If K is a connected, locally connected, compact subset of the plane, then a Riemann map f: C-D \to C-K extends continuously to the unit circle. Thus K induces an equivalence relation on the circle $x\sim y iff f(x)=f(y)$.
One can ask the converse question: given an equivalence relation on the circle, is there a conformal map realizing this equivalence?
This is essentially the conformal welding problem. In this talk we discuss some criteria for the existence for such a map. We give example applications of this criteria for equivalence relations arising in dynamics and probability. Joint with Steffen Rohde.