**October 23, 2020**

*Scalar Curvature and Non-compact Manifolds.*

There has recently been a great deal of activity on Scalar Curvature, both on the index theoretic and metric geometry side. The classic work of Gromov-Lawson and Schoen-Yau is a backdrop for the more modern developments, and here we will describe some recent work with Unger-Yau which seeks to generalize some of the classic results to the setting of non-compact manifolds. Connections with problems in Conformal Geometry, Positive Mass Theorems will be alluded to, and other current developments by other authors will be described. This is based on arXiv:2009.12618.