Heat flow on snowballs
Quasisymmetric maps are fruitful generalizations of conformal maps.
Quasisymmetric uniformization problem seeks for extensions of
uniformization theorem beyond the classical context of Riemann surfaces.
The goal of this talk is to show that quasisymmetric uniformization problem
is closely related to random walks and diffusions. I will explain how the
existence of quasisymmetric maps is equivalent to heat kernel estimates for
the simple random walk on a family of planar graphs. The same methods also
apply to diffusions on a class of fractals homeomorphic to the 2-sphere.
These ideas will be illustrated using snowballs and their graph
approximations. Snowballs are high dimensional analogues of Koch snowflake.