Closed Curves Probing Surfaces and Three-Manifolds

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.