Flows, circles, and dynamics at infinity

摆可以被描述为二维“相空间”上的流动，其坐标由位置和速度给出。描述钟摆运动的数学系统是一个动力学系统。另一个例子是流体动力学，例如，它被用来研究飞机机翼上的空气流动。通常，相空间的结构与它所支持的动力系统有关。例如，球面上的任何流动都有一个固定点，这意味着相应的动力系统具有永远不会改变的状态。这个项目关注的是这个想法的反面：空间上的流动对空间本身意味着什么？我们考虑的空间是三维的，就像我们周围的空间一样；流被用作一种工具，将空间“展平”为更简单的一维和二维空间，其中原始空间的几何结构反映在这些展平的空间的对称性中。除了它在数学上的含义之外，这个项目还应用于应用动力学，其中平坦的空间可以用来理解流体流动的稳定性。拟Anosov流和伪Anosov流是乘积覆盖的，这意味着泛覆盖中的轨道空间是一个拓扑面。这个平面在无穷远处有一个自然圈，带有基本群的作用，它反映了流动的大尺度动力学和流形的几何形状。这个项目使用动力学、三维流形、几何群论和经典分析站点的技术来理解流、其底层流形和无穷远处的圆之间的关系。特别地，它的目的是证明Calegari的猜想，即每个拟偶极流都可以变形为伪Anosov流，刻画来自这种流的圆作用，并利用流来理解双曲三维流形中的立方体和本质曲面。

A pendulum can be described as a flow on a 2-dimensional "phase space," whose coordinates are given by position and velocity. The mathematical system that describes the motion of a pendulum is a dynamical system. Another example is fluid dynamics, which for instance, is used to study the flow of air over an airplane wing. Often, the structure of a phase space has implications for the dynamical systems that it supports. For example, any flow on the sphere has a stationary point, which means that the corresponding dynamical system has a state that never changes. This project is concerned with the obverse of this idea: What does a flow on a space say about the space itself? The spaces we consider are 3-dimensional, like the space surrounding us; the flows are used as a tool to "flatten" the space into simpler 1- and 2-dimensional spaces, where the geometric structure of the original space is reflected in the symmetries of these flattened spaces. In addition to its implications within mathematics, this project has applications to applied dynamics, where the flattened spaces can be used to understand the stability of fluid flows. Quasigeodesic and pseudo-Anosov flows are product covered, which means that the orbit space in the universal cover is a topological plane. This plane has a natural circle at infinity, with an action of the fundamental group, which reflects the large-scale dynamics of the flow and the geometry of the manifold. This project uses techniques from dynamics, 3-manifolds, geometric group theory, and classical analysis situs to understand the relationship between a flow, its underlying manifold, and the circle at infinity. In particular, it aims to prove Calegari's conjecture that every quasigeodesic flow may be deformed into a pseudo-Anosov flow, characterize the circle actions that come from such flows, and use flows to understand cubulations and essential surfaces in hyperbolic 3-manifolds.