Peter Lin - University of Washington, Mathematics Conformal welding is a way of gluing Riemann surfaces along their boundary; however, the existence and uniqueness of the resulting object is in general difficult to determine; this is the conformal welding problem. We prove that a natural conformal welding problem associated to the continuum random tree (CRT) has a (unique) solution, giving rise to a "canonical" random embedding of the CRT in the plane. Joint work with Steffen Rohde.