GEOMETRY AND DYNAMICS ON MODULI SPACE

首席研究员(PI)将致力于TeichMuller空间、Riemann曲面的模空间、多边形和平移曲面上的定向流以及映射类群的几何和动力学问题。这些问题涉及到相当广泛的数学领域，包括复杂分析、动力系统和几何。该项目第一部分的目的是研究模空间上的Weil-Petersson流及其测地线的几何。这一度量是研究模空间性质的基础。Weil-Petersson度量与曲面的双曲几何和动力学之间有许多重要的联系。第二部分研究了Teichmuller测地线的性质及其与曲面上流动的关系，第三部分研究了Teichmuller度量下Teichmuller空间的刚性问题。项目的最后一部分涉及映射类群在TeichMuller空间及其无穷远边界上的作用。如果该项目的目标得以实现，它将增加我们对这些主题的理解。数学的一个主要作用是提供一个框架或语言来讨论科学的过程。这个项目中开发的框架以形状的概念为中心。在数学上，泰希穆勒理论和模空间理论是对曲面可以呈现的形状的研究。空间中的测地线代表了一种在空间中移动或演化的有效方式，而对测地线的研究是几何学的核心。Weil-Petersson测地线和Teichmuller测地线都描述了曲面形状随时间高效演变的自然且截然不同的方式。本项目涉及这两个不同过程下的这种演变的研究。

The principal investigator (PI) will work on problems in the geometry and dynamics of Teichmuller space, moduli space of Riemann surfaces, directional flows on polygons and translation surfaces, and the mapping class group. These problems involve a fairly wide spectrum of mathematics including complex analysis, dynamical systems, and geometry. The objective of the first part of the project is to study the Weil-Petersson flow on moduli space and the geometry of its geodesics. This metric is fundamental in studying properties of moduli space. There are many important connections between the Weil-Petersson metric and the hyperbolic geometry and dynamics of surfaces. In the second part the goal is to study properties of Teichmuller geodesics and their relation to flows on surfaces, and in the third part rigidity questions about Teichmuller space with the Teichmuller metric. The last part of the project concerns the action of the mapping class group on Teichmuller space and on its boundary at infinity. If the goals of the project are met, it would add to our understanding of these subjects.A major role of mathematics is to provide a framework or language to discuss processes of science. The framework developed in this project centers on the concept of shape. In mathematics, Teichmuller theory and moduli space theory is the study of the shapes that a surface can assume. Geodesics in a space represent an efficient way of moving or evolving in the space and the study of geodesics is central to geometry. Both Weil-Petersson geodesics and Teichmuller geodesics describe natural and very different ways in which the shapes of surfaces evolve efficiently over time. This project concerns the study of this evolution under these two different processes.