127 problems

Fejes Toth's conjecture. The triangle packing in the plane, with one packing disk removed, is solid.

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 $G$ be a connected semisimple Lie group with finite center, all of whose simple factors have real rank at least $2$. Let $\Gamma\subset G$ be a lattice, $M$ be a compact manifo…

Katok–Spatzier conjecture. Up to finite cover, the action is $C^\infty$ conjugate to an algebraic action.

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 $X$ be a closed manifold, and let $h_1$ and $h_2$ be Riemannian metrics on $X$ with negative sectional curvature. For each such metric, let $\mathrm{MLS}_h$ assign to every con…

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…

Burns–Katok conjecture. The marked length spectrum map $L$ is injective.

A closed Riemannian manifold $M$ has higher hyperbolic rank if every geodesic $c(t)$ in $M$ has a Jacobi field $J(t)$ such that $J(t)\neq 0$ and the sectional curvature of the plan…

Simpson's motivicity conjecture. Rigid integrable connections are motivic.

Structural decomposition conjecture.

A hyperbolic strictly-convex polyhedron is a strictly-convex polyhedron in hyperbolic space, with dihedral angles measured along its edges. Stoker's conjecture. Every hyperbolic st…

Simpson's conjecture. Every such representation has integral image. This connects rigidity of local systems on projective varieties with arithmeticity; the source presents it as Si…

Let $f$ be an orientation-preserving, real-analytic, Lebesgue measure-preserving diffeomorphism of the 2-sphere $\mathbb{S}^2$ with only two periodic points, which are necessarily…

Let $X=\operatorname{Seg}(\mathbb P^2\times\mathbb P^2)\subset\mathbb P(\mathbb C^3\otimes\mathbb C^3)$. Let $Z^4\subset\mathbb P V$ be a variety, and let $II_{Z,z}$ denote its sec…

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.