Moira Chas | Tantalizing patterns created by curves on surfaces

Four Decades of the Einstein Chair Seminar: einstein-chair...
January 19, 2023
Abstract:
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. Each unbased homotopy class C of closed oriented curves on S determines three numbers: the minimal geometric self-intersection number, the geometric length, and the word length (that is, the minimal number of letters needed to express C as a cyclic reduced word in the generators and their inverses). 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. We will discuss the algorithms to compute or approximate these numbers, and how these computer experiments led to counterexamples to existing conjectures, to formulate new conjectures and (sometimes) to subsequent theorems.
(These results are joint work with different collaborators; mainly Arpan Kabiraj, Steven Lalley and Rachel Zhang)
CUNY Einstein Mathematics Seminar: goo.gl/MsQrHq