Appears in collection : Probability and Geometry in, on and of non-Euclidian spaces / Probabilités et géométrie dans, sur et des espaces non-euclidiens

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.