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

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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 ".