Topology and Dynamics of Geometric Structures

这个项目涉及流形上几何结构的分类。“几何结构”是指一组坐标系统，其中坐标位于具有“经典几何”的空间中：例如欧几里得几何、非欧几里得几何、射影几何、仿射几何或共形几何。自19世纪中期以来，对称性的考虑强调了这样一种观点，即经典几何只是研究在李群的传递作用下不变的性质。与此密切相关的是对称在现代物理学中的重要性，在这项研究中的许多对象都有物理动机。几何结构在固定拓扑上的分类意味着将更严格的几何测量放在松散组织的点集合上，这就是拓扑。一个简单的例子是球体不支持欧几里得几何。这只是对没有精确的世界地图集这一事实的数学抽象。至少有一页世界地图集会扭曲距离。另一方面，环面（甜甜圈、百吉饼或内胎的表面）确实承认欧几里得结构。事实上，将欧几里得几何放在环面上的不同方法有其丰富的几何意义。技术在调查和向公众传播方面都起着关键作用。许多数学家都参与了这个项目，包括同事、博士后、研究生、本科生甚至高中生。研究对象通常是二维和三维的，可以用计算机可视化。在这个项目中开发的所有软件都是公开的，并邀请进一步的教育和实验在这个研究项目。传统的几何结构分类方法涉及商空间，模空间往往是病态的。简单的例子通过混沌群作用得到好的空间商。PI采用的观点是，几何结构的分类实际上是一个动力系统。许多著名的有趣的动力系统都是在这种背景下产生的。特别是Teichmuller测地流在黎曼模空间上向普适特征变化的扩展具有丰富的动态性，影响着低维几何、拓扑和数学物理。PI将把这个动力系统作为发展良好的Teichmuller动力学和他之前在几何结构的特征变化和模空间上映射类群动力学的工作的融合来研究。

This project concerns the classification of geometric structures on manifolds. A "geometric structure" means a set of coordinate systems where the coordinates live in a space with a "classical geometry:" examples include Euclidean geometry, non-Euclidean geometry, projective geometry, affine geometry or conformal geometry. Since the mid 19th century, considerations of symmetry emphasize the viewpoint that a classical geometry is just the study of properties which are unchanged under the transitive action of a Lie group. Closely related is the importance of symmetry in modern physics, and many of objects in this investigation have physical motivations. Classification of geometric structures on a fixed topology means putting the more rigid geometric measurements on a loosely organized collection of points, which is the topology. A simple example is the fact that the sphere cannot support Euclidean geometry. This is just the mathematical abstraction of the fact that there is no metrically accurate world atlas. At least one page of the world atlas will distort distance. On the other hand the torus (the surface of a doughnut, bagel or inner tube) does admit Euclidean structures. In fact the different ways of putting Euclidean geometry on a torus has a rich geometry of its own. Technology plays a key tool both in this investigation and its dissemination to the public. Many mathematicians are involved in this project, including colleagues, postdocs, graduate students, undergraduate students and even high school students. The objects of study are often two- and three-dimensional and can be visualized with the use of computers. All the software developed in the project is publicly available and invites further education and experimentation in this research project.While the traditional methods of classification of geometric structures involve a quotient space, frequently the moduli spaces are pathological. Simple examples lead to quotients of nice spaces by chaotic group actions. The PI adopts the viewpoint that classification of geometric structures is really a dynamical system. Many well known interesting dynamical systems arise from this context. In particular the extension of the Teichmuller geodesic flow to the universal character variety over the Riemann moduli space has rich dynamics, impacting low-dimensional geometry, topology and mathematical physics. The PI will study this dynamical system as the fusion of the well-developed subject of Teichmuller dynamics and his previous work on mapping class group dynamics on character varieties and moduli spaces of geometric structures.