Dynamics in Hyperbolic Geometry and Teichmuller Theory

这个项目是动力学、几何和拓扑学的交叉。该项目旨在研究双曲几何和Teichmuller理论中的各种动力学问题。主要的重点将是继续研究作用于双曲空间的群上的随机漫步和作用于Teichmuller空间的映射类群。我们的目标是进一步扩展这两种情况之间的类比，特别是表明这些空间边界上的相关调和测度具有相似的性质。其他项目包括非经典区间交换的动力学方面的研究和曲线复合体上伪阿诺索夫映射的平移长度的继续研究。本文还探讨了纤维双曲3-流形第二同调上Thurston范数沿纤维面扩张与曲线复平移长度之间的关系。表面理论，如球或甜甜圈的表面，是许多数学领域的基本兴趣。泰希穆勒理论是研究曲面可能呈现的形状的理论。该建议关注这些形状以及它们与表面的某些转换（称为映射类）的相互作用。主要目标是解释形状在随机变换下是如何演变的。其他项目包括理解转换的复杂性。复杂性的一个度量是曲面上通过重复变换得到的曲线的复杂性。提案中概述的问题在几何、拓扑和动力系统等各个领域都很重要。希望该提案将启动富有成效的合作，并提供从博士后到本科生的几个层次的指导机会。

This project lies at the intersection of dynamics, geometry and topology. The project aims to study various dynamical questions in hyperbolic geometry and Teichmuller theory. The main focus will be to continue the study of random walks on groups acting on hyperbolic space and the mapping class group acting on Teichmuller space. The goal is to extend further the analogy between the two situations, specifically to show that the associated harmonic measures on the boundaries of these spaces share similar properties. Other projects include the study of dynamical aspects of non-classical interval exchanges and continuation of the study of translation lengths of pseudo-Anosov maps on the curve complex. The relationship between dilatations and curve complex translation lengths along a fibered face for the Thurston norm on the second homology of a fibered hyperbolic 3-manifold will also be explored.The theory of surfaces such as the surface of the ball or the donut, is of fundamental interest across many fields of mathematics. Teichmuller theory is the study of the possible shapes a surface can assume. The proposal focuses on these shapes and their interactions with certain transformations of the surface called mapping classes. The main goal is to explain how the shape evolves under random transformations. Other projects include understanding the complexity of the transformations. One measure of complexity is the complexity of curves obtained on the surface by repeating a transformation. The problems outlined in the proposal are important in varied fields like geometry, topology and dynamical systems. It is hoped that the proposal will initiate fruitful collaborations and offer mentoring opportunities at several levels from postdoctoral fellows to undergraduate students.