16 problems

Noncontractible-interval conjecture. The image of $\phi$ is exactly the subposet consisting of those closed intervals in $\mathcal{S}_2(n,d)$ whose open interval is not contractibl…

Let $d=2k$ be a fixed even dimension, and let $\mathcal{B}_{2k}(X)$ denote the centrally symmetric polytope constructed from a set $X$ of $n$ equally spaced points on…

UBC for the facets of geometric simplex triangulations. Every geometric triangulation $\mathcal S$ of $\mathbf{s}$ with vertex set $\mathcal V$ has at most

Let $S(n,d)$ be the set of subsets $S\subset\binom{[n]}{d+1}$ corresponding to triangulations of the cyclic polytope $C(n,d)$. Let $S_1(n,d)$ and $S_2(n,d)$ be the two partial orde…

Let $d7c3_{3,2,n}$ be the Schubert exterior polytope, and let $W=d70f(Z)$ be the image of the positive data $Z$ under the twist map. A facet is Schubert if its supporting hyperplan…

Let $C_{k,m,n}(Z)$ be the exterior cyclic polytope constructed from positive data $Z$, and let its $f$-vector record the numbers of faces in each dimension. Constant f-vector conje…

Let $d$ and $n$ be positive integers, let $C^d_n$ be the cyclic $d$-polytope with $n$ vertices, and let $\mathsf{Inv}^d_{d+2}$ denote the algebra of degree-$d$ volume invariants fo…

Amplituhedron sign-characterization conjecture.

Let $S^{2k-1}\subset\mathbb{C}^k$ have its Hopf decomposition into subsets determined by which homogeneous coordinates have maximal absolute value. A Hopf triangulation is a combin…

Let $K$ be a triangulation of the $d$-simplex $\Delta^d$ associated with a $d$-dimensional cyclic polytope, meaning that it is obtained from the boundary of the polytope by removin…

Let $Q$ be the affine semigroup associated with the cyclic polytope construction in the paper, and let $K[Q]$ be its semigroup algebra over the field $K$. Write $n$ and $d$ for the…

An integral cyclic polytope is a cyclic polytope with integral vertices. A lattice polytope is very ample when its associated semigroup is generated up to saturation in the usual t…

Let $C_d(\tau_1,\dotsc,\tau_n)$ and $C_d(\tau_1',\dotsc,\tau_n')$ be integral cyclic polytopes, and let $\Delta_{ij}$ denote the corresponding gaps used in the source. Assume that…

Let $C_4(\tau_1,\tau_2,\dotsc,\tau_n)$ be a cyclic polytope, and write $\Delta_{ij}=\tau_j-\tau_i$ for $i<j$. A lattice polytope is normal when every lattice point in every positiv…

A cyclic polytope is the convex hull of points on the moment curve; here, normal means that every lattice point in every positive dilate is a sum of lattice points of the polytope.…

Let $e_1<e_2<\ldots<e_m$ be positive integers, and define the tail $s^{\mathbf{e}}:[0,1]\rightarrow[0,1]^m$ by … Let $P_{(K_{e_1},K_{e_2},\ldots,K_{e_m});\infty}$ denote the limit…