Statistical, algebraic and combinatorial structures related to curves on surfaces and three manifolds

用基本群及其逆的生成元中的约化循环字来描述有边界可定向曲面上的定向闭曲线的连续变形。自相交数1表示该类的代表的横截自相交点的最小数目。PI旨在研究曲线变形类的自相交、几何长度和组合长度（单词的字母数）相关的统计。第二个目标是加深对定义在曲面上的曲线上的相对较新的代数结构（Goldman-Turaev李双代数）及其与现有结构的关系的理解。第三，研究Goldman-Turaev李双代数到三个流形的推广及其与子流形交结构的关系。曲面是“固体外层”的数学表示。曲面上的曲线，可以被认为是曲面上的“橡皮筋”。我们考虑给定曲面上的所有曲线，其中橡皮筋是拉紧的，并将某些量与每条曲线相关联，这些量与交叉点的长度和数量有关。该项目的数学目标之一是研究这些量之间的统计关系。这个项目的另一个数学目标将是研究曲面上的曲线上定义的代数结构，以及将这种结构推广到三维空间，这些空间是固体的数学表示，例如，甜甜圈或球，以及我们生活的空间。 曲面和三流形在数学的许多分支中起着至关重要的作用，并且有着超越它们的应用-潜在的蛋白质折叠和生物体中的细胞排列。第三个目标是教育：由于这个项目的某些方面不需要广泛的数学知识，并且可能的语句可以通过计算机实验进行测试，因此我们将让本科生参与执行。这将使他们能够“从内部”学习数学。

Oriented closed curves on an orientable surface with boundary are described up to continuous deformation by reduced cyclic words in the generators of the fundamental group and their inverses. By self-intersection number one means the minimum number of transversal self-intersection points of representatives of the class. The PI aims at studying statistics relating self-intersection, geometric length and combinatorial length (number of letters of the word) of deformation classes of curves. The second goal is to deepen the understanding of a relatively new algebraic structure defined on curves on surfaces (the Goldman-Turaev Lie bialgebra) and its relation to existing structures. The third is studying a generalization of the Goldman-Turaev Lie bialgebra, to three manifolds and its relation with intersection structure of submanifolds.A surface is the mathematical representation of the "outer layer of a solid". A curve on a surface, can be thought of as a "rubber band" on the surface. We consider all curves on a given surface, where the rubber band is taut, and associate certain quantities to each curve, related to the length and the number of crossings. One of the mathematical goals of this project consists in studying statistical relations between this quantities. The other mathematical goal of this project will be studying an algebraic structure defined on curves on surface, and a generalization of this structure to three dimensional spaces which are the mathematical representation of solids, like, for instance, a donut or a ball, and also, the space we live in. Surfaces and three manifolds play a crucial role in many branches of mathematics and have applications beyond them - potentially protein folding and cellular arrangements in organisms. The third goal is educational: Since some aspects of this project do not require an extensive knowledge of mathematics, and possible statements can be tested by computer experiments, we will involve undergraduate students in the executions. This will allow them to learn mathematics "from the inside ".