Geometry and Dynamics in Low Dimensional Topology

数学的一个主要作用是为科学研究提供框架或语言。这个数学研究项目的中心是形状及其演变的概念，这是许多科学领域的一个基本研究对象。例子包括从复杂分子的形状，如蛋白质和酶，到太阳系的结构。在数学中，Teichmueller理论研究的是曲面可以呈现的形状。在这个项目中，曲面、它们的测地线（最短路径）和它们的演化将在两种不同的环境中进行研究。在第一种情况下，曲面具有平坦的几何形状，因此曲率为零。在第二种情况下，曲面具有负曲率。该项目的结果将加深和扩展对几何和拓扑的理解。该项目涉及广泛的数学问题，包括复杂分析、动力系统、几何和拓扑等领域。动态系统的重要例子是区间交换变换和平移表面上的流动。这些动力系统的重要方面是遍历性和计数问题。单个平移曲面的性质与所有平移曲面的模空间的研究密切相关。该项目涉及这两个主题的问题。研究者计划继续研究Weil-Petersson测地线流在黎曼曲面模空间上的混合特性。这种流动的知识将极大地扩展我们对黎曼曲面几何的理解。拓扑学和几何学中的一个基本对象是曲面的映射类群；本课题旨在通过研究作用于Teichmueller空间的映射类群的点阵计数问题来进一步理解。

A major role of mathematics is to provide a framework or language for scientific research. This mathematical research project centers on the concept of shapes and their evolution, a fundamental object of study in many parts of science. Examples range from the shapes of complicated molecules, such as proteins and enzymes, to the configurations of the solar system. In mathematics, Teichmueller theory is the study of the shapes that a surface can assume. In this project, surfaces, their geodesics (shortest paths), and their evolution will be studied in two different settings. In the first, the surfaces have a geometry in which they are flat and so have zero curvature. In the second, the surfaces have negative curvature. Results of the project will deepen and extend understanding in geometry and topology.This project addresses questions in a wide spectrum of mathematics, including the fields of complex analysis, dynamical systems, geometry, and topology. Important examples of dynamical systems are interval exchange transformations and flows on translation surfaces. Important aspects of these dynamical systems are ergodicity and counting problems. The properties of an individual translation surface are intimately related to the study of the moduli space of all translation surfaces. The project concerns questions in both these subjects. The investigator plans to continue study of the mixing properties of the Weil-Petersson geodesic flow on the moduli space of Riemann surfaces. Knowledge of this flow will greatly expand our understanding of the geometry of Riemann surfaces. A fundamental object in topology and geometry is the mapping class group of a surface; the project aims to further understanding by studying the lattice counting problem for the mapping class group acting on Teichmueller space.