12 problems

Let $\tau$ be a preorder of size $n$. For $0\leq i\leq n$, let $h_i(\tau)$ count the lattice points of $\mathcal Q_\tau$ with exactly $i$ nonzero coordinates, and write…

Let $\mathcal{P}_{6,48}$ be the centrally symmetric simplicial $6$-polytope whose antipodal quotient of the boundary gives a triangulation of $\mathbb{R}P^5$. A triangulation is ve…

Let $P$ be any point set in the plane, and let $\mathrm{WC}_P$ denote its wiggly complex. Wiggly complex polytopality conjecture. For any point set $P$ in the plane, the wiggly com…

Let $d\geq 4$, let $2\leq i\leq d/2$, and let $P$ be a simplicial $d$-polytope. Assume that $P$ has no missing faces of dimension at least $d-i+1$. Equivalently, let…

Let $P\subseteq \mathbb R^d$ be a simplicial polytope of dimension $d\geq 2k-1$, with $k\geq 2$. Let $G$ be a missing $(k-1)$-face of $\partial P$, and let $F$ be a $(k-1)$-subset…

Stress-sign conjecture. There exists a $(k-2)$-face $F\subset M$ and an affine $k$-stress $\lambda$ on $P$ such that, for every $(k-1)$-face $G=F\cup v_{F,G}$ of $\partial P$, one…

Let $P\subseteq \mathbb R^d$ be a simplicial $d$-polytope. For an affine $k$-stress on $P$, record the signs of the coefficients in its squarefree part on the $(k-1)$-faces; let…

Let $P$ be a centrally symmetric simplicial $d$-polytope. The higher cs lower bound conjecture. If, for some $3\leq r<\left\lfloor\frac d2\right\rfloor$, … then … Here…

Let $\Delta$ be the boundary of a simplicial convex $d$-polytope. For $1\leq r\leq\lfloor d/2\rfloor$, let $g_r(\Delta)$ denote the $r$th entry of its $g$-vector, and call $\Delta$…

Colored SLP conjecture. Any balanced simplicial sphere, or at least any balanced simplicial polytope, has the colored SLP over a field of characteristic $0$.

Let $P$ be a balanced simplicial $d$-polytope, meaning that its underlying graph is $d$-colorable. For $-1\leq i\leq d-1$, let $f_i(P)$ be the number of $i$-dimensional faces, with…

McMullen's conjecture. The sequence $(h_0,h_1,\dots,h_d)$ is the h-vector of the boundary of a simplicial $d$-polytope if and only if $h_0=1$,…