Four Lectures on Asymptotically Flat Riemannian Manifolds

After an introduction to the mathematical description of isolated gravitating systems, we present a proof of the positive mass theorem relying on the compactification trick that lies behind the argument proposed by Schoen and Yau in 2017 to handle the possible occurrence of high-dimensional singularities of minimizing cycles. From there, we describe some rigidity phenomena involving Riemannian manifolds of non-negative scalar curvature, and survey various recent results concerning the large-scale geometric structure of asymptotically flat spaces. In the last section, we outline the construction, due to Schoen and the author, of localized solutions of the Einstein constraints and discuss its implications both on the physical and the geometric side.