Quasisymmetry is a global version of quasiconformality, which is a
generalization of a conformal map. While this notions originate from
complex analysis, it now plays a prominent role in geometric group
theory/rough geometry. In particular quasisymmetric maps appear at the
boundary of Gromov hyperbolic spaces.

The quasisymmetric uniformization problem asks when a given metric space
is quasisymmetrically equivalent to some standard space. Of particular
importance is the question when a metric $2$-sphere is a quasisphere,
i.e., the quasisymmetric image of the standard sphere.

Cannon's conjecture stipulates that a group that "acts topologically" as
a Kleinian group is geometrically a Kleinian group. An equivalent
formulation is that a certain metric sphere (arising as the boundary at
infinity) is a quasisphere.

Thurston's theorem on the characterization of rational maps gives an
answer to the analog question when a map that "acts topologically" as a
rational map "is" a rational map. In particular there are maps that are
not (equivalent to) rational maps. So the analog statement of Cannon's
conjecture is false in this setting.

In this talk I will talk about my joint work with Mario Bonk, which
attempts to describe "expanding Thurston maps" in a fashion similar to
the one arising in the group theoretic setting.