Minsky希望与Brock和Masur合作，将其应用于研究Teichmuller空间的几何性质，并回答一些关于映射类群的群论和算法问题。拓扑空间在数学中以不同的抽象层次出现。例如，地球可以被认为是一个球体，绕着它的轴旋转。这个球体的所有位置的集合，也许可以用纽约一天的时间来参数化，它本身可以被想象成一个圆。地球在绕太阳公转轨道上的位置也构成了一个圆，所有这些信息可以用一个叫做环面的二维表面来描述，因此环面上的每一点都精确地对应着地球-太阳系统的一个位置，我们在空间中的运动产生了绕环面的轨迹。一旦一个表面被研究，我们通常会给它不同的结构：它可以有一个度量，用来测量它的距离；它也可以通过循环以不同的方式填充。所有可能度量的集合本身可以作为一个“空间”来研究，其中这个空间中的运动对应于表面形状的变化。表面上所有循环的集合产生了另一个具有更多组合特征的空间。这些不同层次的抽象概念之间的类比现象，在数学中并不少见，这是一种令人着迷的现象。例如，曲面上的一个度量的性质被称为“双曲性”，以另一种形式再次出现，作为曲面上所有度量的空间和所有环路的空间的性质。这些不同层次之间的相互作用产生了有趣的现象和工具，这些现象和工具可用于解决数学中的许多问题

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