Geometry and Combinatorics of Hyperbolic 3-manifolds

DMS-0203976 Yair Minsky在这个项目中，Minsky将研究双曲三维流形的几何结构与曲面的组合结构之间的联系。该项目的主要重点是解决Thurston的结束层压猜想的程序，该猜想指出双曲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确定的。这个猜想是该领域的中心，并将解决一系列悬而未决的问题的结构deformationspace的所有双曲结构在一个给定的流形。 在这种方法中的principaltool是复杂的曲线，一个组合的对象，它描述了一个给定的表面上的所有基本同伦类的单回路。 这样一个复形，作为度量空间，在Cannon和Gromov意义下是双曲的，描述三维流形几何的不变量是复形的“点集”。对复曲线中测地线的研究使我们能够构造一个仅依赖于其端点不变量的三维流形几何模型。 该计划的一部分是与布洛克和金丝雀联合工作。 曲线的复形也有助于研究表面上双曲结构的Teichmuller空间及其对称群，称为映射类群。复形的双曲性意味着Teichmuller空间和映射类群的一种相对双曲性性质，在与Brock和Masur的合作中，Minsky希望将其应用于研究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。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中的时间来参数化，它本身可以被可视化为一个圆。 地球在绕日轨道上的位置也构成一个圆，所有这些信息加在一起可以用一个二维表面来描述，这个二维表面被称为环面，环面上的每一点都恰好对应于地球-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一个轨道。 一旦我们研究了一个曲面，我们通常会赋予它不同的结构：它可以有一个度量，在它里面度量距离;它也可以用不同的方式被环填充。所有可能度量的集合本身可以作为一个“空间”来研究，在这个空间中的运动对应于表面形状的变化。在表面上的所有循环的集合产生另一个空间，一个更组合的字符。 在数学中常见的一种模糊现象是这些不同抽象层次之间的类比。例如，一个属性的一个单一的度量在一个表面上称为“双曲“再次出现，在另一个幌子，作为一个属性的空间的所有度量的表面和空间的所有循环。这些不同层次之间的相互作用产生了有趣的现象和工具，可以用来解决一些问题，

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