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 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 self-intersection 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. We will discuss the algorithms to compute or approximate these numbers, and how these computer experiments led to counterexamples to existing conjectures, formulations of new conjectures and (sometimes) subsequent theorems.This talk means to be accessible to mathematically young people.These results are joint work with different collaborators; mainly Arpan Kabiraj, Steven Lalley and Rachel Zhang.[-]

Though the term translation surface is relatively new, the notion is quite old (Abelian differential) and in fact predates the notion of hyperbolic surface. The term translation surface emphasizes the (G, X) point of view introduced to their study by Thurston in about 1980. Since then, ideas from both topology and dynamics have revolutionized the study of these structures. In this talk I will describe some of these ideas with preference given to those ideas that connect to both Delaunay triangulations and hyperbolic geometry. This talk is based on joint work with Allen Broughton. See https://arxiv.org/abs/1003.1672[-]

Fock and Goncharov give parameterisations of two different types of moduli spaces of properly convex real projective structures. I'll discuss a number of observations made about these parameterisations, the geometric structures that are parameterised by them, their relationship with representations into SL(3,R), canonical cell decompositions, and compactifications. This includes joint work with Alex Casella, Robert Haraway, Robert Löwe and Dominic Tate.[-]

How much cutting is needed to simplify the topology of a surface? We provide bounds for several instances of this question, for the minimum length of topologically non-trivial closed curves, pants decompositions, and cut graphs with a given combinatorial map in triangulated combinatorial surfaces (or their dual cross-metric counterpart).
Our work builds upon Riemannian systolic inequalities, which bound the minimum length of non-trivial closed curves in terms of the genus and the area of the surface. We first describe a systematic way to translate Riemannian systolic inequalities to a discrete setting, and vice-versa. This implies a conjecture by Przytycka and Przytycki from 1993, a number of new systolic inequalities in the discrete setting, and the fact that a theorem of Hutchinson on the edge-width of triangulated surfaces and Gromov's systolic inequality for surfaces are essentially equivalent. We also discuss how these proofs generalize to higher dimensions.
Then we focus on topological decompositions of surfaces. Relying on ideas of Buser, we prove the existence of pants decompositions of length $O(g^{3/2}n^{1/2})$ for any triangulated combinatorial surface of genus g with n triangles, and describe an $O(gn)$-time algorithm to compute such a decomposition.
Finally, we consider the problem of embedding a cut graph (or more generally a cellular graph) with a given combinatorial map on a given surface. Using random triangulations, we prove (essentially) that, for any choice of a combinatorial map, there are some surfaces on which any cellular embedding with that combinatorial map has length superlinear in the number of triangles of the triangulated combinatorial surface. There is also a similar result for graphs embedded on polyhedral triangulations.
systolic geometry - computational topology - topological graph theory - graphs on surfaces - triangulations - random graphs[-]