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Let X be a countable discrete metric space, and think of operators on L^{2}(X) in terms of their X-by-X matrix. Band operators are ones whose matrix is supported on a "band" along the main diagonal; all norm-limits of these form a C*-algebra, called uniform Roe algebra of X. This algebra "encodes" the large-scale (a.k.a. coarse) structure of X. Quasi-locality, coined by John Roe in '88, is a property of an operator on L^{2}(X), designed as a condition to check whether the operator belongs to the uniform Roe algebra (without producing band operators nearby). The talk is about our attempt to make this work, and find counterexamples.

After an introduction about coarse geometry and Roe algebras, I will explain quasi-locality and a result in the 'positive' direction. Next, I will introduce asymptotic expanders and make a connection with recent results about rigidity of Roe algebras, and counterexamples to the coarse Baum-Connes conjecture. Finally, if time permits, I will talk about Property A and some ingredients of some of the proofs. (Based on joint work with A Tikuisis, K Li, P Nowak, and J Zhang.)