  In dimensions greater than or equal to three, we establish global uniqueness
and obtain reconstruction in the Calderon problem for the Schrodinger equation
with certain singular potentials. The potentials considered are conormal of
order less than 1-k with respect to submanifolds (of arbitrary codimension k).
This gives positive results for (conormal) conductivities which are Holder of
any order > 1. A related problem for highly singular potentials is shown to
exhibit nonuniqueness.
