9 problems

Let $\mathcal{T}_{n,d}$ be the matroid introduced from the combinatorial encoding of arrangements of $d$ generic complete flags in $\mathbb{C}^n$. Let $n\Delta_{d-1}$ be the Minkow…

Let $\mathcal{T}_{n,d}$ be the matroid introduced from the combinatorial encoding of arrangements of $d$ generic complete flags in $\mathbb{C}^n$. Let $n\Delta_{d-1}$ be the Minkow…

Let $n\Delta_3$ be the dilation by $n$ of the standard $3$-simplex. A fine mixed subdivision structure on the boundary facets is boundary data arising from a fine mixed subdivision…

Let $[n]=\{1,\ldots,n\}$ and $[d]=\{1,\ldots,d\}$. A system of permutations records the permutations obtained from the boundary data of a mixed subdivision of $n\Delta_{d-1}$, and…

Let $S$ be a spread-out arrangement of unit triangles. A fine mixed subdivision is a subdivision of a dilated triangle into mixed cells, with all cells fine, and $S$ is completed w…

Let $A,B\subset\mathbb R^2$ be finite two-dimensional sets. A triangulation $T_A$ of $[A]$ uses $A$ as its vertex set, and a triangulation $T_B$ of $[B]$ uses $B$ as its vertex set…

Weak Spread Out Simplices Conjecture. Any $n$ spread out simplices in $n\Delta_{d-1}$ can be achieved as the simplices of an acyclic system of permutations.

Acyclic System Conjecture. Any acyclic system of permutations on the edges of the simplex $n\Delta_{d-1}$ is achievable as the system of permutations of a fine mixed subdivision.

Spread Out Simplices Conjecture. A collection of $n$ simplices in $n\Delta_{d-1}$ can be extended to a fine mixed subdivision if and only if it is spread out.