127 problems

Let $\mathcal{F}$ be a $\mathcal{C}$-formal type of slope $r/h$, and let $\mathscr{O}\subset\operatorname{\mathfrak{g}}$ be a nilpotent orbit. Write…

Forbidden-minor characterization conjecture. The following are equivalent:

One-unit gap conjecture.

Equality conjecture.

Let $(M^n,g)$ be a steady gradient Ricci soliton with positive Ricci curvature. Suppose there are a ball $B_R(0)\subset\mathbb R^n$ and a diffeomorphism from…

Let $(\tau,(W_i))$ be a $d$-dimensional superdense coding protocol. Let $\mathcal{H}_{A'}$, $\mathcal{H}_{A”}$, $\mathcal{H}_{B'}$, and $\mathcal{H}_{B”}$ be the Hilbert spaces in…

Let the $n$-Roberts graph encode the facet adjacencies of the $n$-cube, and let a hinged spanning cycle be a spanning cycle of this graph whose corresponding $2n$ facet adjacencies…

Let $M$ be a closed Riemannian manifold, let $\widetilde{M}$ be its universal cover, and let $\operatorname{Isom}(\widetilde{M})$ be its isometry group containing $\pi_1(M)$. Quant…

Let $(M,g)$ satisfy the hypotheses of the preceding one-endedness conjecture, and suppose that its Ricci curvature agrees with that of a paraboloid for $r\leq r_0$. Paraboloid rigi…

Classification conjecture. Every irreducible higher-rank ${\mathbb Z}^k$- or ${\mathbb R}^k$-Anosov action on any compact manifold is smoothly conjugate to an algebraic action.

Let $G$, $\Gamma$, and $H$ be as in Margulis's orbit-closure conjecture, and let $\mu$ be an $H$-invariant, $H$-ergodic measure on $\Gamma\backslash G$. Suppose that for $\mu$-almo…

Let $G$, $\Gamma$, and $H$ be as in Margulis's orbit-closure conjecture, and let $\mu$ be an $H$-invariant, $H$-ergodic measure on $\Gamma\backslash G$. Suppose that for every…

Let $(M,g)$ be a gravitational instanton whose underlying manifold $M$ is topologically $\mathbb{R}^4$. Topological $\mathbb{R}^4$ rigidity conjecture. The metric $g$ is either the…

Let $\Sigma$ be the closed surface under consideration, let $\mathcal L^*(g)$ denote the relevant set of surface subgroups of genus $g$, and let $A(h)(\Pi)$ be the associated area…

A circle domain $\Omega\subset\widehat{\mathbb C}$ is a domain whose boundary components are points or circles. It is conformally rigid if every conformal map of…

Let $\Sigma^3\subset B^4\subset\mathbb{R}^4$ be a capillary minimal hypersurface in the unit ball; the free boundary case corresponds to contact angle $\pi/2$. Assume that … for al…

Let $B^3\subset\mathbb{R}^3$ be the unit ball, and let $\Sigma_{\mathrm{cc}}$ denote the critical catenoid, the free boundary minimal annulus obtained by scaling the Euclidean cate…

Let $T\subset S^3$ be an embedded minimal torus, and let $T_{\mathrm{Cl}}$ denote the Clifford torus in $S^3$. Lawson's conjecture. There exists an isometry $F$ of $S^3$ such that…

Let $f\colon (\mathcal{X},\mathcal{L})\to S$ be a flat family of $n$-dimensional polarized smooth Calabi–Yau manifolds over a quasiprojective curve $S$. Let $\overline{S}$ be a pro…

Let $M$ be a compact Riemannian manifold with negative sectional curvature and dimension at least $3$. Suppose that, for some $k\geq 2$, there exist infinitely many distinct immers…

Let $(M^n,g_E)$ be a closed Einstein $n$-manifold with $n\geq 3$ and negative scalar curvature $R^*<0$. Let $g$ be another metric on $M$ satisfying … Here $g\leq g_E$ means that…

Volume rigidity conjecture. The metric $g$ is isometric to the hyperbolic metric $g_H$.

Let $\Gamma$ be an irreducible higher-rank lattice, and let a URS mean a minimal subsystem of the space of subgroups of $\Gamma$ with the Chabauty topology. The URS rigidity conjec…

Let $(M,\omega)$ be a complete balanced Hermitian manifold of complex dimension $n$, and suppose that the equality case in the first-eigenvalue estimate of Theorem 1.1 holds, so th…

Let $P$ be a triangulated polyhedron with distinguished vertices $A,B,A',B'$ satisfying … Form the twinned polyhedron $T$ of $P$ around the equator $ABA'B'$, as in the construction…