6 problems

Neoplatonic homotopy conjecture. Every $6$-net $\sigma$ has a family of realizations $\Delta(\sigma,l)$, each unique up to isometry, as undented hyperbolic polyhedra of side length…

Let $X$ and $X_i$ be closed oriented Riemannian manifolds, and let $W_{\alpha_i,\beta_i,\lambda_i}\colon X_i\to X$ denote Riemannian $(\alpha_i,\beta_i,\lambda_i)$-cobordisms, mean…

Let $M_j^3$ be a sequence of three-dimensional manifolds in the class $\mathcal{M}$, and let $m_{ADM}(M_j^3)$ denote their ADM masses. Let ${\mathbb E}^3$ denote Euclidean three-sp…

Gromov's conjecture. The sequence should have a subsequence which converges in some geometric sense to a limit space with some generalized notion of nonnegative scalar curvature.

Let $(M_i,g_i)$ be a sequence of asymptotically flat $3$-manifolds with nonnegative scalar curvature, and let $m(g_i)$ denote their ADM masses, with $m(g_i)\to 0$. Huisken–Ilmanen…

Let a sequence of asymptotically flat spaces have ADM masses converging to zero, and suppose the spaces converge in the Sormani–Wenger sense. Stability conjecture. The rigidity of…