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。 重要的是要有明确的，定量的信息，几何结构，并了解它是如何连接的组合和拓扑性质的流形。 Minsky构造的用于求解Kleinian群中的终结层猜想的模型流形是由组合数据确定的，反映了映射类群的几何性质。这些模型在研究闭双曲三维流形时应该是有用的。 另一方面，在研究闭流形方面已经取得成功的技术，如双曲Dehn手术，变形理论和理想三角剖分，应该有助于进一步改进我们对Kleinian群的理解。 这个项目的目标是联合收割机的分析和几何技术开发了在过去的几年里，由PI和他的合作者与一些新的想法，已经出现在最近的解决方案中的主要programmes在Kleinian groups.The三维流形的研究结合几何，代数和分析工具。 它可以是一个非常平易近人的主题，因为研究的对象包括像结这样的东西，很容易描述和几何可视化。 即使是更深层次的性质，如双曲结构，也可以用当前的图形功能以吸引人的方式呈现。 另一方面，理解纽结和三维流形在数学的其他领域，甚至在物理学和生物化学方面都有重要的意义。

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.