Abstract: Apparent pairs (also known as evident, shallow, close, steepness, Pareto, or minimal pairs) are a fundamental construction at the interface of persistent homology and discrete Morse theory. They play a key role in the context of algorithmic and computational topology. Besides their explicit use in efficient computation of persistent homology, I will show how they can be employed in the study of the geometry and topology of Gromov-hyperbolic spaces and their Rips complexes, and to from a bridge between Morse theoretic approaches to shape reconstruction from point clouds, specifically, relating lexicographically optimal cycles in the Delaunay filtration to Wrap complexes.