Clay Mathematics Institute

Dedicated to increasing and disseminating mathematical knowledge

Chinese dragons and mating trees

Chinese dragons and mating trees: quantum Loewner evolution and conformal matings of continuum random trees
Scott Sheffield (Massachusetts University of Technology)

What is the right way to think of a "random surface" or a "random planar graph"?  How can one explain the dendritic patterns that appear in snowflakes, coral reefs, lightning bolts, and other physical systems, as well as in toy mathematical models inspired by these systems?  How are these questions related to random walks and random fractal curves (in particular the famous SLE curves)?  How are they related to conformal matings of Julia sets?  To string theory?  To statistical mechanics?

To begin to address these questions, I will introduce and explain the "quantum Loewner evolution" (QLE), which is a family of growth processes closely related to SLE.

Joint work with Bertrand Duplantier and Jason Miller.