In this short talk, we will introduce uniform Roe algebras of uniformly locally finite metric spaces as a bridging tool between coarse geometry and operator algebras. We will formulate the (bijective) rigidity problem for uniform Roe algebras and provide its homological characterisation in terms of injectivity of the comparison map from homology of ample groupoïds introduced by Crainic and Moerdijk to the K-theory of uniform Roe algebra.
