The Geometry of 3-manifolds

三维流形几何理论的中心目标之一是理解三维流形的拓扑、组合和几何性质之间的联系。重要的是要有关于几何结构的明确的、定量的信息，并了解它是如何与流形的组合和拓扑性质相联系的。由Minsky构造的用于求解Kleinan群中的End-Lamination猜想的模型流形由组合数据确定，并反映了映射类群的几何性质。这些模型对研究闭双曲三维流形具有一定的参考价值。另一方面，在研究封闭流形方面取得成功的技术，如双曲Dehn手术、形变理论和理想三角剖分，应该有助于进一步完善我们对Klein群的理解。这个项目的目标是将过去几年由PI和他的合作者开发的分析和几何技术与最近解决克莱恩群中的主要猜想时出现的一些新想法相结合。对三维流形的研究结合了几何、代数和分析工具。这可能是一个非常平易近人的学科，因为研究对象包括像结这样的东西，很容易用几何方法来描述和可视化。即使是更深层的性质，如双曲线结构，也可以用当前的图形功能以一种诱人的方式呈现出来。另一方面，理解纽结和三维流形可以在数学的其他领域产生重大影响，甚至在物理和生物化学方面也是如此。

One of the central goals in the geometric theory of 3-manifolds is understand the connection between the topological, combinatorial, and geometric properties of 3-manifolds. It is important to have explicit, quantitative information about a geometric structure and to understand how it is connected with the combinatorial and topological properties of the manifold. The model manifolds constructed by Minsky and used in the solution of the Ending Lamination Conjecture in Kleinian groups are determined by combinatorial data and reflect geometric properties of the mapping class group. These models should be useful in studying closed hyperbolic 3-manifolds. On the other hand, techniques that have been successful in studying closed manifolds, like hyperbolic Dehn surgery, deformation theory, and ideal triangulations should help provide further refinements in our understanding of Kleinian groups. The goal of this project is to combine the analytic and geometric techniques developed over the last several years by the PI and his collaborators with some of the new ideas that have emerged during the recent solutions of major conjectures in Kleinian groups.The study of 3-dimensional manifolds combines geometric, algebraic, and analytic tools. It can be a very approachable subject because the objects of study include things like knots that are easy to describe and visualize geometrically. Even the deeper properties, like hyperbolic structures, can be presented in an inviting manner with current graphical capabilities. On the other hand, understanding knots and 3-manifolds can have significant consequencesin other areas of mathematics and even in physics and biochemistry.