I'll discuss some new tools for studying the scalar curvature of 3-dimensional manifolds, exploiting a relationship between scalar curvature and the topology of level sets of harmonic functions and S^1-valued harmonic maps. These methods share features with the well-known minimal surface and inverse mean curvature flow techniques, while yielding some estimates reminiscent of those arising from Dirac operator methods. We'll describe applications to the study of the Thurston norm of closed 3-manifolds, and the ADM mass of asymptotically flat 3-manifolds. (Based in part on joint work with Hugh Bray, Demetre Kazaras, and Marcus Khuri.)
