Curves on surfaces and three manifolds: string topology and other structures

Moira Chas

Consider an orientable surface S with negative Euler characteristic, a minimal set of generators of the fundamental group of S, and a hyperbolic metric on S. Then each unbased homotopy class C of closed oriented curves on S determines three numbers: the word length (that is, the minimal number of letters needed to express C as a cyclic word in the generators and their inverses), the minimal geometric self­intersection number, and finally the geometric length. Also, the set of free homotopy classes of closed directed curves on S (as a set) is the vector space basis of a Lie algebra discovered by Goldman. This Lie algebra is closely related to the intersection structure of curves on S. These three numbers, as well as the Goldman Lie bracket of two classes, can be explicitly computed (or approximated) using a computer. These computations led us to counterexamples to existing conjectures, to formulate new conjectures and (sometimes) to subsequent theorems.

http://scgp.stonybrook.edu/video_portal/video.php?id=3627

Simons- Program: Poisson geometry of moduli spaces, associators and quantum field theory