26 problems

Let $S$ be a clean three-dimensional lattice simplex, meaning that the only lattice points on its boundary are its vertices, and suppose that $S$ has $k\geq 1$ interior lattice poi…

Let $P\colon K\to\mathbb{R}^n$ be a flexible, not necessarily embedded polyhedron, and let $V_K(P)$ denote its generalized oriented volume, defined by integrating its characteristi…

A complete orientable hyperbolic $3$-manifold is a hyperbolic $3$-manifold with finite volume, and filling means Dehn filling of cusped hyperbolic $3$-manifolds. The hyperbolic com…

Let $M$ be a hyperbolic $3$-manifold containing an embedded geodesic $\gamma$ of length $l$, and write $M_\gamma=M-\gamma$. Let $\operatorname{Vol}$ denote the volume of the comple…

Let $\boldsymbol{\pi}\in\mathfrak{S}_n$ be a permutation, and consider the stack-sorting polytope generated by $\boldsymbol{\pi}$. The source identifies the volume of the polytope…

DCC conjecture. The set of volumes of these orbifold pairs satisfies the descending chain condition, and its minimum nonzero value is

DCC conjecture. The set of volumes of such orbifold pairs satisfies the descending chain condition, and its minimum nonzero value is

Let $(X,B)$ be a projective log canonical pair of dimension $n$, where $B$ is a nonzero reduced divisor and $K_X+B$ is ample. The volume of $K_X+B$ is the top self-intersection num…

Infinite-volume composition conjecture. There exists a sequence of compositions $W_i$ such that

Alternation-free volume divergence conjecture. The restriction that both $K_1$ and $K_2$ are alternating is unnecessary: there exists a sequence of tg-hyperbolic compositions $W_i$…

Let $s_0=2$ and $s_m=s_{m-1}(s_{m-1}-1)+1$ for $m\geq 1$ be Sylvester's sequence. For a positive integer $n$, let $X$ be the $n$-fold given by the canonical Calabi–Yau hypersurface…

Let $s_0=2$ and $s_m=s_{m-1}(s_{m-1}-1)+1$ for $m\geq 1$ be Sylvester's sequence. For each integer $n\geq 2$, set … A general hypersurface $X$ of degree $d$ in … is a terminal Fano…

Let $s_0=2$ and $s_m=s_{m-1}(s_{m-1}-1)+1$ for $m\geq 1$ be Sylvester's sequence. For a positive integer $n$, set … A general hypersurface $X$ of degree $d$ in the weighted project…

Let $\textit{PPerm}(m,n)$ denote the partial permutation polytope. Volume conjecture. The normalized volume of $\textit{PPerm}(n,2)$ (equivalently, $\textit{PPerm}(2,n)$) is equal…

Let $\mathcal{P}(m,n)$ denote the partial permutohedron. Volume conjecture. The normalized volume of $\mathcal{P}(m,2)$ is equal to $3^m-m$. This conjecture gives a closed formula…

Let $d\geq 1$ and $k\geq 1$ be fixed. Let $S_k^d$ be the Zaks–Perles–Wills simplex … where the Sylvester sequence is defined by $s_1=2$ and $s_i=s_1\cdots s_{i-1}+1$. Zaks–Perles–W…

Let $\Gamma$ be the fundamental group of a connected, orientable, irreducible, compact 3-manifold whose boundary is empty or a collection of tori. For finite-index subgroups…

Let $\Sigma$ be a hyperbolic surface, let $\gamma$ be a filling closed geodesic on $\Sigma$, let $\hat{\gamma}$ denote its canonical lift, and let $M_{\tilde{\gamma}}$ be the compl…

Let $n\ge 3$, and let a flexible polyhedron lie either in Lobachevsky space $\Lambda^n$ or in the open hemisphere $S^n_+$. Its generalized oriented volume is defined in the relevan…

A flexor is a flexible polyhedron in $\mathbb{R}^3$, and a flexion is a continuous deformation preserving its combinatorial type and the shapes of its faces. The bellows conjecture…

Let $T$ be an ideal hyperbolic tetrahedron, and let $G$ be its Gram matrix. Denote by $\det(G)$ and $\operatorname{per}(G)$ the determinant and permanent of $G$, respectively. Seid…

A link orbifold is a hyperbolic $3$-orbifold whose singular locus is a disjoint union of closed curves. The volume of an orbifold is denoted by $\operatorname{vol}(\mathcal{O})$. M…

Let $n$ and $i$ be parameters for the interval-vector polytope $\mathcal{P}_n(\mathcal{I}_{1,n-i})$, where $\mathcal{I}_{1,n-i}$ denotes the corresponding family of intervals and…

For each $g>2$, let $\Sigma_g$ be the closed surface of genus $g$, and let $M_g$ be the explicit closed hyperbolic $\Sigma_g$-bundle over the circle obtained by the Dehn fillings d…

Let $Q_n=[0,1)^n$ be the unit cube in $\mathbb{R}^n$, and let $H$ be an $(n-1)$-dimensional hyperplane in $\mathbb{R}^n$ passing through the center of $Q_n$. Good's conjecture. … T…