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.[-]

The crossing lemma for simple graphs gives a lower bound on the necessary number of crossings of any planar drawing of a graph in terms of its number of edges and vertices. Viewed through the lens of topology, this leads to other questions about arcs and curves on surfaces. Here is one: how many crossings do a collection of m homotopically distinct curves on a surface of genus $g$ induce? The talk will be about joint work with Alfredo Hubard where we explore some of these, using tools from the hyperbolic geometry of surfaces in the process.[-]

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[-]