When does the isomorphism of the uniform Roe algebras associated imply coarse equivalence of the underlying spaces?

A recent result of Spakula and Willett gives sufficient conditions in case of coarse metric spaces, These conditions are uniform

discreteness and property A (the `coarse' analogue of amenability). It is not clear whether any of these requirements

(including metrizability) are necessary, and I'll discuss some recent progress. No previous knowledge of coarse spaces, Roe

algebras, or logic is required.

(This is joint work with Bruno De Mendoca Braga.)