Closed Curves Probing Surfaces and Three-Manifolds

在表面上的闭合曲线结构中发现了一些显著的模式。这个项目的目标是在理论上解释尽可能多的这些模式，并发现进一步的结构。本研究将几何学、拓扑学与概率论、统计学相结合。这些工具包括使用PI开发的算法进行计算机实验，寻找模式，然后证明所发现的模式的严格结果。PI将让研究生和本科生参与与该项目相关的研究，她将组织石溪分校女性数学小组。PI还计划创建一系列可供其他研究人员使用的小程序。本课题的主要目标是了解具有负欧拉特征的可定向曲面上闭合曲线的自由同伦类的五个数的渐近统计结构。这些数值是自交点数、字长（即在选定的曲面基本群的一组生成器中表示曲线的单词的最小字母数）、唯一测地线的几何长度以及与其他曲线相互交点数分布的均值和方差。项目的另一部分是发展三维流形的弦拓扑的组合描述，并探索弦拓扑与三个流形几何化的关系。

Remarkable patterns have been discovered in the structure of closed curves lying on surfaces. The goal of this project is explain as many as possible of these patterns theoretically and to discover further structures. This research combines methods from geometry and topology with probability and statistics. The tools include computer experimentation using algorithms developed by the PI, finding patterns, and then proving rigorous results about the patterns that are discovered. The PI will engage graduate and undergraduate students in research related to the project and she will organize the Stony Brook Women in Mathematics Group. The PI also plans to create a series of applets that can be used by other researchers.The main goal of the project is to understand the asymptotic statistical structure of five numbers that can be associated to the free homotopy class of closed curves on an orientable surface with negative Euler characteristic. Those numerical values are the self-intersection number, the word-length (that is the smallest number of letters of a word representing the curve in a chosen set of generators of the fundamental group of the surface), the geometric length of the unique geodesic, and the mean and the variance of the distribution of mutual intersection numbers with other curves. Another part of the project concerns development of a combinatorial description of the string topology of manifolds in dimension three, and exploring a relationship of string topology with the geometrization of three manifolds.