6 problems

Let $S(j,d-1)$ denote the class of $(\mathbb{Z}/2\mathbb{Z})$-homology $(d-1)$-spheres without missing faces of dimension larger than $j$. Nevo's conjecture. For $1\leq j\leq d$, s…

Let $k\geq 2$, and let $\Delta$ be a neighborly $2k$-sphere with $n$ vertices. Write $m_{k+1}(\Delta)$ for its number of missing faces of dimension $k+1$. Missing-value conjecture.…

Let $2\leq i\leq d/2$, let $\Delta\in S(d-i,d-1)$, and let $p$ be a generic or natural embedding of $\Delta$ in $\mathbb{R}^d$. An affine $i$-stress is an element of the affine str…

Let $S(2,4)$ be the class of $(\mathbb{Z}/2\mathbb{Z})$-homology $4$-spheres without missing faces of dimension larger than two, and let $g_1=f_0-6$. Nevo's lower-bound conjecture.…

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…