Explore recent developments in inequalities for the mass-to-capacity ratio on asymptotically flat 3-manifolds with nonnegative scalar curvature in this 47-minute lecture. Delve into new proofs of the positive mass theorem and discover applications that provide sufficient conditions for positive mass using C^0-geometry. Examine the proposal to study manifolds with mass-to-capacity ratios bounded by one, as presented by Pengzi Miao from the University of Miami at the Institut des Hautes Etudes Scientifiques (IHES).