Here we survey the compactness and geometric stability conjectures formulated by the
participants at the 2018 IAS Emerging Topics Workshop on Scalar Curvature and …

In 2014, Gromov vaguely conjectured that a sequence of manifolds with nonnegative scalar
curvature should have a subsequence which converges in some weak sense to a limit …

The null distance for Lorentzian manifolds was recently introduced by Sormani and Vega.
Under mild assumptions on the time function of the spacetime, the null distance gives rise to …

Gromov proved that sequences of Riemannian manifolds with nonnegative sectional
curvature have subsequences which converge in the Gromov-Hausdor sense to Alexandrov …

Given a pair of metric tensors $ g_1\ge g_0 $ on a Riemannian manifold, $ M $, it is well
known that $\operatorname {Vol} _1 (M)\ge\operatorname {Vol} _0 (M) $. Furthermore one …

Abstract By works of Schoen–Yau and Gromov–Lawson any Riemannian manifold with
nonnegative scalar curvature and diffeomorphic to a torus is isometric to a flat torus. Gromov …

We prove results on intrinsic flat convergence of points—a concept first explored by Sormani
(Commun Anal Geom 26 (6): 1317–1373, 2018). In particular, we discuss compatibility with …

We consider rotationally symmetric spaces with low regularity, which we regard as integral
currents spaces or manifolds with Sobolev regularity that are assumed to have nonnegative …

The rigidity statement of the positive mass theorem asserts that an asymptotically flat initial
data set for the Einstein equations with zero ADM mass, and satisfying the dominant energy …

We explore the distinctions between L p convergence of metric tensors on a fixed
Riemannian manifold versus Gromov–Hausdorff, uniform, and intrinsic flat convergence of …