Closed Curves Probing Surfaces and Three-Manifolds

在曲面上闭合曲线的结构中已经发现了显著的图案。这个项目的目标是从理论上解释尽可能多的这些模式，并发现更多的结构。本研究将几何学和拓扑学的方法与概率统计相结合。这些工具包括使用PI开发的算法进行计算机实验，找到模式，然后证明关于发现的模式的严格结果。她将邀请研究生和本科生参与与该项目相关的研究，并将组织石溪女性数学小组。PI还计划创建一系列可供其他研究人员使用的Applet。该项目的主要目标是了解五个数的渐近统计结构，这些数可以与具有负Euler特征的可定向曲面上的自由同伦闭曲线类相关联。这些数值是自交数、字长(即在曲面的基本群的所选生成元集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的相互交数的分布的均值和方差。该项目的另一部分涉及开发三维流形的弦拓扑的组合描述，并探索弦拓扑与三个流形的几何化的关系。

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.