Scalar curvature rigidity and flexibility: two problems motivated by general relativity

I will discuss two results motivated by general relativity. The first concerns Gromov's conjecture that nonnegative scalar curvature and sufficiently fast zeroth-order decay at infinity force a complete metric on Euclidean space to be flat. The second solves and strengthens the Riemannian reverse-Burnett conjecture of Huneau and Luk, showing that every smooth metric with nonnegative scalar curvature is the locally uniform limit of smooth scalar-flat metrics with locally uniform W^{1,\infty} bounds.