349 problems

Let $g\in\frac12\mathbb{Z}_{\geq0}$ and $n\in\mathbb{Z}_{\geq1}$ satisfy $2g-2+n\geq1$, and let $\omega_{g,n}$ be the multidifferential defined by Definition RTR. Refined topologic…

Take the $\{3,5,3\}$ honeycomb in $\operatorname{\mathbb{H}}^3$ and cut it along a horosphere. Consider the union of all icosahedra interior to, or intersecting, the horosphere; th…

Let $a,b,c,d$ be four complex points on the unit circle $S^1$ in this order, with the Euclidean lines $L(a,b)$ and $L(c,d)$ nonparallel. Let $h$ be an arbitrary point on the Euclid…

Let $(M,h_0)$ be a closed hyperbolic manifold of dimension $n\geq 3$, and let $N$ be a closed $n$-manifold with Riemannian metric $g$ whose scalar curvature satisfies…

Let $M$ be a $3$-manifold, let $n$ be a positive integer, and let $\rho$ be a boundary-unipotent representation of $\pi_1(M)$ in $\mathrm{SL}(n,\mathbb C)$ or…

Let $F\subset S^3$ be a quasipositive Seifert surface. A proper arc in $F$ is an arc with endpoints on $\partial F$ and interior in the interior of $F$. Let $A(O,-1)$ be the surfac…

Let the parameter space of hyperbolic monopoles contain a compact subset and a disjoint subset, and let the monopoles have sufficiently small mass. Small-mass asymptotic separation…

Let $\omega$ be the symmetric measure on the sphere at infinity $S^2$ of hyperbolic space ${\mathbb H}^3$. For $f\in C^0_c({\mathbb H}^3)$, define the action … where $x$ is regarde…

Let $\Gamma$ be a Fuchsian group with fundamental class $[\Gamma]$, let $\sigma$ be a multiplier on $\Gamma$ satisfying $\delta(\sigma)=0$, and define … Here a…

McMullen's conjecture. The convex core $C(N)$ does not contain an embedded ball of radius $L$, where $L$ depends on the number of generators of $\pi_1(N)$.

Let $M$ be a hyperbolic manifold containing an incompressible surface $S$. For a manifold $N$ with acylindrical boundary, let $Vol(N)$ denote the volume of the components of $N$ th…

Let $G$ be a finitely generated group with $g$ generators. For a hyperbolic $3$-manifold $N$, write $\operatorname{inj}_N(x)$ for the injectivity radius at $x$, and let the convex…

Let $\operatorname{\mathcal P}(\mathbb X)$ be the scissors congruence group of polytopes in a 3-dimensional geometry $\mathbb X$, and let … be the kernel of the Dehn invariant. Deh…

For an integer $N\geq 3$, let $D_2$ denote the Bloch–Wigner dilogarithm, and consider the real numbers $D_2(e^{2\pi\sqrt{-1}j/N})$ indexed by integers $j$ relatively prime to $N$ w…

Let $N$ be a compact, connected, ${\mathbf{P}^2}$-irreducible $3$-manifold whose boundary is a non-empty collection of incompressible tori and Klein bottles. Assume that every subg…

Let $N$ be a compact, connected, ${\mathbf{P}^2}$-irreducible $3$-manifold whose boundary is a non-empty collection of incompressible tori and Klein bottles. Assume that every subg…

Let $M$ be a closed, connected, ${\mathbf{P}^2}$-irreducible $3$-manifold with infinite fundamental group. Hyperbolization Conjecture. If $\pi_1(M)$ contains no subgroup isomorphic…

Complete saturation and reduction conjecture. Every body $K$ in $\mathbb{E}^d$ (respectively, in $\mathbb{H}^d$) admits a completely saturated packing and a completely reduced cove…

Characteristic-polynomial conjecture. All roots of $f(x)$ are real numbers. If $m=n+2$, the points are in general position in $H^n$, and exactly $s$ of the $\alpha_i$ are positive,…

Let $\mathbf{M}$ be the modular torus, the hyperbolic once-punctured torus described in the paper. A pair of simple closed geodesics is a pair of simple geodesics on $\mathbf{M}$ w…

A once-punctured torus is a hyperbolic surface whose simple closed geodesics have a length spectrum, with multiplicity counting how many distinct simple closed geodesics have a giv…

A hyperbolic manifold is arithmetic when its fundamental group is an arithmetic lattice. A subgroup is geometrically finite if its convex core has finite volume, and a lattice is G…

For an integer $n$, a closed hyperbolic $n$-manifold is a compact manifold locally isometric to $n$-dimensional hyperbolic space, and a closed geodesic is a geodesic that is period…

Let $\pp$ be a compact, oriented, irreducible, atoroidal $3$-manifold with boundary equipped with an angled block structure, and let $\tilde{\pp}$ denote its universal cover. The a…

Let $K$ be a hyperbolic knot in $S^3$, let $R_0$ be the relevant irreducible component of its $SL_2(\mathbb{C})$-representation variety, and let $\mu$ and $\lambda$ denote the meri…