Intersections and Self-Intersections of Arcs and Curves on Surfaces

We give algebraic formulas for the minimal intersection and self-intersection numbers of proper arcs and curves on oriented surfaces. In the case of closed curves, the formulas are given in terms of the Andersen-Mattes-Reshetikhin Poisson algebra of chord diagrams on a surface, and a related operation $\mu$. The operation $\mu$ can be viewed as a generalization of Turaev's Lie coalgebra structure on the vector space generated by nontrivial free homotopy classes of loops on a surface. Some of the work presented is jointwith Vladimir Chernov.