Conjugation of words, self-intersections of planar curves, and non-commutative divergence
MATHEMATICAL PICTURE LANGUAGE
Anton Alekseev - University of Geneva
The space spanned by homotopy classes of free oriented loops on a 2-manifold carries an interesting algebraic structure (a Lie bialgebra structure) due to Goldman and Turaev. This structure is defined in terms of intersections and self-intersections of planar curves. In the talk, we will explain a surprising link between the Gaoldman-Turaev theory and the Kashiwara-Vergne problem on properties of the Baker-Campbell-Hausdorff series. Important tools in establishing this link are the non-commutative divergence cocycle and a novel characterization of conjugacy classes in free Lie algebras in terms of cyclic words. The talk is based on joint works with N. Kawazumi, Y. Kuno and F. Naef.