  We construct a conformal welding of two Liouville quantum gravity random
surfaces and show that the interface between them is a random fractal curve
called the Schramm-Loewner evolution (SLE), thereby resolving a variant of a
conjecture of Peter Jones. We also demonstrate some surprising symmetries of
this construction, which are consistent with the belief that (path decorated)
random planar maps have (SLE-decorated) Liouville quantum gravity as a scaling
limit. We present several precise conjectures and open questions.
