Possible topics for postgraduate research

Conformal welding and Liouville quantum gravity

Given a homeomorphism between the boundaries of two copies of the unit disk, one can always topologically glue the disks together (along their boundaries, according to the homeomorphism) to produce topological sphere. The classical conformal welding problem is to endow this topological sphere with a natural conformal structure. Equivalently, to define a simple loop on the sphere and two conformal maps from the complementary components of the loop to two copies of the unit disk, in a way that agrees with the topological gluing. A simple criteria for the existence and uniqueness of such a triple (loop + maps) is not known in general.

Amazingly it turns out that for some random (and quite exotic!) homeomorphisms, related to the Gaussian free field (see simulation) the existence and uniqueness can be proven, and the law of the random loop on the unit disk can be identified. This can in turn be used to prove “integrability” results related to the Gaussian free field and associated conformal field theories. Although there have been some ground-breaking results of this kind, there are still many open questions and directions to be explored. Please contact me for more details.

Contact: Ellen Powell.