17 problems

Let $k\geq 2$, $d\geq 2k$, and let $\Delta$ be a $(d-1)$-sphere with $g_k(\Delta)=1$. Cell-complex realization conjecture. There exists a $d$-dimensional cell complex $\mathcal{C}$…

Let $d\geqslant 3$ and $1\leqslant\ell\leqslant d-1$. Let $P$ be a $d$-polytope with $2d+\ell$ vertices, and let $T(r)$ denote the $r$-simplex. Pineda-Villavicencio's conjecture. 1…

Let $d=d_1+\cdots+d_m$, let $\bbbeta$ be the block-parameter vector, and let $\mathcal{P}_{d,d}^{\bbbeta}$ be the associated block-beta random polytope. All-face-dimensions conject…

Let the Generalized Upper Bound Theorem mean that, for a simplicial $d$-polytope $P$ and the cyclic $d$-polytope $C(d,n)$, the inequality $f_{d-1}(P) \geq f_{d-1}(C(d,n))$ implies…

Let $\mathcal{T}$ be a triangulation of a closed and connected $d$-dimensional manifold, with $d>0$, $f_0$ vertices, and $f_d$ facets. Spreer–Tobin's sharp vertex bound conjecture.…

Let $(\Delta,\nu)$ be a PL realization of an orientable simplicial $\mathbb{R}$-homology $d$-manifold without boundary in $\mathbb{R}^d$, and let $\Psi(\Delta,\nu)$ be its stress a…

Let $\Delta$ be an orientable simplicial $\Bbbk$-homology $d$-manifold without boundary, and let…

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 $\Delta$…

Let $P$ be a $2$-level $d$-polytope, and let $f_0(P)$ and $f_{d-1}(P)$ denote its numbers of vertices and facets, respectively. Vertex-facet product conjecture. For every $2$-level…

Flag Eulerian upper bound conjecture. For every $1\leq i\leq 2m-1$,

Let $P$ be a $d$-dimensional convex polytope, and let $f_i(P)$ denote the number of its $i$-dimensional faces. A polytope is $k$-neighborly if every subset of $k$ vertices is the v…

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…

For integers $d,i\geq 1$, let $\mathcal{HS}(i,d)$ be the family of $(d-1)$-dimensional homology spheres without missing faces of dimension greater than $i$. For…

Face-number conjecture. For every $d$-dimensional perfect prismatoid $P$, there exists a $d$-dimensional lattice Delaunay polytope $Q$ such that

Let $\Delta$ be a connected triangulated $(d-1)$-manifold without boundary, and let $h_r(\Delta)$ denote its $h$-numbers and $\beta_{j-1}(\Delta)$ its Betti numbers. Bagchi–Datta's…

Let $M$ be a triangulation of a connected closed $d$-manifold. Write $g_j(M)$ for its $g$-numbers, $\beta_i(M;\mathbb F)$ for Betti numbers over a field $\mathbb F$, and let…

Let $d ot obreakgeq 0$ and $0<ileq d+1$ be integers. Let $\mathcal{C}(i,d)$ be the family of $d$-dimensional simplicial complexes with nonzero reduced $d$-homology and no missing fa…