17 problems

Let $M$ be a closed hyperbolic $3$-manifold, and let $b_1(M)$ denote its first Betti number. Volume lower-bound conjecture. If … then … This conjectural lower bound is used to esta…

Let $M_W$ be the manifold obtained by $(5,1)$ and $(5,2)$ surgery on the two components of the Whitehead link. The Weeks manifold smallest-volume conjecture. $M_W$ is the unique or…

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

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 $\mathcal{P}(m,n)$ 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 $\Gamma$ 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…

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 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 $P$ be a flexible polyhedron in Euclidean space, and let its generalized volume be the volume invariant associated with its possibly self-intersecting realization. A bending is…

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…