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Abstract: Given two Riemannian manifolds with suitable curvature conditions and isometric boundary, can we glue them along their isometric boundary, then desingularize the result while preserving the same curvature conditions? A positive answer to such questions enables one to study deep rigidity and flexibility phenomena for the curvature conditions. In this talk, I will focus on related desingularization results concerning such problems for positive scalar curvature and mean convex (or minimal) boundary conditions. This is based on joint work with Alessandro Carlotto.