61 problems

For every finite preorder $\tau$, the $h$-polynomial of $\tau$ equals that of its dual preorder: $h_{\tau}(t)=h_{\tau^{*}}(t)$, where $\tau^{*}$ denotes the dual preorder.

Dickenstein–Nill's Cayley decomposition conjecture. If

Böröczky's conjecture. There is a point $x\in P$ with norm

Volume–Dehn separation conjecture. In Euclidean, spherical, and hyperbolic geometries, do the volume and generalized Dehn invariant separate the scissors congruence classes of poly…

Let $G$ be of type $A_n$, let $\underline{w}_0$ be any reduced decomposition of the longest Weyl-group element, and let $\lambda$ be a weight. Write $Q_{\underline{w}_0}(\lambda)$…

Let $m,n$ be positive integers, let $\xi^m_n$ denote the generator indexed by $m$ inputs and $n$ outputs, and let $\partial$ be the perturbed differential in the minimal model of t…

Polyhedral semistable reduction conjecture. There exists a projective alteration $\Delta_B^1\to\Delta_B$, with induced alteration $\Delta_X^1\to\Delta_X$, and a projective subdivis…

Neoplatonic homotopy conjecture. Every $6$-net $\sigma$ has a family of realizations $\Delta(\sigma,l)$, each unique up to isometry, as undented hyperbolic polyhedra of side length…

Volume-max conjecture. For a prime $6$-net $\sigma$, its ideal neoplatonic realization maximizes volume among all ideal geodesic $2$-cycles with combinatorics $\sigma$.

Ideal-prime conjecture. If the $6$-net $\sigma$ is prime, its ideal neoplatonic realization $\Delta(\sigma,\infty)$ is convex.

Ideal neoplatonic conjecture. Every $6$-net $\sigma$ has a realization $\Delta(\sigma,\infty)$, unique up to isometry, as an ideal neoplatonic.

Neoplatonic conjecture. Any $6$-net $\sigma$ has a realization, unique up to isometry, as an undented Euclidean polyhedron $\Delta(\sigma)$ built from equilateral triangles of side…

Let $G$ be a finitely generated group satisfying the Atiyah conjecture, and let $C_*$ be a finitely generated free $L^2$-acyclic $\mathbb{Z}G$-chain complex. For a cohomology param…

Minimal vertex-count conjecture. (i) For a CCP of a topological torus without self-intersection, the minimal number of vertices is $9$. (ii) For an orientable CCP of genus $g$, if…

Let $P$ be a triangulated polyhedron with distinguished vertices $A,B,A',B'$ satisfying … Form the twinned polyhedron $T$ of $P$ around the equator $ABA'B'$, as in the construction…

Let $C\subseteq\mathbb{R}^{d+1}$ be a polyhedral cone whose cross section with the hyperplane $\{x_0=1\}$ is a polytope $P\subseteq\mathbb{R}^d$. The adjoint polynomial of $C$ is t…

A compact convex set $S\subseteq\mathbb{R}^3$ is Rupert if a second identical copy can pass straight through a hole in the interior of the first with rescaling factor $\mu_S>1$. St…

For a compact convex set $S\subseteq\mathbb{R}^3$, its Nieuwland constant $\mu_S$ is the largest rescaling factor for which a second copy of $S$ can pass straight through a hole in…

A Rupert compact convex set $S\subseteq\mathbb{R}^3$ admits a passage in which a second copy of $S$ passes straight through a hole in the interior of the first, with rescaling fact…

Glazyrin–Pak cobordism conjecture. For every integral curve $\gamma$, there is a unit rhombus $\rho$ and a dome over $\gamma\cup\rho$.

Keel–Yu mirror detropicalization conjecture. (1) In the presence of multiple open tori, the integral points of $\mathrm{sk}(U)$ can be equipped with the structure of a polyptych la…

Norm-extension conjecture. Then $\lVert\cdot\rVert$ is the restriction to $G$ of a norm on $\mathbb Z^3$.

Let $\Sigma$ be a surface endowed with a triangulation $\mathscr{T}$. Let $V$ be the ambient symplectic vector space in which the immersions take values, and let…

Let $N$ be a numerical semigroup with associated face $F_N$, and suppose that $N$ is staircase and Kunz. When $k=3$, the four-facet conjecture. $F_N$ has at most $4$ facets. This c…

An Archimedean solid is a three-dimensional convex polytope whose faces are regular polygons and whose symmetry group acts transitively on its vertices. Archimedean solid perfectio…