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。这个数学研究项目集中在形状的概念和它们的演变，在科学的许多部分研究的基本对象。例子从复杂分子的形状，如蛋白质和酶，到太阳系的配置。在数学中，泰希穆勒理论（Teichmueller theory）是对曲面可以呈现的形状的研究。在这个项目中，表面，他们的测地线（最短路径），他们的演变将在两个不同的设置进行研究。在第一种情况下，表面具有平坦的几何形状，因此具有零曲率。在第二种情况下，曲面具有负曲率。该项目的成果将加深和扩展对几何和拓扑学的理解。该项目解决了广泛的数学问题，包括复分析，动力系统，几何和拓扑学领域。动力系统的重要例子是区间交换变换和平移曲面上的流。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。研究人员计划继续研究黎曼曲面模空间上Weil-Petersson测地线流的混合性质。了解这种流动将大大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;该项目旨在通过研究作用于Teichmueller空间的映射类群的晶格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.