Knot distortion and the geometry of surfaces

Distortion is a type of geometric knot invariant introduced by Gromov in 1983. It measures the smallest bi-Lipschitz constant required to embed the knot in R^3. A basic question about distortion is: given an infinite family of knots, how do we estimate the growth of their distortion? In this talk, I'll describe several results proving lower bounds for knot distortion. I'll also describe some ideas to understand knot distortion, that involve a connection with systolic geometry, as well as a connection with the combinatorics of curves on surfaces.