Stephen McCormick: On bounded scalar curvature fill-ins and positive mass theorems

We review some aspects of the bounded scalar curvature fill-in problem. That is, given some constant C, we ask when can a closed Riemannian manifold (S,g) equipped with a positive function H, be realised as the boundary of some manifold with scalar curvature bounded below by C and having boundary mean curvature equal to H? We show how the existence (and non-existence) of such ``fill-ins'' relates to the positive mass theorem. This is predominantly an overview talk, sprinkled with hopefully a few original thoughts.