Mapping class groups, curve complexes, and Teichmueller spaces

映射类群和Teichmueller空间在我们理解几何和拓扑学中起着重要作用。拓扑学是研究没有几何的形状和空间，所以人们可以自由地弯曲和拉伸感兴趣的形状，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。这些是带孔圆盘的对称性。有几种方法来考虑辫群。想象一下，在一个锅中的粘性流体的表面，杆浸入其中。人们可以互换杆，而不删除他们，这搅拌流体。杆可以回到它们的起始位置，但流体的表面已经改变;它已经混合。流体的表面经历了拓扑对称。我们也可以把辫群看作是三维空间中的弦的缠结。我们可以从垂直的弦开始，而不是流体中的杆，弦的最低点粘在罐子的底部。通过抓住琴弦的顶端，我们可以做和用竿一样的运动和换位，这样就把琴弦缠结起来了：它产生了辫子。这两种不同的视角是等价的。在科学中，三维的观点被应用于聚合物和DNA链。二维的观点被应用到拓扑量子计算和机器人。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：如果数字度量大小，那么群度量对称性。令人惊讶的是，我们可以通过将一个群理解为几何空间的对称性来了解它：这被称为几何群论。Teichmueller空间是研究映射类群的一个有用的几何空间。如今，几何群论的快速发展领域在三维流形的几何和拓扑、复动力学、组合群论、表示论、逻辑和代数几何等各个领域的最新进展中发挥着不可或缺的作用。此外，还有几何群论的概念，用于大数据分析。本项目的目的是实施几何群论的深远技术，以研究映射类群和Teichmueller空间，这是与曲面相关的基本对象。更具体地说，我们的目标是使用的概念，从最新的突破3-流形理论研究映射类组，并使用的概念，如曲线复杂-这提供了重大发展双曲几何-调查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.