  We prove several rigidity results for corona $C^*$-algebras and
\v{C}ech-Stone remainders under the assumption of Forcing Axioms. In
particular, we prove that a strong version of Todor\v{c}evi\'c's $\OCA$ and
Martin's Axiom at level $\aleph_1$ imply: (i) that if $X$ and $Y$ are locally
compact second countable topological spaces, then all homeomorphisms between
$\beta X\setminus X$ and $\beta Y\setminus Y$ are induced by homeomorphisms
between cocompact subspaces of $X$ and $Y$; (ii) that all automorphisms of the
corona algebra of a separable $C^*$-algebra are trivial in a topological sense;
(iii) that if $A$ is a unital separable infinite-dimensional $C^*$-algebra, the
corona algebra of $A\otimes \mathcal K(H)$ does not embed into the Calkin
algebra. All these results do not hold under the Continuum Hypothesis.
