Mapping class groups, curve complexes, and Teichmueller spaces

映射类群和Teichmueller空间在我们对几何和拓扑的理解中起着很大的作用。拓扑学是研究没有它们的几何的形状和空间，所以一个人可以自由地弯曲和拉伸他感兴趣的形状，但不能撕裂它们。映射类群是曲面的拓扑对称。映射类群的最生动的例子是辫子群。这些是有孔圆盘的对称性。关于辫子群有几种想法。想象一下罐子里粘性液体的表面，里面浸泡着棒子。人们可以互换棒子，而不需要移除它们，这会搅动液体。棒子可以回到它们的起始位置，但液体的表面已经改变；它已经混合了。流体的表面已经经历了拓扑对称。我们也可以把辫子群看作三维空间中的弦的纠缠。我们可以从竖线开始，而不是从液体中的杆子开始，竖线的最低点粘在罐子的底部。通过抓住弦的顶端，我们可以执行与我们对杆所做的相同的运动和换位，这将使弦缠绕在一起：它产生辫子。这两种不同的视角是等价的。在科学上，编织基团的三维观点被应用于聚合物和DNA链。二维视点被应用于拓扑量子计算和机器人技术。映射类组是称为组的抽象代数对象的示例。群是对称性的集合：如果数字衡量的是大小，那么群就是衡量对称性的。令人惊讶的是，我们可以通过将群理解为几何空间的对称性来了解群：这被称为几何群论。研究映射类群的一个有用的几何空间是Teichmueller空间。今天，几何群论的快速发展领域在包括三维流形的几何和拓扑、复动力学、组合群论、表示论、逻辑和代数几何在内的各个领域的最新进展中发挥着不可或缺的作用。此外，还有一些几何群论的概念被用于大型数据分析中，本项目的目的是实现几何群论的深远技术来研究映射类群和Teichmueller空间，它们是与曲面相关的基本对象。更具体地说，我们的目标是使用三维流形理论的最新突破的概念来研究映射类群，并使用诸如曲线复形的概念-这些概念提供了双曲几何的主要发展-来研究Teichmueller空间。

Mapping class groups and Teichmueller spaces play a large role in our understanding of geometry and topology.Topology is the study of shapes and spaces without their geometry, so one is free to bend and stretch the shapes one is interested in but not tear them. The mapping class groups are the topological symmetries of surfaces. The most vivid examples of mapping class groups are the braid groups. These are the symmetries of a disc with holes.There are several ways of thinking about the braid groups. Imagine the surface of a viscous fluid in a pot in which rods are immersed. One can interchange the rods, without removing them, which stirs the fluid. The rods can return to their starting positions but the surface of the fluid has changed; it has been mixed. The surface of the fluid has undergone a topological symmetry. We may also regard the braid groups as tangles of string in 3-dimensional space. Instead of rods in a fluid, we can start with vertical strings whose lowest points are glued to the base of the pot. By taking hold of the tops of the strings, we can perform the same movements and transpositions as we did with the rods, and this tangles the strings up: it produces braids. These two different perspectives are equivalent. In science, the 3-dimensional point of view of braid groups is applied to polymers, and strands of DNA. The 2-dimensional point of view is applied to topological quantum computing and robotics. The mapping class groups are examples of abstract, algebraic objects called groups. A group is a collection of symmetries: if numbers measure size then groups measure symmetry. Surprisingly, we can learn much about a group by realizing it as the symmetries of a geometric space: this is called geometric group theory. One such useful geometric space for studying the mapping class group is the Teichmueller space.Today, the fast growing area of geometric group theory plays an indispensable role in the recent advances of diverse fields including the geometry and topology of 3-manifolds, complex dynamics, combinatorial group theory, representation theory, logic, and algebraic geometry. Furthermore, there are notions from geometric group theory that are used in large data analysis.The purpose of this project is to implement the far-reaching techniques of geometric group theory to study the mapping class groups and the Teichmueller spaces, which are fundamental objects associated to surfaces. More specifically, we aim to use notions from the latest breakthroughs in 3-manifold theory to study the mapping class groups, and use concepts such as the curve complex---which have provided major developments in hyperbolic geometry---to investigate the Teichmueller space.