Given a metric space $(X,d)$, one defines a subalgebra of the space of operators on $l_2(X)$ called the uniform Roe algebra of $(X,d)$, denoted $C_u^*(X)$. This is the closure of the algebra of finite-propagation operators. The study of these algebras comes from the fact that $C_u^*(X)$ catches algebraically some of the large scale geometrical properties of $X$. Uniform Roe algebras have therefore an intrinsic relation with coarse geometry and the coarse Baum-Connes conjecture.
In recent year, much work was dedicated to show which geometric properties are preserved by isomorphisms of Uniform Roe algebras. Namely, if $C_u^*(X)$ and $C_u^*(Y)$ are isomorphic, how much do $X$ and $Y$ look alike? We pose the same question for Uniform Roe corona algebras.
Since $C_u^*(X)$ contains all compact operators, we can define the natural quotient $Q_u^*(X)=C_u^*(X)/K(l_2(X))$, the Uniform Roe corona algebra of $X$. Which geometric properties do the spaces $X$ and $Y$ share, when an isomorphism between $Q_u^*(X)$ and $Q_u^*(Y)$ is given? For example, must $X$ and $Y$ be coarsely equivalent, or even bijectively coarsely equivalent? (Two spaces are coarsely equivalent if they look the same when the observer is far from them).
We answer these questions with the aid of some set theory, in particular of Forcing Axioms. Forcing Axioms are generalizations of the Baire category theorem. They are alternative to the Continuum Hypothesis, and they're at the base of many rigidity phenomena observed in the theory of quotients (both discrete such as Boolean algebra quotient, and continuous, as the Calkin algebra or corona $C^*$-algebras). The talk starts with introducing the objects in play. The goal is to state the main results, and at least sketch the salient points of their proofs. We conclude with a list of open questions. This is joint work with Bruno Braga and Ilijas Farah.