37 problems

Let $M$ be a connected matroid, and let $P_M$ be its base polytope. Say that $P_M$ admits a series-parallel subdivision if it can be subdivided into base polytopes of series-parall…

Let $P_M$ be the base polytope of a matroid $M$. A lattice polytope is Ehrhart positive when all coefficients of its Ehrhart polynomial are nonnegative. De Loera–Haws–Köppe conject…

Let $M$ be a connected matroid of rank $k$ on $n$ elements. Let $E_M(t)$ denote its Ehrhart polynomial, and let $M_{\rm min}$ and $U_{k,n}$ denote respectively the minimal matroid…

A positroid is a matroid arising from a cell of the totally positive Grassmannian. A matroid base polytope is Ehrhart positive when all coefficients of its Ehrhart polynomial are n…

Let $M$ be a matroid and let $\mathscr{P}(M)$ be its matroid polytope. If $\mathscr{P}$ is a lattice polytope of dimension $d$, its $h^*$-polynomial $h^*(\mathscr{P},x)$ is defined…

Matroid-base quadraticity conjecture. The toric ideal $I_{P_B(\mathcal{M})}$ is quadratic.

Quadraticity conjecture. The toric ideals of smooth polytopes and matroid polytopes should be generated by quadratic binomials.

Let $M$ be a dyadic matroid, meaning a matroid representable over the dyadic partial field. Let $P(M)$ be its matroid base polytope. Dyadic matroid triangulation conjecture. The ma…

Let $M=(E,\mathcal B)$ be a matroid with matroid base polytope $P(M)$, and let $I_{P(M)}$ be its toric ideal. A triangulation of $P(M)$ is called quadratic when its non-faces have…

Let $Q$ be an MV polytope of type $A$, and let $Q(\theta_I)$ denote the Schubert matroid polytope associated with $I\subseteq [n]$. For collections…

Moment-polytope realization conjecture. Every polypositroid—and more generally every flag matroid polytope—whose vertices are contained in $\operatorname{NC}_n$ is the moment polyt…

Let $a,b$ be positive integers. Let ${\texttt{Pan}}_{b+1,a+b,a+b+2}$ be the panhandle matroid, let $i(M,t)$ denote the Ehrhart polynomial of the matroid polytope of $M$, and let…

Geometry extremality conjectures.

Complete-graph graphic-matroid conjecture. For every $n\geq 1$, the graphic matroid $\mathsf{K}_n$ of the complete graph on $n$ vertices is an extremal matroid.

Direct-sum extremality conjecture. If $\mathsf{M}_1,\ldots,\mathsf{M}_s$ are extremal matroids, then $\mathsf{M}_1\oplus\cdots\oplus\mathsf{M}_s$ is extremal.

Let $\mathrm{U}_{r,n}$ be the uniform matroid of rank $r$ on $[n]$, and let $\Delta(\mathrm{U}_{r,n})$ be its matroid base polytope. A matroidal decomposition is a decomposition of…

Let $\operatorname{Pan}_{r,s,n}$ be a panhandle matroid, with $1\leq r\leq s<n$, and let $\mathcal{P}_{\operatorname{Pan}_{r,s,n}}$ be its matroid polytope. A lattice polytope is E…

Connected series-parallel subdivision conjecture. The polynomial $\wwtilde{g}_{\mathsf{M}}(t-1)$ has non-negative coefficients. This is a stronger positivity statement beyond Schub…

Let $\mathsf{M}$ be a matroid, and let $g_{\mathsf{M}}(t)\in\mathbb{Z}[t]$ denote its $g$-polynomial. Speyer's conjecture. For every matroid $\mathsf{M}$, the polynomial…

For each positive integer $n$, let $\mathsf{S}_n$ be the sparse paving matroid on $[n]$ of rank $\lfloor n/2\rfloor$ with the maximal possible number of circuit-hyperplanes and lex…

For parameters $k,n,r,s$, define the cuspidal matroid polytope … Such polytopes are particular cases of positroids. A lattice polytope is Ehrhart positive when all coefficients of…

Matroid Ehrhart-positivity conjecture. Every matroid is Ehrhart positive.

Notched rectangle matroid Ehrhart-positivity conjecture. Notched rectangle matroids, equivalently cuspidal matroids, are Ehrhart positive.

Positroid Ehrhart-positivity conjecture. Every positroid is Ehrhart positive.

Eight-face conjecture. The facet of $\mathbf{N}_{3,n}$ minimizing $\gamma_{j_1,j_2,j_3}$ has exactly eight codimension-$3$ faces combinatorially isomorphic to