14 problems

Let $APrism_m$ be the $m$-parameter antiprism, and let $Prism(APrism_m)$ denote the prism over it. Write $z$ for the zigzag vector and $Int$ for intersection vectors. Prism-over-an…

Let $C_p$ and $C_q$ be polygons, and set … Write $z$ for the multiset of zigzag lengths and multiplicities, and $Int$ for intersection vectors. Product-of-polygons conjecture. For…

Let $d$ be the dimension parameter, let $\beta_{d-1}$ be the regular $(d-1)$-dimensional complex and let $\alpha_{d-1}$ denote the corresponding simplex; write $Pyr(\beta_{d-1})$ f…

Let $f:\Delta_X\rightarrow\Delta_B$ be a surjective morphism of rational conical polyhedral complexes such that $f^{-1}(0)=\{0\}$. A projective subdivision replaces the complexes b…

Let $F$ be the piecewise-linear function represented by a ReLU network, and let $4\mathcal{C}(F)\4$ be its canonical polyhedral complex. Consider a general training path in the net…

Abramovich–Karu polystable subdivision conjecture. Any morphism $P\to Q$ of rational polyhedral complexes can be made polystable by an appropriate subdivision of $P$ and $Q$ and al…

Let $f(x_1,\ldots,x_n)\in\mathbb{C}[x_{1}^{\pm 1},\ldots,x_{1}^{\pm 1}]$ be a Laurent polynomial. Let $\mathcal{N}_f$ be its Newton polytope, let…

Let $P$ be a planar polyhedral complex, and let F=\max\{n\mid\text{\rm {there is an n-gon in }}P\}. Let $k$ and $r$ be the parameters in the spline-dimension formula of Theorem…

Polyhedral-complex conjecture. If geodesic triangles in orthant spaces $\mathcal{F}_\Omega$ are always closed, then each geodesic polytope $T$ is a polyhedral complex with cells…

Combinatorial distinctness conjecture. The regular polygonal complexes in $\mathbb{E}^3$ are also combinatorially distinct.

The 2-skeleton conjecture. The complex $\mathcal K$ coincides with the $2$-skeleton of $\mathcal P$.

Let $X$ be a locally finite polyhedral complex, let $G=\operatorname{Aut}(X)$, and let $\Gamma$ be a nonuniform lattice in $G$. Non-finite presentability conjecture. The group…

Let $X$ be a right-angled hyperbolic building, let $G=\operatorname{Aut}(X)$, and let $\Gamma$ be a nonuniform lattice in $G$. Non-finite generation conjecture. The group $\Gamma$…

Higher-dimensional Balinski-type connectivity conjecture. The graph ${\mathcal G}_k({\mathcal C})$ is