  This research is motivated by the study of the geometry of fractal sets and
is focused on uniformization problems: transformation of sets to canonical
sets, using maps that preserve the geometry in some sense. More specifically,
the main question addressed is the uniformization of planar Sierpinski carpets
by square Sierpinski carpets, using methods of potential theory on carpets.
  We first develop a potential theory and study harmonic functions on planar
Sierpinski carpets. We introduce a discrete notion of Sobolev spaces on
Sierpinski carpets and use this to define harmonic functions. Our approach
differs from the classical approach of potential theory in metric spaces
because it takes the ambient space that contains the carpet into account. We
prove basic properties such as the existence and uniqueness of the solution to
the Dirichlet problem, Liouville's theorem, Harnack's inequality, strong
maximum principle, and equicontinuity of harmonic functions.
  Then we utilize this notion of harmonic functions to prove a uniformization
result for Sierpinski carpets. Namely, it is proved that every planar
Sierpinski carpet whose peripheral disks are uniformly fat, uniform quasiballs
can be mapped to a square Sierpinski carpet with a map that preserves carpet
modulus. If the assumptions on the peripheral circles are strengthened to
uniformly relatively separated, uniform quasicircles, then the map is a
quasisymmetry. The real part of the uniformizing map is the solution of a
certain Dirichlet-type problem. Then a harmonic conjugate of that map is
constructed using the methods developed by Rajala arXiv:1412.3348.
