The shortest non-simple closed geodesics on hyperbolic surfaces

In joint work with Hugo Parlier and Hanh Vo we prove that, for sufficiently large k, the shortest closed geodesics on any hyperbolic surface with at least k self-intersections lie on an ideal pair of pants and are so-called corkscrew geodesics. These shortest possible geodesics have length 2 arccosh (2k+1). Previously the only known case was k=1, being the figure 8 on the ideal pair of pants. After  giving some background we will outline the proof of this result.