14 problems

Let $M$ be a finite-volume hyperbolic 3-manifold, and let $TV_r(M;e^{\pi i/r})$ be its Turaev–Viro invariant for odd integers $r\geq 3$. The Chen–Yang volume conjecture. The growth…

Let $K$ be a hyperbolic knot in ${\mathbb S}^3$. An ideal triangulation $X$ of ${\mathbb S}^3{\smallsetminus} K$ has a nonempty angle-structure space $\mathcal{A}_X$, and let…

Let $P(u)$ be a flexible polyhedron in the sphere $\mathbb{S}^n$, where $n\ge 3$. Replacing a vertex by its antipode means replacing that vertex of $P(u)$ with its antipodal point…

The generalised Casson conjecture. For every $\alpha\in\mathcal{A}(\mathcal{T},\mathbf{s})$,

Let $M$ be a compact orientable 3-manifold with empty or toroidal boundary. Define … where $r$ runs over all odd integers, and let $\operatorname{Vol}(M)$ be the simplicial volume,…

Let $N$ be a $3$-manifold with non-empty boundary and let $\mathcal T$ be an ideal triangulation with edge set $E$. Let $\{\mathbf b^{(r)}\}$ be a sequence of colorings of…

Let $M$ be a closed oriented $3$-manifold and let $L$ be a framed hyperbolic link in $M$ with $|L|$ components. Let \\{\mathbf a^{(r)}\} be a sequence of colorings of the compon…

Refined volume conjecture. The set $R_M^*$ is infinite, and, as $r$ ranges over $R_M^*$,

Let $\Gamma$ be a planar $3$-connected graph, let $\overline{\Gamma}$ denote the associated maximal-volume proper generalized hyperbolic polyhedron, and let $col$ be any $r$-admiss…

The volume conjecture for polyhedra.

Let a quantum $6j$-symbol be evaluated in a regime corresponding to geometric data of a hyperbolic tetrahedron. Costantino's 6j-symbol volume conjecture. The exponential growth of…

Let ${\mathcal{T}}_\ell$ denote the space of truncated hyperbolic tetrahedra whose edge lengths satisfy the constraint defining ${\mathcal{T}}_\ell$, and let $\Delta_\ell$ be the r…

Let $X$ be a compact Kähler manifold of dimension $n$. Let $\alpha$ be a transcendental pseudo-effective $(1,1)$-class, let $\mathcal{M}$ denote the cone of movable classes, and le…

Let $K$ be a knot, and let ${\cal P}_n(K;a,q,t)$ be its $S^{n-1}$-colored superpolynomial. Set $q=e^{\hbar}$, with $a$ and $t$ fixed, and hold $x=q^n$ fixed as $\hbar\to0$ and…